Testing a model for emergent spinor wave functions explaining elementary particles and gauge interactions

Abstract

A geometric model for wave functions, which also allows deducing
space, general relativity and the standard model of particle physics, is tested against observations. The model is based on Dirac's proposal to describe spin 1/2 particles as tethered objects. Dirac's proposal is condensed into a fundamental principle that defines the quantum of action at the Planck scale. As a consequence, quantum particles are modelled as fluctuating rational 3d tangles consisting of strands with Planck radius whose crossing switches are observable. Time-averaged densities of tangle crossings are shown to have all the properties of wave functions and to follow the Dirac equation, as Battey-Pratt and Racey showed in 1980. Tangles of strands provide the only known model for the emergence of spinor wave functions and quantum field theory.

Several consequences of the tangle model go beyond quantum field theory. Classifying the possible rational 3d tangles yields the
observed spectrum of elementary particles, their quantum numbers, and calculable mass values. Classifying tangle deformations with the three Reidemeister moves yields the three observed gauge interactions, their symmetry groups, and calculable coupling constants. Tangles of strands provide the only known derivation of the particle and interaction spectra from the Planck scale.

The tangle model yields more than 50 tests of quantum theory and particle physics and predicts no effects beyond the standard model with massive Dirac neutrinos. All tests agree with observations. The fundamental principle of the tangle model visualizes qubits, includes general relativity, clarifies the relation to other approaches to relativistic quantum gravity, settles several issues of quantum field theory, and proves that the standard model of particle physics is unique, simple, and elegant.

Full text

Testing a model for emergent spinor wave functions
explaining
elementary particles and gauge interactions
Christoph Schiller ID ∗
Motion Mountain Research, 81827 Munich, Germany
(2024)
Abstract
A geometric model for wave functions, which also allows deducing space, general relativity
and the standard model of particle physics, is tested against observations. The model is
based on Dirac’s proposal to describe spin 1/2 particles as tethered objects. Dirac’s proposal
is condensed into a fundamental principle that defines the quantum of action at the Planck
scale. As a consequence, quantum particles are modelled as fluctuating rational 3d tangles
consisting of strands with Planck radius whose crossing switches are observable. Time-
averaged densities of tangle crossings are shown to have all the properties of wave functions
and to follow the Dirac equation, as Battey-Pratt and Racey showed in 1980. Tangles of
strands provide the only known model for the emergence of spinor wave functions and
quantum field theory.
Several consequences of the tangle model go beyond quantum field theory. Classifying the
possible rational 3d tangles yields the observed spectrum of elementary particles, their quan-
tum numbers, and calculable mass values. Classifying tangle deformations with the three
Reidemeister moves yields the three observed gauge interactions, their symmetry groups,
and calculable coupling constants. Tangles of strands provide the only known derivation of
the particle and interaction spectra from the Planck scale.
The tangle model yields more than 50 tests of quantum theory and particle physics and
predicts no effects beyond the standard model with massive Dirac neutrinos. All tests agree
with observations. The fundamental principle of the tangle model visualizes qubits, includes
general relativity, clarifies the relation to other approaches to relativistic quantum gravity,
settles several issues of quantum field theory, and proves that the standard model of particle
physics is unique, simple, and elegant.
Keywords
Origin of spinor wave functions; origin of elementary particles; origin of gauge groups;
origin of quantum field theory; origin of the standard model; strand tangle model.
∗cs@motionmountain.net

2
Contents
An appetizer: describing nature with high precision and little mathematics 3
Part I: A geometric model for emergent quantum theory 4
1 Nature provides a limit for every observable 5
2 Nature’s limits forbid points, equations of motion and Lagrangians 7
3 Black holes, space and fermions require strands 9
4 The search for emergent wave functions leads to strands 15
5 Crossings of strands resemble wave functions 17
6 Events are quanta of change due to crossing switches of strands 18
7 The fundamental principle of the strand tangle model describes all of nature 19
8 Rotating and orbiting quantum particles emerge from tethers 20
9 Tethers determine the spin of particles composed of fermions 23
Part II: Wave functions, superpositions, Hilbert spaces, and measurements 24
10 Path integrals and rotating arrows emerge from tangles 28
11 Wave function superpositions are described by tangles 30
12 Tangles imply Hilbert spaces 33
13 Tethered particles can pass each other 34
14 Strands lead to the interference of fermions 36
15 An intermezzo: strands lead to the interference of photons 37
16 Measurements, Born’s rule, wave function collapse and decoherence 38
17 Strands imply a finite decoherence time 41
18 Quantum entanglement is due to topological entanglement 42
19 Strands are not hidden variables 45
20 The probability density is limited 45
Part III: Dynamics of states 46
21 The Schrödinger equation emerges from tangles 48
22 Indeterminacy is a consequence of the fundamental principle 49
23 The Klein-Gordon equation emerges from tangles 50
24 Pauli spinors and the Pauli equation are due to tangles 51
25 Rational 3d tangles describe antiparticles 52
26 Tangles lead to Dirac spinors and the free Dirac equation 53
27 A second quantitative derivation of the Dirac equation 56
28 Strands imply the Dirac equation despite the lack of trans-Planckian effects 57
29 Strands predict the lack of other models for wave functions 58
Part IV: Differences to conventional quantum theory 59
30 Particle mass is due to chiral tangles 59
31 Classifying tangles leads to the spectrum of elementary fermions 64
32 Classifying tangle deformations leads to interactions and gauge groups 66
33 Classifying tangles leads to the spectrum of elementary bosons 74
34 All measurements are electromagnetic 77

3
35 Strands make predictions about elementary particle mass values 78
36 The principle of least action follows from strands 80
37 Strands limit the possible interaction vertices of the standard model 80
38 Strands imply experimental tests of the tangle model 83
39 Strands yield Galileo’s book of the universe 85
40 Particle tangles have open issues 85
41 Conclusion 86
42 Outlook 87
Acknowledgments and declarations 88
Appendix 88
A General relativity and all of nature deduced from strands 88
B Strand crossings visualize qubits and Urs 90
C Strands define quantum fields 91
D On preons 93
E On superstrings, supermembranes, and strands 94
F On the Yang-Mills millennium problem 95
G On physics’ axioms and Hilbert’s sixth problem 97
References 98
An appetizer: describing nature with high precision and little mathematics
A description of nature based on the Planck limits ~,c,c4/4Gand kln 2 automatically contains
quantum theory, special relativity, general relativity, and thermodynamics. This result has been
proven in many publications summarized in the following.
However, the Planck limits do not explain the appearance and the details of wave functions,
spinors, elementary particles, and gauge interactions. These aspects of nature can only be deduced
from tangles of strands with Planck radius – and in particular from their three-dimensional topol-
ogy and shapes. Only strands lead to quantum field theory and the full standard model. Arguments,
tests and predictions are given in the present article. They are based on standard topological knot
theory. All consequences of the tangle model are tested and found to agree with all observations.
Likewise, the Planck limits do not explain the appearance and the values of the elementary
particle masses, the coupling constants, the mixing angles, and the number of dimensions. Again,
only tangles of strands with Planck radius deduce them. The published explanations are summa-
rized below. High-precision calculations are still a subject of research.
In short, the fundamental principle based on strands of Planck radius explains all observa-
tions. It is tested in every experiment. All observations in nature – including the standard model
of particle physics with massive Dirac neutrinos, quantum field theory, and general relativity –
are described exactly by the fundamental principle. Only the rotation curves of galaxies need
exploration. Strands imply the lack of additional, unknown laws of nature. Therefore, no new
mathematics is required to describe nature with precision. In the following, strand topology is
shown to be sufficient to deduce wave functions, particles, interactions, and quantum field theory.

4
Fig. 1: In his lectures, Dirac demonstrated the properties of spin 1/2, a pure quantum effect, using a pair
of scissors tied to a chair with strands or with a belt. Any tethered object, such as a pair of scissors, returns
to the original state only after a double rotation, but not after a single rotation. Tethered objects and spin
1/2particles thus behave in the same way. The figure is inspired by a drawing of Penrose, who attended
Dirac’s lectures [2]. The details of Dirac’s trick are illustrated in Figure 2.
Part I: A geometric model for emergent quantum theory
The discovery of the quantum of action ~by Planck [1] led, via the observation of wave-particle
duality, to the use of the wave function to describe the state of a quantum particle or a quantum
system. The wave function is described by one or several complex fields that follow an evolution
equation, such as the Schrödinger equation or the Dirac equation. The present article argues, step
by step, that wave functions, spinors and their evolution equations emerge from a more funda-
mental description, namely from fluctuating tangles of unobservable strands with Planck radius.
The strand tangle model is based on the counterintuitive notion, due to Dirac, that every particle
in nature is tethered, i.e., connected with unobservable strands to the cosmological horizon. Dirac
demonstrated the idea for a matter particle with the arrangement illustrated in Figure 1. The scis-
sors represent the spin 1/2particle and the chair the cosmological horizon – though in nature, both
particles and horizons also consist of strands.
Part I of the present article begins by showing that observations require a description of nature
and of quantum theory based on strands. Then, the demonstration of Figure 1is used to deduce
that the difference between a rotation by 2πand no rotation at all is given by crossing switches
of strands. This result is condensed into a single fundamental principle that implies all results
presented in the following: Crossing switches of unobservable strands with Planck radius define
the quantum of action ~.In particular, the fundamental principle implies spin 1/2and all its prop-
erties. Part II derives wave functions, superpositions, entanglement, decoherence, and quantum
measurements from the fundamental principle. Part III uses the fundamental principle to derive
the Schrödinger equation, antiparticles, spinors, the Dirac equation, and several experimental pre-

5
dictions. For the first time, all observations about wave functions are derived, without measurable
deviations, and shown to emerge from the Planck scale.
Part IV shows that several differences arise from quantum theory. Only these differences make
the model worth exploring. The fundamental principle is shown to restrict the possible elemen-
tary particles, to derive the three particle generations, to derive the particle masses and quantum
numbers, to restrict the possible gauge interactions, and to derive their gauge groups and cou-
pling strengths. Strands imply the complete standard model of particle physics, including massive
Dirac neutrinos, without any deviation. So far, strands provide the only available derivation of
the standard model that does not predict unobserved phenomena. All differences from quantum
theory deduced from strands agree with observations. The differences solve the open problems in
quantum theory, quantum field theory, and the standard model of particle physics.
The presentation is self-contained. Each section ends with a summary paragraph that begins
with ‘In short’ and allows for quick reading. The appendices cover additional aspects of strands,
from general relativity to qubits, preons, strings, the Yang-Mills millennium problem, and the
limitations of axiomatic approaches to quantum field theory.
Strands allow the derivation of more than 50 high-precision experimental tests. They include
predictions about neutrino masses, supersymmetry, and dark matter. The predictions are high-
lighted as Test n. All agree with all observations.
In short, the article tells how the fundamental principle of the strand tangle model leads to
the emergence of wave functions, spinors, Dirac’s equation, elementary particles, gauge interac-
tions and the fundamental constants, including the coupling constants and particle masses. The
Lagrangian of the standard model is derived. All open questions about quantum theory are an-
swered. Experimental tests and predictions are deduced. They agree with all the data. Also
general relativity arises. No other model provides these results.
1 Nature provides a limit for every observable
In 1899, Planck discovered the quantum of action ~=h/2πin light [1]. Around 1910, Bohr
summarized quantum theory [3] as the sum of all consequences of the smallest measurable action
value ~. The action limit is confirmed by all real experiments and thought experiments ever carried
out. Later, the limit given by the quantum of action was found to be invariant.
In 1905, Einstein derived special relativity from the invariant maximum energy speed cob-
served in nature [4]. The principle of maximum speed, together with all its consequences, was
confirmed in all real experiments and thought experiments ever carried out.
In the years around 1926, applying the principle of the quantum of action ~to electrons led
Schrödinger and many others to develop the concept of the wave function, to describe its evolution,
and to use it to describe measurements. In 1928, Dirac incorporated spin 1/2and the energy
speed limit c. Dirac’s equation for electrons and Planck’s description of light led to quantum
electrodynamics, which agrees with all experiments ever carried out.
Around the year 1973, Elizabeth Rauscher discovered that general relativity implies a max-
imum force [5]. Around the year 2000, the work by Gibbons and by the author showed that
Einstein’s field equations can be deduced from the principle of maximum force c4/4Gor the
principle of maximum power c5/4G[6–10]. If preferred, one can also start from the equivalent
principle of maximum mass to length ratio c2/4G[11] or from the principle of maximum mass

6
flow rate c3/4G. All these limits are related to black holes. All these invariant limits agree with all
thought experiments and all real experiments ever carried out [6–47]. So does general relativity.
Based on Planck’s discovery of the Boltzmann constant k[1], also thermodynamics can be
deduced from an invariant limit, namely from the principle of smallest observable system entropy
kln 2 [48–53]. The entropy limit and thermodynamics agree with all thought experiments and
all real experiments ever carried out. Together with the previous limits, the entropy limit allows
deriving Bekenstein’s entropy bound and black hole entropy.
Around the year 2000, the mentioned results allowed summarizing quantum theory as the
consequence of the smallest action ~, special relativity as the consequence of maximum speed
c, general relativity as the consequence of maximum force c4/4G, and thermodynamics as the
consequence of the smallest system entropy kln 2. Ideally, these limits are called corrected Planck
limits because, traditionally, the quantity c4/G is called the Planck force. However, an additional
factor 4 is part of nature’s force limit c4/4G[6,10]. In general, the corrected Planck limits arise
when Gis replaced by 4Gin the traditional Planck units.
As a consequence of observations, all corrected Planck units are limits of nature. In partic-
ular, the corrected Planck limits include the minimum length p4G~/c3and the minimum time
p4G~/c5. Intervals smaller than these two values cannot be measured, cannot be observed, have
no effect, and thus do not exist in nature [54–65]. For this reason, those Planck limits that are
lower limits are also smallest measurement errors. Nature prevents vanishing measurement errors.
Several Planck limits – the Planck energy, the Planck momentum and the Planck mass – are valid
only for a single elementary particle because this condition is necessary to derive them. As a
whole, modern physics predicts that every search for observations or effects beyond the corrected
Planck limits will fail [66]. Indeed, no known observation contradicts the corrected Planck lim-
its, despite intensive search efforts in special relativity [67], in quantum field theory and particle
physics [68], in general relativity [10], in quantum gravity [69], and in thermodynamics [53].
Not all Planck limits are equal. The speed limit cis observable: some systems realize it, such
as light and gravitational waves, and other systems approach it quite closely, such as particles
in accelerators or cosmic rays. The quantum of action ~is observable: some systems realize it,
such as trapped electrons, and other systems approach it quite closely, such as in the photoelectric
effect or the Frank-Hertz experiment. The force limit c4/4Gis observable: some systems realize it,
such as black holes or gravitational horizons, and other systems approach it quite closely, such as
colliding black holes. The non-relativistic quantum gravity limit 4G~is observable: it is realized
in the quantum states of cold neutrons in gravity [70–72].
In contrast, the Planck limits of relativistic quantum gravity – those containing c,4Gas well
as ~– such as the corrected Planck length, Planck time or, for single elementary particles, the
Planck energy, are neither realized nor approached by any system in nature. Nature does not al-
low reaching the limits of relativistic quantum gravity. All experiments and observations about
relativistic quantum gravity are several or even many orders of magnitude away from the limit
values. For example, measurements of the electron dipole moment are a factor 103away from the
corrected Planck length [73,74]. Measurements of elementary particle energy in cosmic rays are
more than a factor 107away from the corrected Planck energy [75]. Measured temperatures, such
as in quark-gluon plasmas, are a factor of 1020 away from the Planck temperature. Measurements
of time intervals, such as the Higgs boson lifetime, are a factor of 1020 away from the corrected
Planck time [68]. The Planck electric field limit is unattainable because of the Schwinger limit,

7
which is over 1040 times lower. The general reason for the unattainability of the relativistic quan-
tum gravity limits is the existence of minimum uncertainties and minimum measurement errors.
In particular, the minimum uncertainties and measurement errors for length [54–65], time, area
and volume prevent attaining the relativistic quantum gravity limits.
In short, the corrected Planck limits express that no trans-Planckian effect of any kind exists
or is observable in nature. In addition,
BIn experiments, nature allows reaching and observing the simple limits c,~and 4Gas
well as the limits due to combinations of two constants, such as c4/4G,~/c,4G~.
BThe minimum uncertainties and minimum measurement errors prevent from reaching
or observing any combination of all three constants, such as 4G~/c3or c7/16G2~.
Any description of nature is correct if it reproduces these observations. Almost none is.
2 Nature’s limits forbid points, equations of motion and Lagrangians
Experiments show that nature is made of particles and space. Therefore, the search for a theory
of emergent quantum mechanics, which describes the emergence of particles and wave functions,
is closely tied to the search for a theory of relativistic quantum gravity, which describes the emer-
gence of space. Nature’s limits guide both searches.
In nature, evolving systems are described by physical action. The action Wcan be defined as
W=F l t =F l2
vor as W=E t =m c2t=m
lc l2,(1)
where Fis force, lis length, vis speed, Eis energy and mis mass. The expressions can be
used to insert the maximum speed c, the minimum action ~, and, for general relativity, either the
maximum force c4/4Gor the maximum mass per length ratio c2/4Gof black holes. Both cases
yield the same statement:
BNature limits length intervals, length uncertainties, and length measurement errors:
l>r4G~
c3≈3·10−35 m.(2)
So far, no experiment disagrees. In other words, the domain of nature where maximum speed,
maximum force, and the quantum of action play a role at the same time – the domain of relativistic
quantum gravity – is characterized by a minimum length. The minimum length is twice the Planck
length [66]. Smaller lengths cannot be achieved, measured or observed. Likewise,
BNature’s corresponding lower limits for area, volume, and time cannot be exceeded.
Again, no experiment disagrees. Historically, scientists took two millennia to define space as a
continuous set of points and to get used to working with real numbers as coordinates. Poking
gentle fun at thinkers who challenged continuity and questioned the infinitely small, such as Zeno
of Elea, is a part of modern science. In full contrast, relativistic quantum gravity implies:
BNature has no point-like physical observables.
Point-like quantities cannot be detected, measured, or play a role in nature. Point-like quantities
– including point particles, δfunctions (δdistributions), or singularities of any type – do not exist

8
in nature. In other words, every point-like concept is an approximation. No point-like observable
has ever been found. Points are the oldest error of fundamental science. The lack of points and the
existence of the minimum length uncertainty have drastic consequences.
BSpace and time cannot be continuous and cannot consist of points.
The minimum length implies that continuity, derivatives, differentials, discrete points and discrete
instants of time do not exist in nature. All these concepts are only approximate: they are due to the
averaging of some random substrate. The intrinsic uncertainties and measurement errors in nature
also imply that an equation can never be valid precisely, i.e., can never be tested without error or
doubt. In relativistic quantum gravity, two quantities – the two sides of an equation – cannot be
shown to be equal by any experiment.
BNo unified theory of physics can make use of equations.
A second argument confirms the conclusion. Any unified equation of motion, any unified La-
grangian, would contain some fundamental fields. The origin of these fields would have to be
defined. However, a unified theory must explain the origin of all quantities it uses. This is impos-
sible in a description that uses Lagrangians.
BNo unified theory of physics can be based on a Lagrangian.
All fields in nature – matter, electric, magnetic, strong, weak, gravitational – must be emergent. In
other words, all equations of motion and all Lagrangians are approximate and due to averaging.
Because of the intrinsic uncertainties and measurement errors, any description of fundamental
constituents must be statistical and must involve many constituents. The additional consequence
that nature cannot have exact parts at all is explored in Appendices Aand G.
Because of the minimum length and the lack of points, any unified description must deduce all
known equations of physics without the use of more fundamental equations.
BUnification means: deducing equations from no equation.
History shows that such a program can be realized. Planck’s discovery that action Wis limited by
W>~≈10−34 Js (3)
led to quantum theory, the derivation of Schrödinger’s equation and Dirac’s equation. The discov-
ery by Rauscher, Gibbons and the author that force Fis limited by
F>c4
4G≈3·1043 N(4)
can be used to derive general relativity and its field equations, as shown in references [6–10]. The
remaining task is to realize the program for the standard model of particle physics, as done below.
In short, because of the minimum length, fundamental physics cannot have points or equations.
The only way left to describe nature and observations is the usual approach of fundamental science.
First, a fundamental principle is distilled. Then, all possible consequences are deduced from the
fundamental principle. These consequences can be equations even if the principle does not contain
any. Every consequence is tested by comparing it to observations. If a comparison fails, the
principle is falsified. If a comparison with observations is positive, the next test is performed.
Comparison with observations is the only criterion for checking the validity of a principle. All
other alleged criteria about fundamental physics are mere habits of thought. Among these habits,

9
the yearning for fundamental equations is particularly persistent. Because of the minimum length,
nature does not fulfil this yearning. Because of the minimum length, nature’s constituents differ
from points.
3 Black holes, space and fermions require strands
Black holes are limit cases for curved space and for densely packed particles. Therefore, the
constituents of black holes are also the constituents of space and particles. Thus, both the search
for relativistic quantum gravity and the search for emergent quantum mechanics require finding
the common constituents of black holes, space, particles, and wave functions.
Black holes yield many details about the constituents of nature. The minimum length appears
in the form of the minimum area 4G~/c3in the expression for their entropy S[76,77]:
S=kc3
4G~A . (5)
Here, kis Boltzmann’s constant, Ais the surface of the black hole, and the factor 4is due to
the black hole limits. Even though the expression has never been confirmed by experiments, its
order of magnitude is without doubt, as it follows from the limits of nature [53]. Because black
holes have entropy, they are composed of constituents that fluctuate. Because black holes have
finite entropy, they are composed of a finite number of discrete constituents. Because the black
hole entropy contains the minimum area, the common constituents of space and particles have
an effective cross-section given by the minimum area. Because space and wave functions are
extended and reach up to the cosmological horizon, this also holds for the common constituents.
Therefore, particles and space are made of common constituents with unobservable Planck radius
that fluctuate and reach the cosmological horizon. The common constituents thus are filiform.
The minimum length and the expression for black hole entropy eliminate many alternative
types of fundamental constituents [66]. The minimum length eliminates all options for space and
space-time based on continuity or discreteness. In particular, minimum length is in contrast with
space lattices, with fractals, additional dimensions, non-commutative space, singularities, con-
formal symmetry, holography, higher dimensions, space-time foam, categories, spatial lattices,
supersymmetry, supergravity, conformal gravity, conformal field theory, anti-de Sitter space, de
Sitter space, twistors, doubly special relativity, Wick rotation approaches, spin networks, and with
any combination of continuous space with additional discrete or continuous mathematical struc-
tures. Of the many approaches to relativistic quantum gravity [78], only a few are not eliminated
by the existence of a minimum length. Black hole entropy further eliminates compact constituents,
filiform constituents of finite length, and filiform constituents that are branched or knotted. The
minimum length and black entropy are incompatible with knots, bands, branched structures, net-
works, graphs, and ribbons. The common constituents can only be filiform.
In the 1990s, it was common lore that black hole entropy can be explained by filiform con-
stituents, as told by Weber [79]. For a recent example, see Verlinde and Visser [80]. In general
relativity, filiform constituents have been explored by Carlip [81] and independently by Botta-
Cantcheff [82]. In relativistic quantum gravity, an interesting approach is the use of causal sets, as
explored by Sorkin and by Bombelli [83]; filiform constituents can be seen as a specific realiza-
tion of causal sets. A crossing of filiform constituents, defined below, also resembles a tetrahedron;

10
particle,
i.e.,
tangle
core
Dirac's belt trick or string trick: Double tethered particle rotation is no rotation.
Resulting observation. Time averaging unobservable tethers but observable
crossing switches leads to a probability density and phase:
phase
move
all
tethers
move
lower
tethers
move
upper
tethers
move
tethers
sideways
like
start
rotate
particle
twice in
any
direction
tethers tethers
probability
density
phase phase
Fig. 2: Upper part: Dirac’s string trick or belt trick shows that a double rotation by 4πof a tethered,
indivisible particle – here an oriented tangle core – is equivalent to no rotation at all. Lower part: the
particle tangle leads to a probability density and a phase once the ideas from Figure 11 and those
explained in the text are used. The belt trick works as long as the tethers – any number of them – are
allowed to fluctuate, untangle, and are assumed to be unobservable, whereas their crossing switches are
assumed to be observable. A single rotation by only 2πdoes not allow untangling to occur. Indivisible
localized cores with 3 or more tethers thus exhibit the properties that characterize spin 1/2particles. As a
result, a tethered particle can rotate continuously. Careful observation also shows that Dirac’s trick couples
rotation and displacement of the tethered core. The coupling of rotation frequency and displacement
determines the inertial mass, as discussed in Section 30. The twists in the tethers are virtual gravitons that
determine the gravitational mass, as outlined in Section 35. Dirac’s trick can be easily reproduced with
real belts, strips of paper, or ropes. Links to animations illustrating the belt trick are given later on.
such structures are used in some approaches to quantum gravity, such as the one by Oriti [84]. The
behaviour of the spatial complement of filiform constituents, the motion of the space between
them, was investigated in detail by Asselmeyer-Maluga [85,86]; in this approach, a universe is
made of space only, albeit an intricately shaped one. The recent bit threads are discussed in Ap-
pendix B.
Not only black holes and space are made of filiform constituents; also matter is. Dirac pro-
vided the first clue. From around 1929 onwards [87], as shown in Figure 1, Dirac used his scissor
trick – also called the string trick or the belt trick – in his lectures [87]. Dirac’s trick, illustrated in

11
The belt trick simplified
Resulting
observation:
A quantum effect:
a sign change in
the wave function.
one core
rotation
tethers
tethers
corecore
crossing switched crossing,
of opposite sign
The essence of the belt trick: a crossing switch
Resulting
observation:
An action ℏ:
a fundamental
observable event.
The basic concept: a skew crossing – the region
of smallest distance between two strands
overpassunderpass
Resulting
observation:
None. Unobservable
when not switching.
Fig. 3: Top: Dirac’s belt trick is based on the idea that a crossing switch of tethers leads to an observable
change in the sign of the wave function. The figure shows that a crossing switch is related to a tangle core
rotation by 2π. Centre: Simplifying further, a crossing switch is the sign change of a crossing. In the
strand tangle model, this sign change is observable even though the tethers themselves are unobservable
due to their small radius. The bottom diagram highlights that all crossings are apparent and skew: a
crossing is the region of smallest distance between two strands that pass each other.
Figure 2, captures the basic properties of spin 1/2: the original situation reappears after a rotation
by 4π– keeping the observable particle, scissor or belt buckle fixed but moving the (assumed un-
observable) belt or strand tethers around. In contrast, the original situation does not reappear after
a rotation by 2π. In experiments, after a rotation by 2πonly, the sign of the wave function of a
spin 1/2particle is changed. It returns to the original sign only after a rotation by 4π. The only
difference about a tethered system before and after a rotation by 2πis the sign of the tether cross-
ings, as illustrated in Figure 3. If a system is not tethered, no difference is detectable between the
situation before and after a rotation by 2π. Tethers, or strands, are the only possible visualization
of spin 1/2in three dimensions. In simple terms, independently of black holes, spin 1/2proves
that in nature, everything – all matter – is tethered and connected to everything else.
A variant of Dirac’s belt trick is the fermion trick, illustrated in Figure 4. It describes the basic

12
move
tethers
only
particles,
i.e., tangle
cores
move
tethers
only
T 
ermion trick: Double tethered particle exchange is no exchange.
exchange
particle
cores
twice
like
start
The trick also works if some or all the strands connect one tangle core to the other core.
tethers
tethers
Fig. 4: The fermion trick shows that a double particle exchange of tethered particles is equivalent to no
exchange at all. The fermion trick works if the tethers – any number of them, even if some connect the
particles directly – are allowed to fluctuate and untangle. Tethers are assumed to be unobservable, whereas
crossing switches are assumed to be observable. In contrast, a single particle exchange does not allow
untangling. Indivisible tangle cores made of several tethers thus show all the properties that characterize
fermions. The fermion trick is easily reproduced with long strips of paper, ropes, or belts.
property of fermions: after a double particle exchange, two tethered fermions return to the original
situation – but not after a single exchange. Again, crossing changes are observable through their
effects on the sign of wave functions, even if the tethers themselves are unobservable. Again,
no other visualization of fermion behaviour exists. The effects of tethers have been visualized
in numerous internet videos. Examples are found in references [88–92]. The fermion trick even
works if the two fermions are connected directly. For example, the two observable ends of a belt,
assumed to be unobservable, reproduce the fermion trick. The fermion trick proves that in nature,
everything is tethered and connected to everything else.
Not only fermions, also photons have been visualized as helically deformed filaments [93].
This will be explained later on.
Dirac’s belt trick and the fermion trick were the first hints that matter should be described with
extended, filiform, unobservable constituents, for which only crossing switches are observable.
Fifty years later, in 1980, Battey-Pratt and Racey showed that tethers imply the full free Dirac
equation [94]. Their argument is presented below. In simple terms, Battey-Pratt and Racey proved
that Dirac’s trick implies Dirac’s equation. They wrote to Dirac about their discovery, but he never
answered. In other words, Dirac’s equation proves that in nature, all fermions are tethered.
Everything is tethered. However, no tethers are observed. The tethers must be unmeasurably

13
thin. The tethers must have Planck radius. It comes natural to think that nature is completely made
of filiform constituents of Planck radius. They are called strands in the following.
As Dirac explained [87], his trick also demonstrates that a spin value smaller than ~/2is not
possible. Figure 3shows: because of the Dirac trick, crossing changes of tethers are related to
~.This connection was most clearly stated by Kauffman in the 1980s [95,96]. In Dirac’s belt
trick – and the rest of this article – strands never intersect. For strands with Planck radius, the
lack of intersections visualizes the minimum length. All strand crossings seen in diagrams are
apparent or skew and are recognizable as ‘crossings’ only in a two-dimensional projection. In
three dimensions, ‘crossings’ are defined and recognizable by the shortest distance between the
two involved strand segments. A crossing switch, or crossing change, is the change of sign of
a crossing, i.e., the change of which strand passes under the other. Dirac’s belt trick implies
that crossing switches are observable – even if the tethers themselves and their crossings are not.
Specifically, Dirac’s trick implies that the quantum of action ~is due to a crossing switch. This
will lead to a definition of wave functions with crossings explored below.
If everything in nature is composed of strands, what exactly are particles? The straightforward
conjecture that particles might be modelled as open knots, i.e., as knots in infinitely long tethers,
as illustrated in Figure 5, is unsuccessful. Despite intense attempts in this and similar directions
[78,98–113], open knots, prime knots, closed knots, bands, branched structures, loops, networks,
or graphs do not fully explain the particle spectrum or the appearance of wave functions. Above
all, none of these topological structures explains gauge interactions or particle reactions.
Between 2008 and 2014 it became clear that modelling elementary particles as rational 3d
tangles – i.e., as unknotted tangles – does explain wave functions, the particle spectrum, particle
reactions, and particle interactions. The concept of a rational 3d tangle – a three-dimensional braid
– is illustrated in Figure 5. As shown later on, the classification of rational 3d tangles yields the
spectrum of elementary particles, with their observed charges and other quantum numbers. The
classification of strand deformations using the three Reidemeister moves yields the three known
gauge groups [114,115]. Also this result is presented below. The classification of Feynman
vertices yields all particle interactions. With strands of Planck radius, unique values for the particle
masses, coupling constants and mixing angles arise. Strands yield the Lagrangian of the standard
model with massive Dirac neutrinos.
The present article is a prequel to the strand papers on particle physics. It describes wave func-
tions, basic quantum mechanics, and quantum fields as consequences of the fundamental principle
of the strand tangle model.
For strands with Planck radius, gravity and general relativity also arise [116,117]. In particular,
strands yield the Lagrangian of general relativity. Together, these results allow deriving both
gravity and particle physics from a single fundamental principle inspired by the crossing switch
illustrated in Figure 3. The presentation of the strand tangle model in reference [118] was followed
by extensive tests of the consequences for particle physics, including quantum electrodynamics
and quantum chromodynamics [119–121], and by tests of the consequences for general relativity,
black holes and quantum gravity [116,117]. All consequences of strands agree with observations.
Until recently, no approach to quantum gravity has deduced a model for wave functions, for
particles, or for gauge interactions. Only the strand tangle model achieved this.
In short, the results by Planck, Einstein, Dirac, Battey-Pratt and Racey, Rauscher, and Gibbons
prove that three ingredients are necessary to describe all observations in nature: corrected Planck

14
Knotted tangles
K
nots
t nnt
t nnt
core
tethers
an open knot
a prime tangle
trefoil
knot
g


knot
av
e
tether
above
paper plane
av
e
b
w
b
w
tether
below
paper plane
Rational 3d tangles – or 3d braids
Links
t nnt
t
tethers
core
nnt
H

f
link
B
n
rings
Fig. 5: The differences and overlaps between rational 3d tangles – or 3d braids – knotted tangles, links,
and knots are illustrated. Rational 3d tangles are formed by braiding their tethers, shown with dashed
lines. In contrast, knotted tangles cannot be formed in this way. Equivalently, rational tangles, unlike
knotted tangles, can be disentangled by moving their tethers in space. Rational 3d tangles owe their name,
for the case of two strands in the plane, to their close relation to rational numbers [97].
limits, unobservable strands connecting everything, and observable crossing switches. These three
ingredients are combined in the fundamental principle of the strand tangle model by using fluc-
tuating strands with Planck radius, defining ~with crossing switches, and limiting time intervals
to twice the Planck time. In the present century it was shown that the strands and Planck limits
in the fundamental principle are also sufficient to describe black hole thermodynamics, general
relativity, quantum theory, and the standard model of particle physics – and thus all observations
in nature [114–121]. The rest of this article explains why and how crossing switches of strands are
observable and how 3d tangles of strands lead to wave functions, particles, quantum fields, and
interactions.

15
4 The search for emergent wave functions leads to strands
The search for an emergent description of quantum theory has been a research topic for almost
a century. The efforts of de Broglie, Schrödinger, Bohm, Bell, Kochen, Specker and many other
researchers in the twentieth century had a lasting impact that fuelled hope on the one hand and
limited possibilities on the other. In this century, books like those by Adler [122] and by Peña,
Cetto and Valdés [123] have examined the subject. They concluded that an emergent descrip-
tion of quantum theory using a statistical basis is possible, but they did not propose any specific
model. Many other scholars have studied the topic over the past two decades [124–137]. Emergent
quantum theory is also a topic of regular conferences. The collection [138] is a recent example.
Despite all the efforts, no explicit model for emergent wave functions that agrees with obser-
vations has been proposed thus far. There are two reasons. First, already the ancients stated that
nature consists of particles and void. As a result, any model for emergent wave functions must
at the same time be a model for relativistic quantum gravity, which is not straightfoward. The
second reason is that the options left over by the mentioned investigations about emergent quan-
tum theory are hidden and different from habits of thought. Indeed, the strand tangle model for
wave functions is based on the counterintuitive notion of tethers, that is, on unobservable connec-
tions between every quantum particle and the cosmological horizon. Even though tethers were
introduced around 1929 by Dirac, as depicted in Figure 1, to explain spin 1/2[87], it took over
fifty years before somebody took unobservable tethers seriously. In 1980, Battey-Pratt and Racey
used tethers to visualize and derive Dirac’s equation in their inspiring paper [94]. However, they
did not explicitly explore wave functions. The strand tangle model extends the idea of tethers by
modelling elementary particles as tangles and wave functions as crossing densities. Continuing
the exploration requires a definition.
BAstrand is defined as a smooth curved line surrounded by a cylinder with a tiny, Planck
radius. More precisely, the line is a one-dimensional, open, continuous, everywhere
infinitely differentiable subset of R3(or of the spatial part of a curved Riemannian
space) without self-intersection, unknotted, and without endpoints. In addition, the line
is surrounded by a volume defined by a perpendicular disk of Planck radius p~G/c3
at each point of the line. The entire strand volume is not allowed to self-intersect; thus
the curvature radius of the curved line is never smaller than the Planck length.
From a mathematical viewpoint, the definition of strands is the usual definition used in knot theory,
for example, in the ropelength calculations of tight knots [139,140]. From a physics viewpoint,
strands differ in several ways from flexible ropes, cables, or cooked spaghetti. First of all,
BStrands randomly fluctuate in shape and length.
The origin of the shape fluctuations is the continuous pushing of strands against strands – including
those that make up the vacuum, as told in Appendix A. This pushing is a consequence of their
impossibility to intersect. Due to the impossibility of measuring or preparing lengths smaller than
the minimum length,
BStrands cannot be cut.
In other words, contrary to habits of thought, strands are not made of parts. Strands are indivisible.
Strands are not made of anything else. In further contrast to everyday life,

16
BStrands have have no ends.
Strands reach up to the cosmological horizon, as visualized in Appendix A. But the main difference
between strands and any rope of everyday life is their Planck radius. The radius of strands is so
small that it is negligible in almost all situations – except in the domain of relativistic quantum
gravity. In particular,
BStrands are unobservable and have no observable properties.
BOnly specific changes in the tangling of strands, crossings switches,are observable.
In particular, strands do not have mass, colour, energy, tension, momentum, charge, or any quan-
tum number. Among others, strands have no fixed length, and cannot exert forces: strands cannot
pull anything, and they do not offer any resistance when they are deformed. The reasons and
consequences will be clarified throughout this article.
The concept of a rational 3d tangle is inspired from topology [97,141] and is defined intu-
itively in Figure 5as a collection of braided strands.
BRational 3d tangles or 3d braids are unknotted; their tethers lead to the cosmological
horizon along specified average directions in three-dimensional space.
The region of space in which the strands are tied up and tangled is called the tangle core. The
term tether is used for those strand segments that connect the core to the cosmological horizon.
Thus, a 3d tangle consists of a core and tethers. As will be shown below, tethers are responsible
for the quantum of action ~, for spin 1/2, for fermion behaviour, and for all other quantum effects,
including wave functions and their evolution equations. The classification of rational 3d tangles
and their deformations leads to the observed elementary particles, the observed gauge interactions,
and the observed particle reactions.
In physics, filiform particle constituents avoid all arguments against constituents inside elemen-
tary particles. Experimentally, all the known elementary particles lack particle-like constituents
[68]. Also theory argues against particle-like constituents. Such constituents would have to be
confined inside an extremely small volume, resulting in extremely large kinetic energy and thus
in particle mass values that are much higher than those observed. In contrast, extended filiform
constituents suggest that the region where they are tangled is the region where the wave function
of the quantum particle is localized. As shown below, rational 3d tangles of extended filiform
constituents can be counted, exhibit fermion behaviour, and have spin 1/2, parity, chirality as well
as all other properties of quantum particles. However, tangles are also challenging: mechanisms
must be found that allow mass and gauge interactions to emerge from tangling.
In short, already a long time ago, the counter-intuitive approach that quantum particles are
tethered to the cosmological horizon with unobservable strands was used to explain spin 1/2
and the Dirac equation. Strands are tiny, fluctuating and uncuttable connections spanning across
nature. The idea that particles consist completely of tangled strands of Planck radius suggests
that an emergent description of quantum theory may be possible. Proving the suggestion requires
several tasks: specifying a strand tangle model for wave functions, deriving conventional quantum
theory, explaining the origin of different particles and their masses, explaining gauge interactions,
reproducing quantum field theory, and continuously comparing tangles and their differences from
conventional quantum theory with experiments.

17
   !" # $
ci
%
es the
position
,
the shortest distance s de
%
nes the
amplitude
,
and the angle 
  #
phase.
crossing axis
phase
A
 &
'
 $,
( )*+(-. /+0))1-2
has the same properties as a
w(3e 45-/*10-6
7  '
e an
(89:
1*5.e  ;< 
f
9p
()e)
.
Same
!<$$&
,
'

wed along the
shortest distance. The phase is oriented
<!; <=
to the paper plane.
crossing
axis
s
>
?
>
?
>
?

phase
s
   !" # $  @ C =
along the shortest distance s
C
een
the two strands.
orientation
around axis
phase is the orientation
around the axis
Same
!<$$
&
,
'

wed along
the crossing axis. The skew
strand arrangement is
'
$D
crossing
axis
, 


small s yields a
large amplitude
Fig. 6: A strand crossing consists of two skew strand segments separated by the shortest distance s. Due
to strand impenetrability, the shortest distance sis never shorter than the minimum length. A strand
crossing allows defining amplitude and phase(s) around an axis. Therefore, a strand crossing at a point has
the same properties as a wave function at a point. The drawing on the top right defines the crossing axis as
the sum of the two tangent vectors at the two endpoints of s. (This requires defining a positive direction on
each strand.) The (main) phase αdescribes the direction in which the shortest distance vector points
around the axis, once a sequence is defined among strands. The dotted direction defining vanishing phase
around the axis is a matter of choice, as usual for phase. Regardless of the choice of vanishing phase,
phase differences are always defined uniquely. The angles or phases β,γand δplay a role in spinors.
5 Crossings of strands resemble wave functions
Spin 1/2proves that in nature, everything is connected to everything else. Dirac’s trick shows that
crossings of these connections are essential in the study of quantum particles. In this article, the
term crossing is used in the sense of mathematical knot theory [142–144].
BAstrand crossing is the region of the smallest distance between two skew strands.
In three dimensions, the strand segments are always at a distance, as illustrated in Figure 6. Equiv-
alently, as usual in knot theory, a crossing always implies a skew geometry of two strand segments.
Therefore, a literal ‘crossing’ appears only when the configuration is projected in two dimensions.
The existence of a crossing, which is the region of the smallest distance between two strands, is
independent of the observer. Crossings can occur between any two strand segments. In particular,

18
crossings can occur in two segments of the same strand or of different strands. Crossings also arise
between extremely distant strands. But such long-distance crossings have no physical importance
because crossing switches – which define all physical observables – will not arise for them.
Crossings are consistent with the minimum length in nature. The definition of crossing assumes
that in the region of the smallest distance, both strands have a radius of curvature that is larger than
the Planck length. Due to the Planck-sized radius of strands, the shortest distance at a crossing is
always larger than the minimum length.
Crossings have mathematical properties that resemble wave functions. The simplest wave
function appears in the Schrödinger equation. It is defined by an amplitude Rand a phase ϕ:
ψ(x, t) = R(x, t) eiϕ(x,t).(6)
The more involved Pauli spinors have three phases, whereas Dirac spinors have seven. Figure 6
shows that also strand crossings are described by amplitude and phase(s). The amplitude is large
when the shortest distance between the two strands is small. The (main) phase describes the
orientation of the shortest distance around the crossing axis. Both quantities are defined precisely
in Part II, which will use the density of tangle crossings to define wave functions. The other angles
illustrated in Figure 6play a role in spinor wave functions, as shown below.
Apart from the ideas by Kauffman [95,96], there seems to be no research literature on strand
crossings in quantum theory. In particular, there seems to be no research on the similarity between
crossings and wave functions.
In short, the geometric properties of strand crossings – amplitude and phase(s) – closely re-
semble the mathematical properties of wave functions. No other model does.
6 Events are quanta of change due to crossing switches of strands
Traditionally, in quantum theory, the concept of ‘event’ plays a negligible role. This changes in
the strand tangle model. The fundamental principle, illustrated in Figure 7, states:
BA crossing switch exchanges which strand passes over the other and defines a funda-
mental physical event.
In mathematical language, a crossing switch is the change of sign of a crossing. In the strand
tangle model of nature, events are thus discrete and countable. Every event is a local process,
defined with a position error of the order of the Planck scale.
The fundamental principle states that every observed process in nature – motion, interaction,
decay, particle transformation, observation, or measurement – is composed of crossing switches.
In particular, every observation and every type of change is due to and is composed of crossing
switches. Crossing switches are the irreducible ‘building blocks’ of change.
BEach crossing switch yields a quantum of action ~.
BEach crossing switch is a quantum of change.
These statements are the essence of Dirac’s trick. Across physics, change is measured with action.
Thus, the statements about crossing switches agree with the quantization of action. Crossing
switches visualize the quantization of action. They define quantum theory. No other model does.
Wave functions are unobservable. Also crossings are unobservable. But both can be used to

19
Strand description: Resulting
observation:
E
fundamental
e
F
ent
G IJLMINO
ed
in spac
PQRNSP
UNRVNW XIMW LY
scales
The fundamental, Planck-scale principle of the strand tangle model
t t + ∆t
W=~
∆l>p4~G/c3
∆t>p4~G/c5
S=kln 2
Fig. 7: The fundamental principle of the strand tangle model, deduced from Dirac’s trick, describes the
simplest observation possible in nature: a fundamental event. In the tangle model, a fundamental event
occurs at the Planck scale and is almost point-like. Every fundamental event results from a strand crossing
switch. A crossing switch is the simplest observable process in nature; it defines ~as the unit of the
physical action W. The crossing switch is a deformation of strands in space, illustrated by the circling
arrows. The deformation results, for example, from rotating the two segments on the right-hand side
against the two segments on the left-hand side, around the west-east axis in the paper plane. The location
of the crossing is defined by the centre of the shortest distance between the two strand segments. The
strands, best imagined as elastic threads, have a Planck-size radius, are not observable, are uncuttable, and
are impenetrable. The Planck length and Planck time describe the most localized and the most rapid
crossing switch possible. As shown below, the fundamental principle also implies the indeterminacy
relation, wave functions, particle tangles, the canonical commutation relation, the Dirac equation, gauge
interactions, measurement via electromagnetism, the standard model, general relativity, qubits, and the
lack of axioms in physics.
define physical observables. In quantum theory, the simplest observable is the probability density.
In the strand model, the simplest observable is a crossing switch. As will become clear shortly,
crossing switches define the probability density – and all other observables. The final reason why
crossing switches are observable – and that nothing else is – is given below, in Section 34, where
it is shown that crossing switches couple to the electromagnetic field.
Similar event-based approaches to nature have been explored by Krugly [145], by Powers and
Stojkovic [146], and by Giovannetti, Lloyd and Maccone [147]. Many approaches to relativistic
quantum gravity yield a similar description [78,116].
In short, in the strand tangle model, a crossing switch is a process and a fundamental event.
Each crossing switch yields a quantum of action ~, the observable quantum of change. The re-
mainder of this article will prove that crossing switches of strands make up all known quantum
processes, visualize them, and yield quantum theory. No other model does.
7 The fundamental principle of the strand tangle model describes all of nature
The strand tangle model states that matter, radiation, space, and horizons – thus all systems ob-
served in nature – consist of strands that fluctuate [118,119]. More precisely, the complete strand
tangle model is based on one statement:
BCrossing switches of strands with Planck radius determine the quantum of ac-
tion ~and the minimum time, as illustrated in Figure 7.

20
This statement is the fundamental principle of the strand tangle model. Several statements follow.
BAlthough strands are themselves unobservable, crossing switches are observable, be-
cause of their relation to ~,c,kand 4G.
BCrossing switches define all physical units and observables.
BPhysical space is an aggregate of untangled strands. (See Figure 38.)
BHorizons – including black hole horizons, Rindler horizons and the cosmological hori-
zon – are weaves of strands. (See Appendix A.)
BParticles are rational 3d tangles, i.e., 3d braids of strands. (See Figures 25 and 34.)
BWave functions are densities of tangle crossings. (See Figure 11.)
BProbabilities and intensities of specific fields are due to the number of crossing switches
occurring in specific tangles. (See Figure 11.)
BPhysical motion minimizes the number of observable crossing switches of unobservable
fluctuating strands. This is the principle of least action. (See Section 36.)
Every change in nature, be it a nuclear reaction, a supernova, or a human life, is due to crossing
switches of strands. As will become clear step by step, the fundamental principle implies both the
standard model of particle physics and general relativity, including their Lagrangians [116–121].
The definition of the quantum of action in the fundamental principle covers and explains all
quantum processes, from entanglement to the quark model. Doing so, the strand definition of ~
eliminates the concept of quantization from all domains of theoretical physics, including gravita-
tion, in all the nuances that are found in the literature. Strands simplify quantum physics.
As is usual in quantum mechanics, physical space is assumed to be flat throughout. The strands
making up empty space are mostly not discussed in the following. However, vacuum strands play
a role in vacuum fluctuations, in particle creation and in annihilation.
In short, the fundamental principle of the strand tangle model answers Wheeler’s question
“How come the quantum?” [148]. Crossing switches of fluctuating strands with Planck radius
define the quantum of action. Crossing switches imply particle physics and general relativity. They
imply all laws of physics. In this article, the strand tangle model is tested against observations in
the domain of quantum physics. The first test concerns the properties of fermions.
8 Rotating and orbiting quantum particles emerge from tethers
As Dirac demonstrated in his lectures, tethers explain spin 1/2behaviour and fermion behaviour.
These consequences of Dirac’s trick, which goes back to Magnus [149], are illustrated in Figure 2
and Figure 4. Both results apply whenever the number of tethers is three or larger. But the two
figures show more.
Dirac’s belt trick implies that a tethered particle can rotate continuously. This is best observed
in animations. A video showing the continuous spinning of a particle with six tethers has been
produced by Hise [88]; still images from the video are shown in Figure 8. A video of a spinning
particle with four tethers (or two attached belts) was produced by Martos [89]; still images from
that video are shown in Figure 9. Videos of rotating particles with large numbers of tethers are also
found on the internet [91,92]. In other words, when unobservable tethers are allowed to fluctuate,
particle rotation can continue forever, without any obstacles, despite the tethering. The relation
between spin and rotation is not new and was regularly pointed out in the past [150]. In other

21
Z [
\ ]
^ _
Fig. 8: The particle rotation for a (free and moving) lepton can be visualized with the animation produced
by Jason Hise, available at [88]. The rotating (spinning) central cube symbolizes the tangle core, i.e., the
tangled region where the particle is localized with the highest probability. The continuous rotation of a
tethered particle is possible. (The figure is modified from reference [119].) The cube rotates twice before
returning to the starting configuration. The factor of 2 appears in many expressions involving the rotation
angle and the quantum phase of fermions.
words, spin can be visualized as the rotation of tangle cores.
A related statement can be made for systems consisting of two particles. The fermion trick
implies that two tethered particles can orbit each other continuously. Martos published a further
video showing the fermion behaviour of two tethered particles [90]; still images from the video
are shown in Figure 10. For example, Figure 10 can be seen as visualizing an electron orbiting
a proton in a hydrogen atom. In particular, the video shows that two tethered particles can orbit
each other forever – if tethers are allowed to fluctuate. Again, the number of tethers is not limited.
In other words, electrons orbiting nuclei can be visualized as tangles orbiting tangles.
The mentioned properties imply that the tethers of two typical quantum particles do not disturb
each other, as long as strands are unobservable, are infinitely flexible, and have no fixed length, no
tension and no mass. This is true also for more than two particles. As a result, the many tethers
filling empty space usually lead to no interaction between particles. Exceptions related to gauge
interactions and virtual particle effects are discussed below.

22
` c
d f
Fig. 9: The return to the original configuration after a double rotation can also be visualized with the
animation produced by Antonio Martos, available at [89]. The rotating central belt symbolizes the tangle
core. The continuous rotation of a tethered fermion is possible.
h i
j k
Fig. 10: Fermion exchange can be visualized with the animation produced by Antonio Martos, available at
[90]. The rotating central belt buckles symbolize the two tangle cores, i.e., the regions where the two
particles are localized with the highest probability. (In the strand tangle model, tethers and the belts they
form are unobservable.) The first image shows two particles whose positions were exchanged twice. The
other images show that the shape changes of the belts, each consisting of two or more tethers, bring back
the original, unexchanged situation. Therefore, the continuous exchange of the two tethered particles is
possible. The animation thus illustrates, among others, the motion of an electron (red) continuously
orbiting a proton (blue) in a hydrogen atom. The sub-figure 1 on the top left can also be taken as the
defining configuration for a composed system.

23
In the strand tangle model, every quantum particle is a tangle tethered to the cosmological
horizon in which the strands fluctuate continuously.
BTangles with their tethers reproduce spin 1/2as tangle core rotation.
BRotation and orientation of tethered cores reproduce particle spin, including the orien-
tation of the spin axis in space, and thus the flag model of spin.
BTethers thus show that spin is angular momentum.
BTethered cores orbiting each other reproduce orbiting particles.
BTethered tangle core exchange reproduces fermion behaviour.
BTethers thus reproduce the spin-statistics theorem. Only spin 1/2particles are fermions,
and vice versa [151,152]
BNo explanation of spin 1/2and fermion properties is possible without tethers.
These results are valid generally. In particular, the results are valid at all measurable energies.
In short, tethered tangles are essential for describing and understanding spin 1/2, particle rota-
tion, orbiting particles, particle exchange, and fermion behaviour. This understanding is impossi-
ble without tethers. The result suggests that every other motion in the quantum domain – including
translation, interference, scattering and interactions – can also be described with tethered tangle
cores. This is indeed the case, as argued in the remainder of this article. However, before doing
so, it is worth confirming the effects of tethers in the case of composed particles.
9 Tethers determine the spin of particles composed of fermions
In nature, when two spin 1/2fermions form a composite, the composite is observed to have either
spin 0 or spin 1 and to be a boson. This behaviour can be reproduced with tethers.
In the strand tangle model, a system is composed when it forms a unique tangle core connected
to the cosmological horizon by tethers. Examples showing composite systems most clearly are the
second graph from the left in Figure 4or sub-figure 1 in Figure 10. Composed systems usually
have a different spin value than the particles they are made of. The two figures show that a system
composed of two tangle cores – two belt buckles – returns to itself exactly after a rotation by 2π
– the double exchange. The composite system does not return to itself after rotation by π, that
is, after a simple exchange. These are the defining properties of integer spin. Thus, a composite
system of two spin 1/2tangles behaves like a particle with integer spin.
The analogy can be made more precise by recalling that in the strand tangle model, each of
the two fermions in Figure 10 continuously spins along its axis. The simplest situation is that
each particle continuously spins around an axis along the straight belts. If the rotation axes and
rotation sense of the two fermions and that of the double exchange agree, the situation is described
by S= 1 and z-component Sz= +1. If the rotation axes of the two fermions are the same but
that of the double exchange is opposite, one has S= 1 and Sz=−1. If the rotation axes of the
two fermions are the same, but that of the double exchange is perpendicular to them (e.g., in the
direction of the line connecting the two cores), one has the situation S= 1 and Sz= 0.
The defining property of spin 0 is that a rotation by any angle keeps the system unchanged.
If the rotation directions of the two fermions are opposite to each other, then the total system has
S= 0 and Sz= 0, independently of the rotation axis of the double exchange. A composite for
which two cores rotate in opposite directions has a total spin 0. Any tangle core that does not

24
rotate when strands are randomly deformed has spin 0.
The exploration shows that a composite of two tethered spin 1/2particles cannot have any
other spin value: apart from the four mentioned cases, no other behaviour under system rotation
is possible. In other words, composing two particles of spin 1/2can only yield a composite with
spin 0 or spin 1. For tangle cores with integer spin, core rotation by 2πis equivalent to no rotation,
in contrast to the case of a half-integer spin core. In these arguments, the two particles do not need
to be identical or elementary. Even if their tangle cores differ or if they are themselves composed,
the results about spin composition still hold.
The above arguments can be extended to composites of more than two spin 1/2particles.
Particles composed of an odd number of spin 1/2particles are fermions. Particles composed of
an even number of spin 1/2particles are bosons. Tethers reproduce the observed behaviour of
fermions under composition.
In nature, particles with integer spin behave differently from fermions: they are bosons. The
strand model reproduces this result. When the positions of two identical tangle cores with integer
spin are exchanged, no partial orbit of one core around the other core is required – in contrast
to the case of half-inter spin cores. No double exchange is needed to restore the sign of the
wave function. Indeed, in the strand tangle model, bosons, whether elementary or composed, can
exchange positions without hindrance. For composites, exchange is achieved by the cores passing
through each other. In other words, tethers explain why all particles with integer spin are bosons,
and why all bosons have integer spin.
In short, tethers explain why all particles, whether elementary or composed, are either
fermions, with a spin given by an odd multiple of 1/2, or bosons, with an integer spin. Teth-
ers imply the full spin-statistics theorem.
Test 1: Strands predict the lack of elementary particles with non-standard statistics.
This is observed. Tethers thus explain the existence, locality, spin, statistics, composition, rotation,
and orbits of quantum particles. These aspects cannot be explained without tethers – except if
tethers are hidden in the Dirac equation. Because tethers explain all these aspects of quantum
physics, the next task is to use tethers to define wave functions.
Part II: Wave functions, superpositions, Hilbert spaces, and measurements
Tethers define the quantum behaviour of particles. The strand tangle model goes one step further
than the tether idea by Dirac and the tethered particle model by Battey-Pratt and Racey. In the
strand tangle model, quantum particles themselves are made of tethers, i.e., of strands.
BQuantum particles are tangles of strands with Planck radius.
The starting point of the strand tangle model for quantum theory is the basic similarity of crossings
and wave functions:
BWave functions are short-time-averaged crossing densities of spinning fermion tangles.
Crossings of strands that do not belong to the same fermion do not contribute to the wave function.
The short-time average of tangle fluctuations is taken over a few Planck times. This averaging time
is much shorter than any time interval that plays a role in observations and measurements. The
implied averaging time is sufficient because strands and strand shapes are not observable, and thus
there is no ‘speed’ limit for their shape fluctuations.

25
The strand tangle model for
lmo
e functions
tangle
core
Step 3: Tak
q rsq uxyz{q |}~yyu
of the wave function to get
rsq {}zr ~qur
.
Observed
probability
density
crossing
midpoint
spin
axis
Crossing
midpoints
with their amplitudes
and local phases
Step 1: Take the crossing
midpoints and their phases
o
rsq z}
ve tangle.
Step 2: Take the time average
of all crossing midpoints and phases
to get the wave function.
Wave function
amplitude
and central
phase
central
phase
sq z

dot speci

es the crossing position

the shortest distance s determines the
crossing amplitude

and the angle 
de

nes the crossing phase.
s amplitude

s amplitude
position
Spinning
electron tangle
Spin S = 1/2
tether
spin
axis

ag

ag

ag
crossing axis
phase orientation
around crossing axis
, 

Fig. 11: In the strand tangle model, the wave function in the Schrödinger picture is the time-averaged
tangle crossing density, and the probability density is the time-averaged crossing switch density. The
figure illustrates how a tangle – an electron in this case – defines crossings and local phases, how these
fluctuating crossings lead to a wave function and a central phase, and how the observable crossing switches
lead to a probability density. The figure contains a simplification: in nature, the phase of the wave function
depends on the position, as illustrated in the following Figure 12.
Two roads lead from tangles to wave functions: Schrödinger’s approach and Feynman’s ap-
proach. They are illustrated in Figure 11 and Figure 13. The two approaches differ in how they
perform the time average that leads from a given tangle to its wave function.
Schrödinger’s approach starts with a loose tangle and averages crossings over all possible
strand shape fluctuations realized during the averaging time.
BFluctuations are due to the jiggling strands making up the vacuum.
As a result, each segment of a particle tangle continuously changes shape. The crossings in a
particle tangle continuously move around in space. Crossings continuously appear or disappear.

26
Averaging all crossings at a point in space yields a local value for the amplitude and phase of the
wave function. The average yields the common Schrödinger picture of quantum mechanics.
For example, the spinning tangle for a free particle illustrated in Figure 8leads, after averaging
the shape fluctuations (not shown in the figure), to a rotating cloud with a rotating phase. Figure 2
already gave an impression of the rotating cloud resulting from a spinning tangle.
The averaging procedure leading from crossings to probability densities is visualized in Fig-
ure 11 in three steps. The first step consists of reducing the tangle to its crossing midpoints,
crossing amplitudes, and respective crossing phases, leaving out the unobservable strands. The
second step consists of averaging all crossing midpoints, amplitudes and phases over the strand
shape fluctuations occurring during a few Planck times. It yields the wave function. Strands and
their crossings imply that the quantum phase varies from point to point, a feature not shown in
Figure 11, but in Figure 12. The third step consists of averaging crossing switches. It leads to the
probability density that describes a quantum state.
As the inset at the top right of Figure 11 recalls, at a given point in space, a crossing can be
described by one positive real number that specifies the smallest strand distance, and by up to four
angles or phases. Also a wave function at a point is described by one positive real number and
several angles or phases. In the following, wave functions will be deduced from crossings. Three
types of wave functions are important in this context: wave functions for the spin-less case, Pauli
spinors, and relativistic Dirac spinors. They take different numbers of phases into account.
The spin-less wave function used in the Schrödinger equation or in the Klein-Gordon equation
is the simplest case. It is described by a single complex field ψthat can be written as
ψ=Reiϕ =√ρeiα/2.(7)
Here, Ris the (positive) modulus or amplitude,ρ=R2is the probability density,ϕis the phase,
and α= 2ϕis the tangle core rotation angle due to the belt trick. The similarities between
crossings and wave functions lead to the following definition:
BThe amplitude due to a crossing varies inversely with the shortest strand distance s.
The probability density increases when the strand density increases. More precisely,
the amplitude or modulus R(x, t)of the wave function at a point xis defined as
R(x, t) = 1
s3/21
√n.(8)
The average – denoted as hi– is taken over all possible strand shape fluctuations during
a few Planck times. The crossing distance sis defined in Figure 11 as the shortest dis-
tance between two strand segments. The average of the crossing distance is normalized
with the help of the number n, the so-called (minimal) crossing number of the tangle.
The crossing number nis the smallest number of crossings that arise if a tangle is laid
down on paper. The crossing number nis a topological invariant; for example, n= 3
for the electron tangle presented in Figure 24. In particular, the crossing number nis
observer-invariant; it is a constant factor introduced in the modulus to normalize the
wave function.
As a result of the definition, strand fluctuations imply that the amplitude R(x, t)is a continuous
and differentiable positive real function of space and time, as expected. In the same way, the inset

27
Fig. 12: In a moving wave packet, the phase depends on the position, as seen in this illustration by Bernd
Thaller from his website [153], which visualizes phase with colour and amplitude with saturation. (Used
with permission.)
at the top right of Figure 11 leads to the definition of the quantum phase.
BThe (quantum) phase is due to the average orientation of the shortest distance saround
the crossing axis. More precisely, the quantum phase ϕ(x, t)of a spin-less wave func-
tion at point xis half the time-averaged local strand crossing phase αof the particle
tangle – the rotation around the crossing axis – at that point:
ϕ(x, t) = hα(x, t)/2i.(9)
The factor of 1/2is due to the belt trick. Again, the average is denoted by h i. It is
taken over a few Planck times and thus over the underlying strand shape fluctuations.
BThe crossing axis is defined with the two unit tangent vectors of the strands at the
endpoints of the shortest distance s, as illustrated in the inset of Figure 11. In particular,
the crossing axis is given by the sum of the two tangent vectors. The axis is always
perpendicular to the shortest distance vector.
BThe sign of a crossing is the direction in which the right hand turns when the thumb
and index follow the two strands along their orientations: clockwise hand rotation cor-
responds to a positive sign.
In simple words, the phase of the crossing specifies the orientation of the shortest distance around
the crossing axis. The phase angle αis measured against a predefined direction, indicated by the
vertical black dotted line in Figure 11. The other angles describing the crossing are ignored in
the case of the Schrödinger equation and the Klein-Gordon equation. As usual, there is freedom
in the definition of the direction that corresponds to the vanishing phase of a particle tangle. In
Figure 11, the freedom is the ability to choose the direction of the black dotted line. In the case of
tangles yielding gauge fields instead of wave functions, the freedom to choose the orientation of
the zero phase is related to the freedom of gauge choice, as will become clear below.
An instructive visualization of spin-less wave functions uses colour for the phase and to use
its saturation for the amplitude, as shown in Figure 12. This is done in the fascinating quantum
theory books by Thaller [154,155] and in the animations on his website [153].
The second type of wave function, the non-relativistic Pauli spinor, also takes into account the

28
angles βand γof a crossing, as visualized in Figure 11. These angles describe the orientation of
the crossing in space and are thus related to the spin orientation. A non-relativistic spinor used in
the Pauli equation has two complex components and can be written [156,157] as
Ψ = √ρeiα/2 cos(β/2) eiγ /2
isin(β/2) e−iγ /2!,(10)
Again, ρdenotes the probability density. In all situations without interactions, the tangle core can
be approximated and imagined to be a rigid spinning body. The three angles α,βand γare the
Euler angles that describe the orientation of the rigid rigid body in three dimensions. The factors
1/2are due to Dirac’s trick. In other words, a Pauli spinor can be described by one positive real
number and three phases. Alternatively, a Pauli spinor is a flagpole with a rotating flag. The
additional sign required in the flag model specifies the tangling state of the tethers.
The third and last type of wave function, the relativistic Dirac spinor with four complex com-
ponents, can be described by one positive density, three real parameters, and four phases. Dirac
spinors take Lorentz boosts and the angle δof Figure 11 into account. The angle describes the
relative weight of particles and antiparticles. Dirac spinors are explored later on.
In short, nature is neither perfectly continuous nor perfectly discrete.
BQuantum effects and measurements are due to crossing switches.
BContinuous quantities arise through time averages, over a few Planck times, of strand
shape fluctuations.
BWave functions are crossing densities resulting from the averaging of strand shape fluc-
tuations in spinning fermion tangles. Wave functions the blurred strand crossings of a
particle tangle. They are rotating and diffusing clouds of tangle crossings.
BTangles are the fluctuating skeletons of wave functions.
Strands imply that wave functions describe systems completely. The local amplitude is the time-
averaged local density of tangle crossings. The local phase is the time-averaged local orientation
of tangle crossings. As a result, a spin-less particle is described by a complex-valued field. The
next sections will prove that crossing densities have all the known properties of wave functions.
10 Path integrals and rotating arrows emerge from tangles
Feynman’s path integral approach is presented in his beautiful book QED [158]. He described the
motion of a quantum particle as an advancing and rotating arrow. Wave functions arise when the
effects of all possible paths are superposed. In particular, the phase and amplitude for all paths
arriving at a point must be added.
The strand tangle model directly leads to Feynman’s path integral approach. The approach is
based on tight rotating tangles, keeping in mind that strands are unobservable. Its three steps are
illustrated in Figure 13. The first step is to neglect the radius of the unobservable strands and to
imagine that the tangle core is tightened to a point-like region by ‘pulling’ the tethers. This yields a
point particle with a phase arrow attached to it. In the spinning tangle for a free particle illustrated
in Figure 8this implies neglecting the unobservable tethers and zooming out until the central cube,
the tangle core, has negligible size. In the strand tangle model, the continuous rotation of the tangle
core visualizes Feynman’s rotating arrow. The second step is to imagine that the tight tangle core,

29
The strand tangle model for a fermion in the path-integral formulation
Step 3:
  
erage of crossing
switches is taken, yielding the

density.
spin axis
spin axis
Step 1:
  
tightened
and tethers are neglected,
yielding a position and a phase.
Step 2:
  
erage of the
fluctuating point and of its phase
is taken,

  
e function.
Wave function
amplitude
and central
phase
central
phase
core
Observed
probability
density
crossing

Tight tangle
  
flag
Spinning
electron
tangle
Spin S = 1/2
tether
flag
flag
Fig. 13: The wave function and probability density in Feynman’s path integral formulation are due to
time-averaged fluctuating (almost) point-like particle tangles. The figure illustrates how a tangle that is
“pulled tight” defines the position and the local phase. The fluctuations of the (almost) point-like tangle
core then lead to a wave function with phase. In particular, an advancing particle is an advancing rotating
arrow. For particles with spin, the tangle can be represented by a rotating flagpole with rotating flag. The
arrow or the flagpole represents the phase. As usual, the modulus of the wave function leads to a
probability density. Also this figure is simplified: in reality, the phase of the wave function of a localized
particle depends on the position, as shown in Figure 12.
which is continuously spinning, randomly changes its position, taking different paths. As before,
the fluctuations are due to fluctuating strands making up empty space. Averaging over all possible
paths of the tight tangle core yields the wave function. The third step is, again, the definition of
the probability density as the crossing switch density.
In the strand tangle model, the fluctuations of the tangle motion represent the effects of all
the possible paths. With the definition of tangle addition given below, path combination occurs
exactly as in Feynman’s path integral formulation of quantum theory. Contracting electron tangles
– and the photon tangles introduced below – to a point and ignoring their unobservable tethers
fully reproduce all the details of Feynman’s approach. The two possible rotation directions of
advancing cores distinguish right-handed from left-handed particles. The two mirror versions of a
chiral core distinguish particles and antiparticles.

30
Feynman’s approach for deriving wave functions from strands was implied and used already
by Battey-Pratt and Racey [94]. Mathematically, the approach yields the same result for amplitude
and phase that arise in Schrödinger’s approach. However, defining the phase using the orientation
of a tight tangle corresponds more directly to Feynman’s idea of a rotating arrow. In addition, the
appearance of half angles in the definition of spinor wave functions (and the g-factor 2) becomes
more intuitive. In any case, the description of the quantum state using a fluctuating tight tangle is
equivalent to the description with a fluctuating loose tangle. Both ways lead to a wave function
described by an amplitude and one or several phases, that is, to one or several continuous complex
functions of space and time.
How can one be sure that the tight tangle path averaging process results in the correct wave
function ψ(x, t)? The answer is the same as that of the original path integral formulation. Because
an advancing rotating tight tangle reproduces the fermion propagator, its path integral reproduces
the free particle wave function and, as will appear below, its full evolution equation [158].
In short, the strand tangle model of quantum particles reproduces the path integral formula-
tion of quantum mechanics once particle tangle cores are approximated as tight chiral tangles of
strands with vanishing radius and the unobservable tethers are ignored. Averaging tight tangles
that fluctuate over space yields the usual wave function. In other words, the wave function of
a particle is either a rotating and diffusing cloud of crossings or a cloud due to its continuously
fluctuating tight tangle core.
11 Wave function superpositions are described by tangles
Crossing densities can only be models for wave functions if they form a Hilbert space. A Hilbert
space is a vector space with an inner product and several technicalities. To show that crossing
densities from a vector space, linear combinations or superpositions of two wave functions need
to be defined. This requires the definition of two operations: scalar multiplication and addition.
The first, simple way is to define the operations for wave functions as in quantum mechanics:
BThe scalar multiplication aψ and the addition ψ1+ψ2of wave functions ψiare defined
by applying the respective operations on the complex numbers at each point in space,
i.e., on the local values of the wave function. In the strand tangle model, this implies
performing the operations on the corresponding crossing densities.
This first definition is sufficient to show that crossing densities form a vector space. The same
approach can be used to define an inner product and then show that the crossing densities form a
Hilbert space, as expected. Also the time evolution of the wave functions follows. One can then
continue directly with Part IV of this article, starting on page 59, which explores the differences
between the tangle model and quantum theory.
However, a second way to define a vector space for wave functions is more intuitive and strik-
ing. Addition and multiplication can be defined as operations on the tangle describing a quantum
system – the fluctuating skeleton of its wave function. The short-time average – the blurring – can
be taken after the operations on the underlying tangle are performed. In this way, suitable opera-
tions on tangles will be shown to imply the defining axioms for vector spaces, inner products, and
Hilbert spaces for their crossing densities.
BThe scalar multiplication aψ of a state ψby a complex number a=reiδ, with 06r6

31
¡¢£ ¤¥
erage
of crossing
switches
Scalar multiplication of tangle states
Strand tangle
¢¦§£¨©
ª«¬£¥
ed
®¦«¤«¡¨¡¯
§£°¬¡¯©
untangled
addition
region
√0.2ψ
√0.8ψ
ψ
Fig. 14: The scalar multiplication of a localized tangle by two different real numbers smaller than 1 is
illustrated. The resulting thinning of the corresponding wave function ψis visualized.
1is formed by taking the underlying tangle, rotating its tangle core by the angle 2δ, and
then ‘pushing’ a fraction 1−rof the tangle to the cosmological horizon, thus keeping
the fraction rof the original tangle at finite distances. Thus, scalar multiplication with
r61is a process of tangle core rotation and thinning. Time averaging leads to the
wave function aψ =reiδψ.
A simple case of scalar multiplication for a tangle representing a state ψis illustrated in Figure 14.
The figure illustrates the idea of thinning a tangle core. Pulling a tangle core apart such that
no crossings arise in between corresponds to dividing the tangle into two fractions. The relative
size of the two fractions is determined by (the square root of) the relative volume integrals of the
probability densities. The mentioned process of tangle core thinning is visualized by the untangled
strand segments in the so-called addition region.
The tangle version of scalar multiplication is effectively unique. Indeed, even though there is a
choice about which specific fraction rof tangle crossings is kept and which specific fraction 1−r
of crossings is sent away, this choice is only apparent. The resulting crossing density, defined as

32
x1x2
x1x2
x1x2
x1x2
±²³´ µ¶
erage
of crossing
switches
Linear combination of tangle states
·¸¹´º¶
ed
»º¼¸µ¸²½²±¾
¿´À¹²±¾Á
Two quantum states localized at different positions
A linear combination
untangled
addition
region
±²³´ µ¶
erage
of crossing
switches
·¸¹´º¶
ed
»º¼¸µ¸²½²±¾
¿´À¹²±¾Á
ψ1ψ2
√0.8ψ1+√0.2ψ2
Fig. 15: A linear combination of two states representing a particle localized at two different positions
visualizes the superposition of wave functions in the strand tangle model.
the average over fluctuations, is independent of this choice because the tangle topology, which
specifies the particle, remains intact.
The tangle version of scalar multiplication is associative: the relation a(bψ)=(ab)ψholds by
construction. The scalar multiplication of strands also behaves as expected for the factors 1 and
0. Finally, strand multiplication by −1is defined as the rotation of the full tangle core by 2π, as
required by the belt trick. In other words, scalar multiplication by complex numbers can indeed
be modelled as tangle core rotation and thinning. Also addition can be defined for tangles.
BThe addition of two tangles a1ψ1and a2ψ2, for which |a1|2+|a2|2= 1 and for which
ψ1and ψ2have the same topology, is defined by directly connecting the crossings not
pushed far away during the scalar multiplication by a1and a2. The connection of
tangles must be performed in such a way as to maintain the topology of the original
tangles; in particular, the connection occurring in the spatial addition region must not
introduce any crossings. Time averaging then leads to the superposition a1ψ1+a2ψ2.
An example of superposition for the case of two quantum states at different positions in space is
shown in Figure 15. No strand is cut or re-glued during connecting – although imagining doing so
might help for visualizing the operation. The addition region shown in the figure will be important
later, in the exploration of entanglement.
The definition of linear combination requires the final tangle a1ψ1+a2ψ2to have the same

33
topology and the same norm as each of the two normed tangles ψ1and ψ2that are combined.
Physically, this means that only states for the same type of particle can be added; this also means
that particle number is preserved. Tangles thus automatically implement the corresponding super-
selection rules of quantum theory. This property is welcome because, in conventional quantum
mechanics, the superselection rules need to be added by hand. In contrast, the strand tangle model
contains them automatically.
One notes that the sum of two tangles is unique, for the same reasons given for the case of
scalar multiplication. Tangle addition is commutative and associative, and there is a zero state,
or identity element, given by the trivial, i.e., the untangled tangle. Tangle addition also implies
distributivity with respect to the addition and scalar multiplication of states.
In short, using tangle rotation, thinning and connecting to define the scalar multiplication and
addition of crossing densities of tangles proves that the crossing densities, as expected from wave
functions, form a vector space.
12 Tangles imply Hilbert spaces
To form a Hilbert space, crossing densities deduced from tangles must allow the definition of an
inner (or scalar) product. This implies reproducing these properties from quantum mechanics:
BThe inner (or scalar) product between two states ψ1(x, t)and ψ2(x, t)is defined as
hψ1|ψ2i=Rψ1(x, t)ψ2(x, t) dx.
BThe norm (or modulus) of a state is kψk=phψ|ψi.
BThe probability density ρof a state is ρ(x, t) = hψ|ψi=kψk2.
In the strand tangle model, the conjugate tangle ψis formed from the tangle ψby exchanging
the sign of each crossing, i.e., by exchanging underpasses and overpasses. Conjugation arises by
switching crossings. As a consequence,
BThe inner product hψ1|ψ2iis a complex number whose phase is given by the average
crossing rotation between the two states and whose magnitude is the spatial average
ratio of switched crossings in the tangle ψ1that also appear in the tangle ψ2.
This inner product has all the required properties: it is Hermitian, sesquilinear, and positive def-
inite. The required technicalities about the completeness of the norm are also realized by tangles
of strands. As a consequence, wave functions defined with strand tangles form a Hilbert space.
Being related to crossing switches, the inner product is a physical observable, in contrast to the
states themselves. This property is as expected.
The inner product of wave functions allows defining the norm of a wave function. In conven-
tional quantum mechanics, the norm is the square root of the integral of ψψ, taken over all space.
In the strand tangle model, the norm of the wave function is defined in the same manner. The
resulting integral has the value 1 for every quantum state defined with tangles. Thus, one-particle
wave functions are normalized in the tangle model.
Anticipating the results presented below, the investigation of electrodynamics [120] shows that
the (appropriately signed) minimum crossing number of a tangle is the electric charge of a particle
in units of e/3. In the tangle model, the probability density thus reproduces the charge density.
All processes, shape fluctuations and strand exchanges of strand tangles conserve charge.

34
As mentioned above, the probability density is the crossing switch density. Because the defi-
nitions of the probability density and the inner product involve crossing switches, both quantities
are physical observables. This is as expected.
Thus, wave functions and Hilbert spaces arise as consequences of ~. This result retraces the
history of quantum theory. Schrödinger derived his evolution equation from de Broglie’s matter
waves. More precisely, he derived his equation from the expression λ=~/mv for the wavelength
of quantum particles. The quantum of action ~and the mass mof the electron are the only two
foundations of the Schrödinger equation for free electrons. Because the equation is linear, its solu-
tions also yield and describe superpositions, interference and entanglement. All quantum effects,
Hilbert spaces, all other mathematical aspects, and all its counter-intuitive aspects disappear if ~
is set to zero. The simplicity of the historical basis for quantum theory explains why assigning ~
to a crossing switch allows deducing all of quantum theory.
Once states and their Hilbert space are defined, operators on that Hilbert space can be defined
using strands. Operators deform tangles. In quantum theory, unitary operators preserve inner
products. In the strand tangle model, unitary operators, such as the time evolution operator, de-
form tangles by retaining the norm of the wave function, that is, by retaining both the topology
and the shape of the tangle core. Unitary operators deform tethers but not the tange core. In quan-
tum theory, self-adjoint or Hermitian operators are important because they conserve probabilities,
have a real spectrum, and describe physical observables. In the strand tangle model, Hermitian
operators leave the tangle topology invariant but also deform the tangle core. Further aspects of
field operators are explored in Appendix C.
The position operator Xlocalizes a particle tangle core at a ‘point’, i.e., it tightens the core
into a given region of Planck size. The momentum operator P‘localizes’ a particle tangle core at a
momentum value, i.e., it deforms the core into an infinitely long helix with negligible diameter and
given wavelength. Both conventional quantum theory and the tangle model lead to the operator
equation
P X −XP =i~.(11)
This well-known canonical commutation relation was first deduced by Born and Jordan in 1925
[159]. In 1987, Kauffman suggested that the commutation relation is due to a crossing switch
[95,96]. However, at that time, no one took up this suggestion. The strand tangle model does.
In short, if strand tangles are used to define wave functions as crossing densities, then they re-
produce superpositions, form Hilbert spaces, and allow deducing unitary and Hermitian operators.
Probability densities, defined as crossing switch densities, behave as expected from conventional
quantum theory and observations. The next checks are whether tangles of strands reproduce inter-
ference, measurements, and entanglement.
13 Tethered particles can pass each other
In everyday life, we all can imagine two balls, both connected to the border of space by six
mutually perpendicular steel tethers under tension. Two such balls cannot move past each other,
because at a certain relative position, their steel tethers will touch and thus will prevent any further
movement.

35
orientation
phase
phase
phase
position
orientation
phase
phase
no crossing,
ÂÃÄÅÅÆÇÈÉÊ
e
Total
tangle
Total
tangle
ËÉ
ÌÍÎ
tangle
fraction
Second
tangle
fraction
position
position
ËÉ
ÌÍÎ
tangle
fraction
Second
tangle
fraction
Constructive interference
Destructive interference
position
position
Fig. 16: The strand explanation at the basis of interference: two crossings connected by strands either
superpose constructively (top) or destructively (bottom).
In the strand tangle model, the situation for two tethered fermions differs markedly. Because
tethers have no observable properties, they have no tension, are extremely flexible, have no fixed
length, produce no forces, and have no mass. Therefore, tethers do not hinder each other in any
way, nor do they have any effect on the motion of the tangled cores to which they are attached.
The touching of unobservable, extremely flexible, and extremely extendible tethers does not affect
the motion of the particle cores to which they are connected.
In short, for all practical purposes, the touching of tethers far away from particle cores can
be ignored. Fluctuating tethers do not hinder the free motion of non-interacting particles across
space. This result is useful for the description of interference with tangles.

36
Constructive
interference
for an electron
Destructive
interference
for an electron
Starting
electron
ÏÐÑ
ctron
ÒÓ ÒÔÔÕÖÒÐ
Starting
electron
V
Ò×ØØÙ
ÒÓ ÒÔÔÕÖÒÐ
Fig. 17: The tangle explanation of interference is illustrated for an electron tangle passing a double slit.
Depending on the phase difference arising in the two paths, the electron tangle interferes constructively
(left) or destructively (right) with itself. The tethers of the particles making up the screen can be ignored,
as explained in Section 13.
14 Strands lead to the interference of fermions
The observation of interference of quantum particles – such as electrons, neutrons, atoms and
molecules – is due to the superposition of coherent quantum states with different phases at one
position in space. Interference results from the linear combination of wave functions. Such super-
positions are a central feature of quantum physics and highlight wave-particle duality.
As expected, the strand tangle model reproduces interference. Using the definition of superpo-
sitions given above, an equally weighted sum of a tangle and the same tangle with a phase rotated
by π/2(thus with a core rotated by π) results in a tangle whose phase is rotated by the intermediate
angle, thus with a phase rotated by π/4. This example of constructive interference is illustrated in
the upper half of Figure 16.
The most interesting observation is destructive interference, or extinction. In experiments,
the multiplication of a wave function ψby −1yields the negative of the wave function, i.e., its
additive inverse −ψ. The local sum of a wave function and its exact negative vanishes. This is the
explanation of extinction in conventional quantum theory.
In the strand tangle model, the negative of a fermion tangle has a core rotated by 2π. Using the
strand definition of linear combinations, the sum of a fermion quantum state with its exact negative
requires rotating half of the core by 2πand then connecting it to the other, unrotated half, without
crossings between them. This is impossible without untangling the core. Equivalently, this case
yields untangled strands at that position. This is shown in the lower half of Figure 16. Therefore,
the result has a vanishing crossing density in the spatial region where a state is added to its exact

37
Constructive interference
for a photon
Destructive interference
for a photon
Starting
photon
ÚÛÜÝ
on
Þ
ÝÞßßàáÞâ
Starting
photon
V
Þãääå
ÞÝÞßßàáÞâ
Fig. 18: The tangle explanation of photon interference is illustrated. Depending on the relative phase
between the two paths, a photon passing a double slit interferes constructively (left) or destructively (right)
with itself.
negative. Vanishing crossing density is the defining characteristic of the vacuum. Strand tangles
thus explain extinction as a consequence of particle tangle superpositions.
As a consequence, tangle superpositions describe and visualize the double-slit experiment.
Depending on the two paths taken to the region of superposition, the phases add up or cancel out.
This behaviour is visualized in Figure 17. As explained in the preceding section, the influence of
the tethers of the particles in the screen with the slits can be ignored.
In short, fluctuating strand tangles describe and explain both the constructive and destructive
interference of matter particles. This result suggests to explore also the case of photons.
15 An intermezzo: strands lead to the interference of photons
In the strand tangle model, a photon is a single strand with a rotating loop or twist, as illustrated
in Figure 18. The graphic summary of the relation of photon tangles to electrodynamics is given
in Figure 19. The relation between photons and the other gauge bosons in nature is shown below,
in Section 32. Topologically, a photon is a propagating first Reidemeister move.
The twist is the tangle core of a photon. The size of the twist is the photon wavelength. The
energy of a photon is localized in the curved sections of its strand. For any photon, only its
crossing switches are observable; its tethers are not. When a photon advances, its core advances
and rotates. The phase of a photon is determined by the pointing direction of its twisted core.
(This is in contrast with fermions, for which the phase is given by half the pointing direction of
the core.) As illustrated in Figure 19, the photon, like the electron, can therefore be seen as an
advancing rotating arrow. Thus, the strand tangle model simply adds a twist and two unobservable

38
tethers to the conventional description of the photon as a rotating arrow.
The tangle model shows that a photon whose core has rotated by 2πis equivalent to a photon
with an unrotated core: the tethers can fluctuate from one case to the other. This is the – much
simpler – spin 1 version of Dirac’s belt trick and is illustrated in the top left of Figure 19. An almost
insignificant ‘boson trick’ corresponding to the ‘fermion trick’ mentioned above also exists: after
a single core exchange, two nearby photons yield the same observables as before. Thus, photons
have spin 1 and show boson behaviour. The photon tangle is topologically trivial; as explained
below, this implies a vanishing mass. Electric and magnetic fields are densities and flow densities
of twists. In particular, electric charge, its motion slower than light, and its conservation appear
naturally. Also, Figure 19 shows that strands imply Coulomb’s law. Mathematical theorems based
on these properties by Heras [160–164] and by Burns [165] then prove that Maxwell’s equations
hold, including the Lagrangian of classical electrodynamics. In particular, strands imply minimal
coupling. The way of visualizing Maxwell’s equations and quantum electrodynamics with the
help of twisted strands has been extensively explored elsewhere [120].
Figure 18 visualizes the interference of a photon – always only with itself, as told by Dirac
[166] – in the case of the double-slit experiment. The ‘negative’ of a photon state is a strand
whose looped twist points in the opposite direction. The addition of half a twist with its other
negative half yields a strand without crossing. This situation visualizes extinction and is shown on
the right-hand side of the figure. In the case of extinction, photon states with opposite phases add
up to vacuum strands.
Photons resemble waves: they have frequency and wavelength, and they can interfere. Photons
also resemble particles: they can be counted and have angular momentum, linear momentum, and
energy. The tangle model of photons – and of electrons – thus visualizes wave-particle duality.
A literature search shows that a similar description of a photon with a curved strand, such as the
‘corkscrew model’ of the photon [93], is part of physics lore. However the approach has never
been expanded to include the quantum of action, nor has it ever been published.
In short, the tangle model of the photon is a rotating twist. The model reproduces electromag-
netism. The model also describes and explains the quantum interference of a photon, always with
itself, because only crossing switches are observable. The rotating twist model of the photon, with
its implied U(1) gauge symmetry, is confirmed in more detail below.
16 Measurements, Born’s rule, wave function collapse and decoherence
Experiments show that every measurement of a quantum state ψhas two effects. First, every
measurement yields a real eigenvalue aof the operator Adescribing the variable being measured.
Secondly, every measurement changes, projects, or collapses, the quantum state into the corre-
sponding eigenstate ψa(in the situation without degeneracy). The collapse occurs with a proba-
bility given by |hψa|ψi|2, the squared inner product between the quantum state and the eigenstate.
This is known as Born’s rule.
In nature, every measurement apparatus is a device that produces, stores and displays measure-
ment results. This is possible because every measurement apparatus is a device with a memory,
and thus a classical device. All devices with memory contain at least one bath, i.e., a subsystem
described by a temperature [167]. Thus, every measurement apparatus couples a bath to the system
that is measured. The detailed coupling depends on and defines the observable being measured

39
æçèé
e
paper
plane
ç
êë
è
w paper
plane
æçèé
e
æçèé
e
ç
êë
è
w
.
ìíîïð ñòóôðîõ ö÷ îíñøöó ùúûõùõ í÷÷
ect
only topologically chiral tangles:
üýð ûóþðñðùîíôûÿûù
T
of strands
leads to twist transfer.
ç
êë
è
w
ü
opologically
chiral tangles are
affected, i.e., they are
electrically charged.
ü
opologically
achiral tangles are
not affected, i.e.,
they are neutral.
A
óö
v
ûñï ùúûõù ûóþÿûðõ ó
 
S = 1,
=

for photons,
íñø í

ïíòïð ÷îððøöó
e
phases
üýð ùúûõù óö
v
e, applied to two interacting
tangles, yields quantum electrodynamics
:
üýð
twist, or first Reidemeister
óö
v
e, yields a
óöøðÿ ÷
or the photon
íñø ûùõ óöùûöñ
:
f
ðîóûöñ
photon
v
í

òòó
f
ðîóûöñ
E
óûõõûöñ íñø îðíôõöîþùûöñ ö÷ ñòóðîöòõ
îíñøöó ùúûõùõ
v
ûî
tual phot
öñõ

ô
y chiral
tangles leads to Coulomb's law:
phase phase
phase
electron photon
E
óûõõûöñ íñø îðíôõöîþùûöñ
v
ûî
tual photons
yields an 1/r2 dependence.
phase
tethers
fluctuate,
phase
rotation
goes on
Fig. 19: A graphical summary of the tangle model for electromagnetism is given. The motion of a photon
with its rotating phase, the absorption of a photon, the origin of electric charge from tangle chirality and
the origin of Coulomb’s law are visualized. More details are found below and in reference [120].
by the apparatus. Every coupling of a quantum system to a bath results in decoherence. Decoher-
ence leads to wave function collapse and probabilities. Therefore, collapse and probabilities are
necessary and automatic consequences of decoherence in conventional quantum theory [168,169].
The strand tangle model describes the measurement process in precisely the same way as con-
ventional quantum theory. In addition, strands visualize the measurement process.
BAmeasurement is a specific deformation of the system tangle, induced by the bath
in measurement apparatus, that deforms the strands of the quantum system into the
resulting eigenstate.
BThe deformation of the tangle of the quantum system by the bath strands is the collapse
of the wave function.
BThe storage of the result in the memory of the measurement apparatus involves the
tangle of the quantum system and the tethers in the bath of the apparatus.
An example of measurement for the case of a spin superposition is illustrated with the help of

40
Superposition (one of two equivalent configurations)
either
'
up
'
Observed spin:
Spin measurement
Basis states
always
up
always
down
or
'w'
untangled
addition
region
Spin
mm
direction
tm 
erage
of crossing
switches
+
Observed spin:
Spin
mm
direction
Fig. 20: The measurement of a spin superposition makes the addition region disappear either outwards or
inwards.
strands in Figure 20.
BWhen a measurement is performed on a superposition of two spin states, the untangled
‘addition region’ is made to shrink or expand into disappearance by the bath.
When this occurs, one of the underlying eigenstates ‘gobbles up’ the other eigenstate: the wave
function collapses. This process is triggered by the strands in the bath inside the measurement ap-
paratus (not shown in the figure). The strands of the apparatus make the addition region disappear
either towards the outside or the inside. The choice is determined by the details of the state of the
bath coupled to the system being measured. Thus, the bath fluctuations determine the outcome of

41
the measurement.
In the strand tangle model, when the bath in the measurement apparatus selects an outcome, the
bath tangles change the original quantum state being measured into the eigenstate of the outcome.
This change occurs by deforming all the strands of the quantum state being measured into the
strand configuration of the eigenstate. In other words, a large number of bath tangles causes
the quantum state to collapse by pushing the strands of the quantum system. Therefore, in any
measurement, the initial quantum state and the final quantum eigenstate differ only in their strand
shapes, and not in their topologies. This is observed.
The probability of measuring a particular eigenstate will depend on the (weighted) volume that
the eigenstate takes up in the superposition because the bath will choose the eigenstate with the
higher weight more often. This is Born’s rule.
For example, whenever the position of a charged particle is measured, an initially delocalized,
spread-out quantum state is localized at the measured position. When the particle charge triggers
the particle detector, the tangle crossings of the particle interact with the crossings of the charges
inside the detector. This interaction localizes the quantum state of the particle, and thus the tangle
core, at the spot where the detector is triggered. This happens because whenever charges interact
via virtual photon exchange, their cores are automatically localized.
Thus, the strand description of measurement is a specific realization of wave function collapse
induced by the environment. This approach has been championed by Zeh [168] and many after
him. The details of contextual collapse are still being refined [170]. In simple words, strands state
that there is a stochastic aspect in the environment of every quantum system. This description
appears to resolve the issues raised by Adler [171].
The strand visualization of the wave function collapse triggered by the bath also implies that the
collapse takes time. This is observed, as explained in the next section. The collapse time is the time
taken by the bath to trigger the collapse. The strand visualization of the wave function collapse
also clarifies that the collapse is not limited by any speed limit, as no energy and no information
are transported, and thus no signal is transmitted. Indeed, the collapse occurs because the strands
of the system are deformed by the strands from the bath. If one disregards the bath, collapse seems
superluminal. If the bath is taken into account, the limit for signal speed is respected. Likewise,
the quantum Zeno effect results [172].
In short, the strand tangle model describes measurements in the same way as conventional
quantum theory as bath-induced decoherence and collapse. Any measurement apparatus – which
includes a bath by definition – interacting with a quantum system automatically enforces Born’s
rule through the many bath tangles. In particular, strands visualize the collapse of the wave func-
tion as a shape deformation from a superposition tangle to an eigenstate tangle, enforced by the
many bath tangles. This description agrees with the usual description of collapse as a consequence
of environmentally induced decoherence. In the tangle model, quantum probabilities reflect strand
fluctuation probabilities.
17 Strands imply a finite decoherence time
In nature, the decoherence process takes time. Numerous experiments have confirmed this. [173–
178]. The value of the decoherence time is due to the interaction with the bath in the apparatus or
environment. Generally speaking, the denser and the more energetic the microscopic degrees of

42
freedom of the involved bath, the faster the decoherence.
The decoherence time can be estimated using various methods. A simple estimate arises for
baths composed of many scattering particles. In this case, the decoherence time typically is of
the order of the time interval between two scattering events due to the bath. The resulting spatial
localisation is of the order of the (thermal) de Broglie wavelength of a typical bath particle. More
precisely, the decoherence time is given by the particle flux and the interaction cross-section, as
explained by Joos and Zeh [168] and by Tegmark [179]. In most – but not all – practical situations,
the resulting decoherence time is extremely short and also leads to localization within a tiny spatial
domain.
A second estimate of the decoherence time uses the relaxation time and temperature of the
bath [169,180]. In all everyday situations, unmeasurably small values for the decoherence time
result. Long decoherence times are possible if interactions with baths are minimized. This requires
careful experimental setups in specialized laboratories. Such setups are commonly realized in
research experiments.
Also in the strand tangle model, scattering processes occur. Also in the strand tangle model, the
relaxation time at the origin of decoherence arises due to microscopic processes in the bath. Also
in the strand tangle model, the decoherence time is an effective interaction time. The decoherence
time is the time that the bath strands take to project the particle tangle onto the eigenfunction of
the measured value of the observable.
The combination of decoherence and minimum length that characterizes the strand model of
quantum mechanics is also found in the Montevideo interpretation of quantum mechanics pro-
posed by Gambini and Pullin [181,182]. Investigating the common aspects is wortwhile.
In short, strands reproduce decoherence in all its aspects: decoherence takes time, destroys
macroscopic superpositions, and is a result of fluctuations in the baths of the environment or
measurement apparatus.
18 Quantum entanglement is due to topological entanglement
In nature, two or more particles can be entangled. Entangled states are coherent N-particle states
that are not separable, that is, they cannot be written as the product of single-particle states. Entan-
gled states are a fascinating and central aspect of quantum physics. Entangled states do not exist
in classical physics but are observed and explored in many quantum experiments.
In conventional quantum theory, an N-particle wave function is usually described by a single-
valued function in 3Ndimensions. However, the strand tangle model naturally defines Nwave
function values at each point in space: each particle is described by its tangle, and each tangle
yields, via short-term averaging, its own complex value(s) at each point in space. Because separate
particles have separate tangles, the state of Nparticles can be described in 3 spatial dimensions –
despite the recurring prejudice that this is impossible. Therefore, 3 dimensions are also sufficient
to describe entanglement.
An entangled state is a non-separable superposition of several particle states. A state is entan-
gled or non-separable if it is not a product of separate particle states. A well-known example of
entanglement is the spin entanglement of two identical, but distant fermions in a spin 0 state. The
example was introduced and discussed in detail by Bohm [183]. It was experimentally tested, for
example, by Fry [184].

43
First separable basis state
Strand model:
Second separable basis state
Observation:
x1
x1x2
x2
|↓↑i
|↑↓i
Fig. 21: Two examples of two distant particles with spin in separable,unentangled and incoherent states
are illustrated, both in the strand tangle model and in the corresponding observed probability densities.
In the strand tangle model, two distant fermions in a separable state are modelled as two
distant, separate tangles of identical topology. Figure 21 illustrates two such separable basis states
in the strand tangle model. In this case, they are two states with total spin 0, given by |↑↓i and by
|↓↑i. Such states (with their linked tethers) are typical outcomes of spin measurement experiments.
However, other states are also of interest. Using the definition of tangle addition, a superposition
such as √90 % |↑↓i +√10 % |↓↑i of the two spin-0 basis states looks as illustrated in Figure 22.
Such a state is not a product state, and therefore it is not separable, but entangled.
The strand tangle model allows a simple definition and visualization of entanglement.
BIn the strand tangle model, a state is non-separable whenever the tethers of the par-
ticles remain topologically entangled even if the tangle cores are pulled apart. More
precisely, as illustrated in Figure 22, two states are entangled if their strand addition
region surrounds both tangle cores.
In the strand tangle model, the entanglement of two particles is visible in the addition region. When
the spin orientation of one of the particles is measured, the untangled ‘addition region’ disappears.
The result of the measurement will either be the state favoured by the inside of the addition region

44
either this eigenstate
(90%)
or this eigenstate (10%)
Entangled state
untangled
addition
region
Strand model: Observation yields:
√90 % |↑↓i +√10 % |↓↑i
x1
x1
x1
x2
x2
x2
Fig. 22: An entangled,inseparable and coherent spin state of two distant particles is illustrated. The
addition region around both particles prevents either particle from being seen as a separate system that is
independent of the rest of its environment.
or the state favoured by the outside. Because the tethers of the two particles are linked, after the
measurement, independently of the outcome, the spins of the two particles will always point in
opposite directions. This happens independently of particle distance. After the measurement, the
state is separable. Despite this extremely rapid and seemingly superluminal collapse, no energy
travels faster than light. Thus, the strand tangle model fully reproduces the behaviour of entangled
spin 1/2states as it is observed in experiments.
The similarity of quantum entanglement and topological entanglement has been noted for a
long time [185–193]. Tethers provide a basis for this analogy. For example, suitably connecting
the tethers at spatial infinity should recover the results of the classification of multi-particle en-
tanglement by Quinta and André [189]. !!!2 Needs to be detailed (entanglement): Likewise, the
description of entanglement with tethers in Figure 22 reproduces the demonstration of this aspect
of quantum mechanics that was provided by Mermin [194]. This would not be possible with-
out tethers. Also descriptions of two-particle or two-qubit entanglement with two Bloch spheres,
as given by Filatov and Auzinsh [195], are simplified when using tethers. Their necessary extra
mathematical conditions describing entanglement can be seen as effects of unobservable tethers.
In short, using the definition of wave functions as tangle crossing densities, tangles of strands
reproduce quantum entanglement through topological entanglement of tethers. Entanglement thus
follows from the fundamental principle. This leads to a topic of basic importance.

45
19 Strands are not hidden variables
At first sight, the strand tangle model seems to introduce hidden variables into quantum theory.
One is tempted to argue that the fluctuating shapes of strands play the role of hidden variables.
Non-contextual hidden variables are impossible in quantum theory, as shown most clearly by the
Kochen-Specker theorem, which is valid for sufficiently high Hilbert space dimensions [196]. In
real-life systems, the conditions of the theorem are always satisfied.
Despite the first impression, the strand tangle model does not contain hidden variables. First,
strands and their shapes are not observable. They are not variables. Strands and their shapes are
hidden, but they have no measurable properties: strands have no momentum, no energy, no spin,
no charge, and no quantum numbers. Therefore, it is impossible to define a measurable position
or shape for strands. All illustrations in this article are unphysical, because they depict strands in
certain positions and shapes, and suggest that strands can be counted. The figures become physical
only once they are modified to include an indeterminacy of time and an indeterminacy of position,
for every strand segment. In nature, only crossing switches are observable, as shown in Section 34.
Crossing switches are due to intrinsic probabilities.
Secondly, strand shapes evolve in a manner dictated by the influence of the environment, which
consists of all other strands in nature, including those of empty space itself. Therefore, the evolu-
tion of strand shapes and crossing switches is contextual.
For the two reasons just given, the strand tangle model does not contradict the Kochen-Specker
theorem. The strand tangle model provides no observables beyond those of quantum theory. As
expected from any model that reproduces decoherence, the strand tangle model leads to a contex-
tual and probabilistic description of nature. In particular:
Test 2: Strands predict the lack of measurable deviations from the linear behaviour of wave
functions.
No deviation has ever been found [197–199]. The results of the strand tangle model also agree
with the research results on probabilities in quantum mechanics. The tangle model reproduces
the approach to quantum theory as a theory of extrinsic properties, as explained by Kochen [200].
Strands confirm the investigations of Colbeck and Renner [201–203], who showed the lack of
alternatives to quantum theory, and in particular, the lack of ‘more complete’ theories that are in
agreement with observations and the freedom of choice. The results of the strand tangle model
also appear to agree with the Pusey–Barrett–Rudolph theorem, which states that wave functions
are more than only information about a quantum state [204].
In short, it was shown that the strand tangle model does not make use of non-contextual hidden
variables. Again, strands imply that wave functions describe systems completely. Nevertheless, in
the strand tangle model, quantum theory emerges from fluctuating tangle shapes. The fluctuations
of the strand vacuum are the origin of probabilities of quantum physics. The probabilities in
quantum physics are macroscopic results of the smallest measurement errors in nature. Quantum
theory remains as fascinating as ever.
20 The probability density is limited
In the tangle model, quantum probabilities reflect strand probabilities. Thus, all probabilities in
quantum physics are due to the smallest measurement errors in nature. This connection between

46
probabilities and strands implies a limit.
Test 3: Strands limit the value of probability density to
||ψ(x, t)||26c3
4G~3/2
≈3·10103 m−3.(12)
The upper bound for the probability density is the inverse of the minimum volume. Specifically,
the limit contradicts the existence of Dirac’s δdistribution. Finding an exception to the limit would
falsify the strand tangle model. No observation contradicts the limit for probability density, the
corresponding limit on charge density and the equivalent limit for the amplitude of wave functions.
The Planck limits for wave function amplitude and probability density have not been discussed
in the research literature. The main reason for this is that the limits are extremely large, and,
as argued in Section 2, they cannot be achieved nor approached in any experiment. There is no
corresponding Planck limit for the phase or phase difference. A phase angle is a length ratio, and
length ratios are not limited by Planck limits.
In short, strands imply that the probability density is limited by the inverse of the minimum
volume in nature. Strands further imply that this limit cannot be approached in any experiment.
This result concludes Part II of this article, which has shown that crossing densities of particle
tangles are wave functions and visualize interference, entanglement, decoherence, and measure-
ments. In simple words, wave functions are blurred tangles, and tangles are fluctuating skeletons
of wave functions. This result allows exploring the evolution over time.
Part III: Dynamics of states
This part of the article explores the time dependence of crossing densities, i.e., of wave functions.
The basic idea follows naturally from the exploration so far:
BA moving particle is described as a fluctuating, spinning and advancing tangle core.
The tangle model combines the descriptions of a particle as a rotating phase, rotating arrow, or
rotating flagpole developed by Schrödinger [205], Feynman [158], Hestenes [206–208], Penrose
[2] and Steane [209,210].
Why do advancing particle tangles rotate? This is the only situation in this article in which
it is necessary to recall that the vacuum is also made of strands. When the vacuum strands get
in contact with the asymmetric, chiral tangle core, they lead to a rotation of the core around the
particle path, similar to a (tethered) propeller moving through a fluid. A moving propeller in a
fluid rotates because its shape is chiral. Because the tangle core of a fermion is chiral as well, the
fermion core and the attached phase arrow rotate when moving through the vacuum. The rotation
speed of a propeller advancing through a fluid depends on the geometrical shape of the propeller,
in particular on the angles of its fins. In the strand tangle model, the rotation speed of a core
depends on its average shape, which in turn depends on its topological structure.
In short, in the strand tangle model, a quantum particle moving through vacuum behaves like
a tethered propeller moving through a fluid. Dirac’s belt trick couples tangle core propagation and
rotation. The continuously rotating crossings of a tangle behave like a spinning propeller.

47
Motion of particles in the strand model and in observations
t1t2t1t2
Rapidly moving particle
t1
t1
t2
t2
Slowly moving particle
rotation, precession,
diffusion and displac

t1t2
t2
 
erage
of crossing
switches
Observed
probability
density over time:
Particle at rest
rotation,
precession,
diffusion
rotation
, precession,
diffusion and displac

Fig. 23: Tangles of free fermions at rest (top), in slow motion (middle) and in fast motion (bottom) are
shown. No vacuum strands are shown even though they are responsible for the spinning. The relativistic
flattening of the fermion core is also illustrated, as is the increasing alignment of the spin axis with
increasing momentum.

48
21 The Schrödinger equation emerges from tangles
The fundamental principle illustrated in Figure 7defines the quantum of action ~with a crossing
switch. As shown now, the fundamental principle also explains how the crossing density of a free
particle tangle evolves in time.
Experiments show that free fermions propagate, spin and diffuse. For the strand tangle model, a
qualitative visualization of the motion of a free fermion is given in Figure 23. In the tangle model,
a localized fermion with constant speed is described by a localized tangle core that advances.
While advancing, the tangle core rotates, i.e., spins, and precesses. Figure 2already showed that
Dirac’s trick couples the spinning of the core and its displacement. The spinning and precession
are due to the deforming and moving of the tethers around the advancing particle core in Dirac’s
trick. On average, after every second core rotation, the fluctuations lead to a rearrangement of the
tethers. The probability of this process will depend on the complexity of the core, the average
shape of the core, and its number of tethers.
While advancing, the fluctuations will also spread out the tangle core. The spreading yields the
diffusion of the probability density. The diffusion of a tangle, like the spinning, is a consequence
of the impenetrability of the strands: the various fluctuating strand segments continuously push
against each other, thus spreading out the core over time.
Every tangle core rotation leads to crossing switches. The fundamental principle states that
each crossing switch yields a quantum of action ~. A rapid core rotation results in many crossing
switches per time, whereas a slow rotation results in a few crossing switches per time. The quantity
defined by action per time of a free particle is commonly called (kinetic) energy. In other terms,
BParticles with high (kinetic) energy have rapidly rotating tangle cores; particles with
low energy have slowly rotating tangles.
Using the fundamental principle that relates crossing switches to ~, the kinetic energy Eof a
tangle is automatically related to the angular frequency ωof its core rotation by the relation
E=~ω . (13)
In other words, the local phase of the wave function ψfor a moving core rotates. This implies
ω=i∂t,(14)
a relation that will be used shortly. The change of phase with time needs to be combined with the
change of phase over the distance covered when the core is advancing and rotating.
BRapidly moving tangles show many crossing switches per distance; slowly moving
tangles show few crossing switches per distance.
The fundamental principle implies that the natural observable in this case is action per distance:
BThe momentum of a moving tangle is the number of crossing switches per distance.
Due to the fundamental principle, the momentum pof a tangle is related to the wave number
k= 2π/λ of the core rotation by
p=~k . (15)

49
This implies
k=−i∂x.(16)
The imaginary unit idescribes the rotation of the core phase during propagation. This second
relation completes the description of the phase of propagating wave functions.
An advancing particle tangle continuously performs a Dirac trick after the other. The greater
the particle momentum, the more its rotation axis must align with its direction of motion. This
effect is illustrated in Figure 23. The helix followed by the endpoint of the tangle phase depends on
the momentum and energy of the particle. Given the speed vof the particle, both momentum and
energy allow defining the inertial particle mass m. The inertial mass thus describes the tightness
of the phase helix. The value mdepends only on the details of the tangle core structure, i.e., on
the topology and the average shape of the core. For non-relativistic motion, the average core shape
is spherical. In contrast, in relativity, the fluctuating core is deformed into an ellipsoidal shape.
Figure 23 illustrates both cases.
In the non-relativistic case, one has
E=p2
2mand ω=~
2mk2.(17)
This dispersion relation reflects how the frequency and wave number increase with increasing
speed vof the particle tangle. Schrödinger’s argument from 1926 can be repeated [205]. Substi-
tuting the differential relations into the dispersion relation, the evolution equation for the crossing
density of wave function ψis
i~∂tψ=−~2
2m∂xxψ . (18)
This is Schrödinger’s equation for a free particle wave function in one spatial dimension. The free
Schrödinger equation describes how a free fermion tangle advances, how its phase rotates, and
how its crossing density and crossing switch density diffuses.
In short, in the strand tangle model, the tangle core of a matter particle is localized. The belt
trick implies that core displacement and core rotation are related, thus allowing the definition of
an inertial mass value. As a further consequence, the wave function, or crossing density, of a free
non-relativistic spin-less particle obeys the free Schrödinger equation. In contrast to usual quantum
mechanics, the mass value is not a free parameter but uniquely determined by the tangle structure.
Several tasks remain: confirming the indeterminacy relation for crossing densities, including the
speed limit c, including fermion spin, including gauge interactions, and calculating tangle mass
values.
22 Indeterminacy is a consequence of the fundamental principle
The Schrödinger equation, being a wave equation, implies Heisenberg’s indeterminacy relation,
which is also called the uncertainty relation. It has been confirmed in every experiment to date.
Also in the strand tangle model, the indeterminacy relation can be deduced from the Schrödinger
equation. In addition, the indeterminacy relation is a direct consequence of the fundamental prin-
ciple illustrated in Figure 7. The fundamental principle implies that the smallest indeterminacy of

50
every action measurement is half a crossing switch. When a strand configuration corresponds to
the middle case of Figure 7, it is unclear which of the two outer configurations it belongs to. The
two outer configurations are separated by ~. Therefore, for any tangle configuration,
∆W6~/2and thus ∆x∆p6~/2.(19)
Due to strands, the indeterminacy product for any two observables whose product is an action and
whose operators do not commute is given by half a crossing switch, and thus half a quantum of
action.
In short, in the strand tangle model, the fundamental principle implies that quantum particles
obey the indeterminacy relation.
23 The Klein-Gordon equation emerges from tangles
In 1980, Battey-Pratt and Racey [94] showed that tethered, relativistic and spin-less particles are
described by the Klein-Gordon equation. Their approach can be used in the strand tangle model.
The spin-less case effectively assumes a spin orientation constant in time and space. The only
difference to the derivation of the Schrödinger equation is the relativistic behaviour. Including the
speed limit cyields time dilation. Time dilation, combined with the belt trick, changes (half) the
core spinning frequency ωas a function of the core speed v:
∇2ψ=ω2v2
c4(1 −v2/c2)ψ . (20)
Inserting the relativistic expression for the observed angular rotation frequency of the tangle phase
ω/p1−v2/c2(the core rotates with twice that frequency) yields
∇2ψ−1
c2∂ttψ=ω2
c2ψ=m2c2
~2ψ , (21)
where mass mis again introduced as a constant that relates the translation speed and the angu-
lar frequency of the belt trick. This is the well-known Klein-Gordon equation. In other words,
Battey-Pratt and Racey showed that the Klein-Gordon equation follows from Figure 23. The re-
sult confirms that in the strand tangle model, the core spinning frequency ω=mc2/~due to the
belt trick reproduces what Schrödinger called the Zitterbewegung.
The differences between the ideas of Battey-Pratt and Racey and the strand tangle model are
small. First, the definition of the quantum phase with crossings explains that the phase varies all
over three-dimensional space, and is not defined only at the location of the rotating core. Secondly,
the crossing density also yields an amplitude that is a function of position. In this way, the strand
tangle model solves the issues mentioned by Battey-Pratt and Racey in their paper [94]. In other
words, the crossing density of a fluctuating strand tangle yields a complete description of the state
of a quantum particle with a wave function ψ(x, t)that fully behaves as expected from special
relativity.
In short, the strand tangle model reproduces the result by Battey-Pratt and Racey: for relativis-
tic spin-less matter particles, tethers imply the Klein-Gordon equation when wave functions are
defined by crossing densities. The next task is to add spin.

51
24 Pauli spinors and the Pauli equation are due to tangles
Any model of wave functions must include spin and the variation of its orientation over space and
time. The results about tangles derived so far can be used to realize this requirement. This section
treats the non-relativistic case.
To extend the wave function by including the orientation of the rotation axis, it is most practical
to use the Euler angles α,βand γ[156,157]. They allow describing the crossing density as
Ψ(x, t) = √ρeiα/2 cos(β/2)eiγ /2
isin(β/2)e−iγ /2!.(22)
As in the spin-less case, the crossing density is the square root of the probability density ρ(x, t).
Again, the angle α(x, t)/2describes the phase, i.e., half the rotation around the axis. The newly
added orientation of the spin axis is described by a two-component matrix that uses the two angles
β(x, t)and γ(x, t)shown in Figure 6and Figure 11. Due to the half angles, the two-component
matrix is a spinor, as Ehrenfest named it, in analogy to the terms ‘vector’ and ‘tensor’. For the
case that β=γ= 0 at all times and all positions, the wave function ψused in the Schrödinger
equation is recovered.
As described by Payne [211], by Penrose and Rindler [2], and by Steane [209], a Pauli spinor
can be visualized as a flag on a pole and a sign. The length of the flagpole is described by a positive
real number √ρ. It is the amplitude of the wave function. The direction of the flag around the pole
is described by the first angle α. The flag angle αis the phase of the wave function; it corresponds
to the rotating arrow used by Feynman in his book QED [158]. These two parameters are the same
as those for the spin-less case. The spin orientation, i.e., the orientation of the flagpole is described
by two new, additional angles βand γ. The sign of the spinor indicates whether the flag has been
rotated by an even or uneven multiple of 2π.
The strand tangle model for a fermion naturally reproduces spinors and their flag visualization.
As before, it is assumed that the tangle core rotates rigidly. As illustrated in Figures 6and 11,
every fluctuating tangle core has a local crossing density, a local average phase, and a local average
orientation of the crossing axis. The crossing density of the tangle core – the ‘size’ or ‘density’ of
the core – reproduces the positive real number √ρ. The orientation of the tangle core – the flag –
is described by three angles, as if the flagpole were glued into the tangle core. Because of the belt
trick, the expression for the axis orientation naturally contains half angles. The twistedness of the
tethers reproduces the sign of the spinor.
The final required ingredient is the usual description of the orientation of particle spin during
particle motion [212]. For a propagating wave function, the spin orientation is described by the
wave vector k=−i∇multiplied by the spin operator σ. Here, the spin operator σ, for a spin
1/2matter particle, is defined as the vector built from the three specific matrices
σ= 0 1
1 0!, 0−i
i0!, 1 0
0−1! ! .(23)
These three matrices are called the Pauli matrices. They are deduced from strands below, in
Figure 32. The description of the spinning motion and the spin axis can be inserted into the non-

52
relativistic dispersion relation ~ω=E=p2/2m=~2k2/2m. This yields the wave equation
i~∂tΨ = −~2
2m(σ∇)2Ψ.(24)
This is Pauli’s equation for the evolution of a free, non-relativistic quantum particle with spin 1/2.
Anticipating the inclusion of electrodynamics presented below, it is well-known how to include
the electric and magnetic potentials in the Pauli equation [212]. This step makes use of the minimal
coupling to the electromagnetic field, also deduced below. Minimal coupling implies substituting
i~∂twith i~∂t−qV and substituting −i~∇with −i~∇−qA. This substitution introduces the
electric charge qand electromagnetic potentials Vand A. A bit of algebra involving the spin
operator then leads to the Pauli equation for a charged particle
(i~∂t−qV ) Ψ = 1
2m(−i~∇−qA)2Ψ−q~
2mσBΨ,(25)
where the magnetic field B=∇×Aappears explicitly. The equation is famous for describing
the motion of non-relativistic silver atoms, which have spin 1/2, in the Stern-Gerlach experi-
ment. In that experiment, an inhomogeneous magnetic field splits a beam of silver atoms into two
beams. This effect is due to the new, last term on the right-hand side, which does not appear in the
Schrödinger equation. Depending on the spin orientation, the sign of the last term is either positive
or negative. Thus, it acts as a spin-dependent potential. The two options produce the upper and
lower beams of silver atoms observed in the Stern-Gerlach experiment. The spin term implies a
g-factor of 2 that is due to the half angle in Dirac’s trick.
In short, a non-relativistic tangle core that rotates continuously and rigidly visualizes a spinor
and its flag model. Such a tangle also reproduces the Pauli equation for non-relativistic particles
with spin 1/2and a g-factor of 2. The next step is the exploration of the relativistic case.
25 Rational 3d tangles describe antiparticles
The explanatory power of the strand tangle model is impressively confirmed for the case of antipar-
ticles. In the tangle model, antiparticles are mirror tangles that rotate in the opposite direction.
Because particle tangles are also chiral, the strand tangle model correctly models the opposite
handedness of particles and antiparticles, their opposite charge, and their equal mass.
Figure 24 illustrates antiparticles using the rational 3d tangles for the electron and the positron.
Their tangles will be introduced later on, in Section 31. All quantum numbers are reproduced.
The figure allows deducing that the two tangles can be continuously transformed into each other by
moving and rearranging the tethers in space. This possibility explains how rational, i.e., unknotted
tangles can reproduce Dirac spinors, for which the particle-antiparticle content can vary from one
position to the next. Such a continuously varying anti-particle content is observed in experiments
and is an essential aspect of the Dirac equation. For example, the transformation of particles into
antiparticles is at the basis of Klein’s paradox [213]. The ability to reproduce this transformation
is a further indication that the tangle model agrees with nature.
Acontinuous transformation between an electron and a positron can only be reproduced using
rational 3d tangles. As explained in Figure 5, a tangle is called rational if it is unknotted and if
it arises purely through the braiding of the tethers. Topologically, the transition between particles

53
 !"# !"$% &(#% )
or an electron
*,&-% !" $%.
,& !/
erage
of crossing
switches
phase
spin
 !"# !"$% &(#% )
or a positron
*,&-% !"$%.
-(p!p,%,0
density
Observation
Positron
Q 1 23
-(p!p,%,0
density
Observation
spin
a456
e
a456
e
paper
plane
a456
e
4b7
5
w
4b7
5
w
4b7
5
w paper
plane
phase
Electron
Q =
83
phase
a456
e
a456
e
paper
plane
a456
e
4b7
5
w
4b7
5
w
4b7
5
w paper
plane
phase
spin
spin
transf
(& ," 
o
!9 (  p
y
t
 p !,#,"$
,& !/
erage
of crossing
switches
Fig. 24: The simplest tangle for an electron (left) can be continuously deformed into the simplest tangle
for a positron (right). One way to proceed is to untwist two neighbouring tethers, shift the third strand, and
properly twist the other two tethers of the first two strands. Another way to proceed is to flatten all tethers
into one plane, rotate three tethers together against the other three, and bend all six tethers out of the paper
plane again.
and antiparticles is a change in tether braiding. Open knots, prime tangles, and all other knotted
tangles do not allow a smooth transition between particles and antiparticles. Only rational 3d
tangles, the simplest of all tangles, are natural candidates for describing antiparticles.
In short, only rational 3d tangles allow visualizing the continuous transition between particles
and antiparticles. The transition makes use of tether deformations and mirror tangles.
26 Tangles lead to Dirac spinors and the free Dirac equation
The free Dirac equation, describing spin and relativistic effects, follows from strand tangles. This
section summarizes a published quantitative argument and adds two qualitative arguments.
The first quantitative argument is due to the work by Battey-Pratt and Racey from 1980 [94].

54
They showed that for the relativistic motion of a tethered spinning particle, the average size of the
belt trick is contracted by the Lorentz factor. When the Lorentz contraction is taken into account,
the Dirac equation appears.
The strand tangle model uses the approach of Battey-Pratt and Racey, with the only difference
that the tethered quantum particle is also made of strands. Thus, a particle is a tangle. Each particle
tangle defines a complex 4-component Dirac spinor ψ(x)as follows:
BAveraged over a few Planck times, the density of the strand crossings defines the am-
plitude of the wave function, and its square the probability density.
BAveraged over a few Planck times, the position of the centre of the core yields the
maximum of the probability density.
BAveraged over a few Planck times, the orientation of the core yields the spin orientation.
BAt each position x, the upper two components of the spinor ψ(x)are defined by the
local average of finding, at that position, the (tight) tangle, with a given orientation and
phase. These components determine the orientation of the flag of the particle spinor.
BAt each position x, the lower two components of the spinor ψ(x)are defined by the
local average of finding, at that position, the (tight) mirror tangle, i.e., the antiparticle,
with a given orientation and phase. These components determine the orientation of the
flag of the antiparticle spinor.
In the strand tangle model, a free fermion advancing through space is described by a constantly
rotating rational 3d tangle core moving through space along a straight line. Particles are teth-
ered tangle cores; as such, they behave as assumed by Battey-Pratt and Racey [94]. Using their
argument, rational 3d tangles of strands follow the Dirac equation.
Using the fundamental principle of the strand tangle model, the conclusion of Battey-Pratt and
Racey [94] can be rephrased in the following concise way:
BThe free Dirac equation is the differential version of Dirac’s trick on a fluctuating ra-
tional 3d tangle.
A more precise formulation is as follows:
BThe belt trick implies the γµmatrices and their Clifford algebra, i.e., their geometric
algebra properties [206–208].
BThe first two components of the γµmatrices describe the rational 3d tangle core, i.e.,
the particle, whereas the last two components of the γµmatrices describe the rational
mirror tangle, i.e., the antiparticle.
The gamma matrices arise from tethers. This is best seen by comparing their Dirac representation
with the Pauli representation of SU(2). Both sets of matrices are due to strands. (Reference [114]
shows this for SU(2).) It is worth noting that strands imply, together with Part IV of this article,
that generalizing Clifford algebras or adding more dimensions does not help in understanding
particle physics.
As mentioned, rotating advancing tangles reproduce Feynman’s description of a quantum par-
ticle as an advancing and rotating arrow [158] once the tangle is imagined to have a negligible core
size. Likewise, a rotating tangle visualizes the description given by Hestenes [206–208] of the free
Dirac equation. A rotating tangle also visualizes the flag description of spinors [2,209,210] if one
idealizes the chiral tangle core as a flag.

55
A second, qualitative argument helps to understand the appearance of the free Dirac equation
from strands. The free Dirac equation
i~γµ∂µψ=mcψ (26)
is due to five basic properties of nature:
1. The action limit given by ~, which yields wave functions ψ.
2. The speed limit for massive particles is given by c, which yields Lorentz transformations
and Lorentz invariance.
3. The spin 1/2properties in Minkowski space-time.
4. Particle–antiparticle symmetry, where this and the previous point are described by Dirac’s
γµmatrices.
5. A particle mass value mthat connects phase rotation frequency and wavelength using the
imaginary unit i.
These five properties are necessary and sufficient to yield the free Dirac equation. (The connection
between the γµmatrices and the geometry of spin was first worked out about a century ago by Fock
and Iwanenko [214].) The strand tangle model reproduces these five properties:
1. All physical observables are due to crossing switches, which imply a minimum observable
action ~and thus the existence of wave functions.
2. Tethers constrain the tangle cores to advance less than one Planck length per Planck time,
which is slower than c(see Figure 2).
3. Tethers reproduce the spin 1/2properties for rotation, exchange and boosts, including
fermion behaviour.
4. Tethers yield the γµmatrices [94], with tangles and mirror tangles corresponding to parti-
cles and antiparticles.
5. Through the belt trick, tethers connect tangle core rotation and tangle core displacement
and lead to a finite mass value m.
In other words, both in nature and in the strand tangle model, the inability to observe action
values below ~leads to wave functions and probability densities. Both in nature and in the strand
tangle model, the inability to observe speed values larger than cleads to Lorentz invariance and the
relativistic energy-momentum relation. Both in nature and in the strand tangle model, a finite mass
value, the spin 1/2properties, and the γµmatrices arise. This implies the Dirac equation for free
particles. For conventional quantum theory, this argument was made by Simulik [215–217]. In the
strand tangle model, all of these properties are due to the tethers and the fundamental principle.
The strand tangle model also explains quantum motion in a third, qualitative way. In nature,
all motion, also quantum motion, can be described with the principle of least action: motion mini-
mizes action. The Dirac Lagrangian specifies how to determine and how to minimize the value of
the action of a relativistic fermion. In the strand tangle model, action denotes the number of cross-
ing switches. The principle of least action then becomes the principle of fewest crossing switches.
In the strand tangle model, motion indeed minimizes crossing switches: strand deformations with

56
few crossing switches are simpler and preferred. After spatial averaging, crossing switch mini-
mization for spinning and advancing fermion tangles leads to the free Dirac Lagrangian and the
free Dirac equation. Still, it is an open challenge to deduce Schwinger’s quantum action principle
directly from the strand tangle model.
In short, the ideas of Battey-Pratt and Racey imply that rational 3d tangles represent relativistic
fermions and follow the free Dirac equation. The simplicity of the fundamental principle and the
principle of least action confirm the conclusion.
27 A second quantitative derivation of the Dirac equation
Over the last decades, several derivations of the Dirac equation have been published. They can be
reproduced with the help of strands. To see this, it is useful to rewrite a general Dirac spinor, with
its four complex components, in the way given by Loinger and Sparzani [218], as
Ψ = √ρeiδ L(v)R(α/2, β/2, γ /2) .(27)
Here, √ρis the amplitude, δis the fraction of particle and antiparticle. Ris a matrix with three real
parameters describing the orientation and phase of the spin (or flag), and Lis a matrix with three
real parameters describing the boost transformation. All quantities depend on space and time.
Most parts of a Dirac spinor wave function can be visualized using a relativistic spinning top. In
particular, Loinger showed in the 1960s that the amplitude and six further parameters follow from
spinning tops [218]. Only δis not reproduced.
Once Dirac spinors are defined, the simplest derivation of the Dirac equation might well be the
one given by Lerner in 1996 [219]. His derivation is based on only two conditions: conservation
of the spin current and Lorentz covariance. Lerner showed that together, these two conditions
completely and uniquely imply the Dirac equation.
In the strand tangle model, a Dirac spinor is described by the geometry of its rotating tangle
core, which can be imagined, because the tethers are unobservable, as a spinning top. The geom-
etry of a spinning top suggests describing a Dirac spinor using the following quantities derived
from a spinning tangle core: one strand density, three angles specifying the rotation or spin axis
and the phase of the core (its flag direction), three parameters that describe the contracted shape
of the tangle core and thus specify the boost direction and magnitude, and one angle that specifies
the relative weight of particle and antiparticle.
For the strand tangle model, density, rotation, and (flag) orientation of the tangle core were
defined and visualized in Part II using crossing densities, in particular in Figure 11. In the rela-
tivistic case, the spherical core is changed to an ellipsoid, as illustrated in Figure 23. The matrix
Ljust mentioned thus describes and yields the ellipsoidal shape of a tangle core boosted in a gen-
eral direction. The relation between rotation frequency and displacement reproduces the mass of
the particle. In addition to the visualization that uses spinning tops, a spinning rational strand
tangle also explains the last phase δ. This phase describes the relative fraction of particle and anti-
particle, i.e., of core and mirror core. The two extreme cases for the phase value are visualized
in Figure 24. Thus, the geometry of strand tangles and their crossing densities yields eight real
parameters that correspond to the eight real parameters that describe Dirac spinors. These eight
real parameters describing tangle cores correspond to the four complex numbers used by Dirac

57
when he wrote down his equation [218,220,221].
Thus, because free quantum particles behave as tethered spinning tops, spinning rational 3d
tangles reproduce Dirac spinors. In other words,
BFree fermions are rotating tangle cores.
Equivalently, a free fermion (and antifermion) can be imagined as a spinning flag (pair).
In the strand tangle model, an advancing particle in a vacuum is visualized either as a tethered
propeller or, more precisely, as a tethered spinning top. The definition of spin with tethers implies
that particles move in a vacuum at a constant speed, that is, tangle cores move at a constant speed
and keep rotating with a constant rotation frequency. In other words, strand tangles imply that spin
current is conserved over space and time. (Indeed, there is no friction or any other mechanism in
the tangle model that can destroy spin current conservation. The strands that make up the vacuum
yield a constant effect on free particle tangles: their rotation speed stays constant.) Thus, the first
condition used by Lerner is fulfilled by tangles. In addition, the definition of spin using tethers
implies the Lorentz covariance of spin, i.e., the proper behaviour under rotations and boosts. This
property was already confirmed by Battey-Pratt and Racey [94] in 1980. In particular, Lorentz
covariance arises because under boosts, tangle cores change shape (in a way that resembles the
change of a sphere to an ellipsoid), and the belt trick therefore changes accordingly. In total, both
conditions used by Lerner are reproduced by the strand tangle model. Therefore, strand tangles
follow the free Dirac equation.
The quantitative derivation of the Dirac equation just given, like the three arguments in the
previous section, implies:
Test 4: Strands predict the lack of measurable deviations from the free Dirac equation.
Deviations from the free Dirac equation have been searched in detail. So far, for all measurable
energies and length scales, none has been found, not even in thought experiments [122,197–199].
In short, using the result of Lerner, the conservation of spin current by the belt trick, together
with the relativistic invariance, implies the free Dirac equation. Dirac’s equation is due to Dirac’s
trick. As a consequence, the strand tangle model with its automatic appearance of antiparticles
allows visualizing all relativistic quantum effects. They include the Zitterbewegung, which is due
to core rotation. Thus, the strand tangle model agrees with the free Dirac equation and with all
measurements so far. Several additional predictions arise.
28 Strands imply the Dirac equation despite the lack of trans-Planckian effects
The derivation of the free Dirac equation from tangles of strands implies the lack of any observable
deviation from relativistic quantum theory.
Test 5: Strands predict that the Dirac Lagrangian for free particles is valid at all measurable
energy scales.
As just mentioned, no deviations have been observed. As mentioned below, the same applies if
gauge interactions and their quantum properties are included. In the case of electromagnetism, this
prediction includes Klein’s paradox. In apparent contrast, the strand tangle model also predicts the
absence of any trans-Planckian effect in nature.

58
Test 6: Strands imply that no elementary particle with energy beyond or equal to the corrected
Planck energy p~c5/4G≈6·1018 GeV/c2≈1 GJ will be observed. A similar upper
limit also applies to the linear momentum and the mass of every elementary particle.
Test 7: Strands imply that no system or particle will ever exceed or achieve the limits for fre-
quency, acceleration, density, pressure, area, volume, temperature, length or time given
by the corrected Planck limits of relativistic quantum gravity.
As explored in Section 1, in contrast to the everyday Planck limits, no limit of relativistic quan-
tum gravity has ever been approached by less than three orders of magnitude. Therefore, the
Planck limits due to relativistic quantum gravity are not in contrast with the Dirac equation in any
observation, even though the corrected Planck energy poses a limit to boosts.
In short, strands predict both the lack of measurable deviations from the Dirac equation and
the lack of trans-Planckian effects. These predictions agree with all observations, despite their
apparent contradiction, because the Planck limits cannot be achieved.
29 Strands predict the lack of other models for wave functions
Strands imply that wave functions are blurred tangle crossings and tangles are fluctuating skeletons
of wave functions. This statement was deduced and tested in the previous sections. However, step
by step, without noticing, strands made a stronger statement.
Test 8: The strand tangle model predicts that no alternative, inequivalent emergent model of
wave functions that also describes gauge interactions is possible. If any inequivalent
alternative would be found, the tangle model would be falsified.
The prediction results from the explanation of spin 1/2, of fermion behaviour, and of the free Dirac
equation. Describing the motion of the space between strands, as proposed by Asselmeyer-Maluga
[85,86], would be an equivalent microscopic model. Strands claim to be the unique model for
emergent wave functions. This uniqueness claim is phrased so provocatively for two reasons.
First, the uniqueness claim aims to motivate the search for a proof or a counterexample. On the
one hand, the lack of alternative descriptions of wave functions so far should not discourage future
searches. On the other hand, the uniqueness of strands was already implicit in Section 3, where
it was suggested that all of nature can be described by unobservable strands, observable crossing
switches, and Planck limits. The uniqueness claim implies the possibility to prove that each step
taken in this article is logically unavoidable, without any alternative. Some specific aspects of
strand uniqueness are discussed in Appendices D,E,F, and G.
Secondly, the uniqueness claim contains a precise experimental prediction.
Test 9: If any other model for wave functions differs in its consequences from the strand tangle
model, observations are predicted to falsify the other model.
Equivalently, there are no measurable deviations from the strand quantum theory. This prediction
aims to motivate precision experiments of every possible type.
The literature on emergent quantum theory and pre-quantum approaches is vast. The ap-
proaches by Adler [222,223], ’t Hooft [224], Elze [225], de la Peña et al. [226], Blasone et
al. [227], Grössing [228,229], Acosta et al. [133], Hollowood [230] and Torromé [231], and
those cited in Section 4lead to several conclusions. First, few approaches are specific enough
to be tested by observation. Secondly, several approaches are compatible with the strand tangle

59
model. Thirdly, no approach contradicts the uniqueness claim. These conclusions should regularly
be tested in future research.
In short, the strand tangle model claims to be the only possible model for emerging wave func-
tions and emerging quantum theory. However, despite the agreement of tangles with observations,
several differences from quantum theory arise. These differences are explored next.
Part IV: Differences to conventional quantum theory
Thus far, the strand tangle model deduced from the fundamental principle has simply reproduced
conventional quantum theory. If this were the only result of the strand tangle model, the model
would be unnecessary: Occam’s razor would speak against it. However, the strand tangle model
and quantum theory differ. The differences are due to the possibility of classifying both particle
tangles and their core deformations. Classification is possible because elementary particles are the
simplest possible tangles of strands:
BElementary particles are rational 3d tangles of one, two or three strands.
The concept of rational 3d tangle was introduced in Figure 5. Rational 3d tangles arise by braiding
tethers, i.e., by moving tethers around in space. Rational 3d tangles are three-dimensional general-
izations of braids. Therefore, particle tangles do not contain knotted regions but do contain tangled
cores. It will appear that modelling elementary particles as rational 3d tangles explains and fixes
their spectrum, spins, charges, quantum numbers, masses, and all other particle properties.
It will also turn out that the interactions of particle tangles can be classified. All observed
gauge interactions arise in this way. It will appear that rational 3d tangles derive and fix the gauge
interactions, their Lie groups, and their coupling constants.
The derivation of particles and interactions provided by strands is hard to vary, thus realizing
Deutsch’s requirement for a good scientific explanation [232]. Tangles of strands have an explana-
tory power that continuous wave functions lack. This explanatory power does not arise in any other
proposal in the research literature and provides the reason for exploring the tangle model.
In short, the strand tangle model promises to restrict quantum field theory to a specific set
of allowed elementary particles and to a specific set of allowed interactions. The strand tangle
model can be tested by comparing the allowed particle properties, including mass values, and the
allowed interaction properties, including the gauge groups, with experiments. This comparison is
performed in the following.
30 Particle mass is due to chiral tangles
According to experiment and quantum theory, when a particle advances, its quantum phase rotates.
BThe (inertial) mass mdescribes the coupling between translation and phase rotation.
Alarger mass value implies, for a given momentum or energy value, a slower translation.
Whenever a chiral body moves through a viscous fluid, it starts to rotate. For example, this
occurs when a chiral pebble falls through water or a maple seed falls through the air. This rotation
is due to the asymmetrical shape of the body [233–236]. Also a tethered chiral body – such as
a tethered propeller or a tethered spinning top – moving with a low Reynolds number through a

60
fluid will rotate, as long as the tethers are elastic and their motion is not disturbed. The motion of
a particle through the vacuum is similar, though it takes place without friction.
In the strand tangle model, particle translation and phase rotation are modelled by the transla-
tion and rotation of the tangle core. Strands imply that chiral tangle cores rotate when they move
through the vacuum. Almost all of the tangle cores of the elementary particles deduced below are
chiral. Chirality also applies to the d quark once the Higgs is taken into account. A similar process
applies to the Higgs boson itself. The combination of chirality and tethers yields the coupling be-
tween translation and rotation. Figure 2gives an impression of how core translation and rotation
are coupled. Thus, the strand tangle model implies
Test 10: Tethered, non-trivial, topologically chiral, rational 3d tangle cores, such as localized
cores composed of two, three, or more strands, are predicted to be massive particles. A
tangle is localized or localizable – or simply non-trivial – if the core collapses to a tight
tangle when the tethers are imagined to be ropes that are ‘pulled outwards’.
Test 11: The mass value for tangles made of a few strands is not arbitrary but is uniquely deter-
mined by the tangle structure and by the resulting average core shape.
Test 12: Elementary particle masses are thus calculable – when the tangle topology is known.
Test 13: The more complex the tangle core – for the same number of tethers – the slower the
translation per rotation, and thus the larger the predicted mass value.
Test 14: Unlocalized, or trivial tangles – which disappear if their ends are pulled outwards – are
predicted to be massless, even if they are geometrically chiral.
These predictions about particle mass sequences agree with the fermion tangles of Figure 25 and
with the boson tangles of Figure 34. Above all, these predictions agree with observations. In
particular, the predictions agree with the quark model and the observed mass sequences of mesons
and baryons [121]. The mass of neutrinos also arises.
In the strand tangle model, the highest possible energy value for an elementary particle, the
corrected Planck energy, is one crossing switch per corrected Planck time. The spontaneous belt
trick of a tangle core is less probable by far.
Test 15: The complex motion of the belt trick implies a low probability for the involved strand
deformations. Strands therefore predict that the mass mof elementary particles is much
smaller than the corrected Planck mass:
mp~c5/4G= 6.1·1027 eV/c2.(28)
Thus, the strand tangle model solves the unsolved mass hierarchy problem [237]. The mass in-
equality agrees with the maximon concept introduced long ago by Markov [238]. In addition,
Test 16: The product of Lorentz factor γand the elementary particle mass mis always small:
γm p~c5/4G= 6.1·1027 eV/c2.(29)
Both inequalities agree with observations, including those of high-energy cosmic rays [75].
In short, the strand tangle model explains the origin of particle mass and solves the hierarchy
problem. Mass is due to the chirality of tangle cores. Calculations of mass values require three
steps: tangles must be classified, the assignment of tangles to the known elementary particles must
be clarified, and a precise calculation method for the probability of the belt trick for each moving
tangle must be developed.

61
L
;<=
ons
> ?@BCD@F GHIJKLM NHOL P
f three strands along cordinate ax
LM R
only
MDNSKLMG UHNDKV NLNCLWMX
Y
arity
Y Z [\] ^
HWVPI IBNCLW ^ Z [\o_]
Spin S = 1/2
`c
HWJL d Z f\o_
ghijkl
> ?GLGWHcLOWHKF GHIJKLM NHOL P
f two strands with four t
LG
cLWM R
only
MDNSKLMG UHNDKV NLNCLWMX
`c
HWJL d Z [no_
q rhijk
suxy
e plane
tether in
paper plane
u
z{
x
w paper
plane
u
z{
x
w paper
plane
h rhijk
suxy
e
u
z{
x
w
u
z{
x
w
tether in
paper plane
l rhijk
suxy
e
u
z{
x
w
u
z{
x
w
c
rhijk
suxy
e
u
z{
x
w
u
z{
x
w
| }~

e


w


w
 rhijk
suxy
e
u
z{
x
w
u
z{
x
w
tether in
paper plane
;;
ctron
Q =
f\]
S = 1/2
suxy
e
suxy
e
paper
plane
suxy
e
uz{
x
w
u
z{
x
w
u
z{
x
w paper plane
;;=j ;h=j
Q =

, S = 1/2
u
z{
x
w
paper
plane
u
z{
x
w
u
z{
x
w
suxy
e paper plane
suxy
e
suxy
e
h
Q =
f\]
S = 1/2
suxy
e
suxy
e
paper
plane
suxy
e
u
z{
x
w
uz{
x
w
u
z{
x
w paper plane
h ;h=j
d Z ]
S = 1/2
uz{
x
w
paper
plane
uz{
x
w
u
z{
x
w
suxy
e
suxy
e
suxy
e paper plane
=ih
Q =
f\]
S = 1/2
suxy
e
suxy
e
paper
plane
suxy
e
uz{
x
w
u
z{
x
w
u
z{
x
w paper plane
 
  
S = 1/2

w
paper
plane

w

w
¡¢
e
¡¢
e
¡¢
e paper plane
Fig. 25: The figure shows the simplest conjectured tangle for each elementary fermion. Elementary
fermions are rational 3d tangles that are formed by braiding tethers. All elementary fermion tangles
consist of two or three strands. The tangles yield spin 1/2and fermion behaviour. The cores are localized,
realize the belt trick, and thus yield positive mass values. The localized cores lead to additional, more
complex tangles in addition to those shown here. They are illustrated in Figure 26 and Figure 27 and
explain the three generations. At large distances from the tangle core, the four quark tethers follow the
axes of a tetrahedron and the six lepton tethers follow the coordinate axes. The tangles of the quarks, the
electron, muon, and tau are topologically chiral and thus electrically charged. Neutrino cores are twisted
strand triplets: they are geometrically chiral, but not topologically chiral; thus, they are electrically neutral.
No additional elementary fermions appear.

62
£ ¤¥¦§¨
©ª«¬®¯ª°
°±²³®¯´
The origin o
µ ¶·¸ ¹ ¤¥¦§¨
generations
c
¤¥¦§¨
©ª«¬®¯ª°
°±²³®¯´
º¤¥¦§¨
©ª«¬®¯ª°
°±²³®¯´
»¤¥¦§¨
©ª«¬®¯ª°
tangle, with
²
¼ ½
«³³ª´
¶ ¤¥¦§¨
©ª«¬®¯ª°
°±²³®¯´
¥ ¤¥¦§¨
©ª«¬®¯ª°
°±²³®¯´
one additional
crossing
»¤¥¦§¨
with
one
½
«³³ª ¾¼
ª¼²
©ª¯
c
¼²¿ ª«¬®¯ª°
°±²³®¯´
¥ ¤¥¦§¨
with
one
½«³³ª ¾¼
ª¼²
©ª¯
c
¼²¿ ª«¬®¯ª°
°±²³®¯´
one additional
crossing
one additional
crossing
one additional
crossing
one additional
crossing
one additional
crossing
one additional
crossing
»¤¥¦§¨
with
t
À
o
½
«³³ª ¾¼
ª¼²ª
©°Á«Â¿ ª«¬®¯ª°
°±²³®¯´
six additional
crossings
d quark
tangle
ñ¬«®Ä
s quark
tangle
ñ¬«®Ä
u quark
tangle
ñ¬«®Ä
c quark
tangle
ñ¬«®Ä
b quark
tangle
ñ¬«®Ä
t quark
tangle
ñ¬«®Ä
etc.
Thick
¿±ªÁ¯¿ ¯²¿Å
±¾¼
Æ
e paper plane.
Thin
¿±ªÁ¯¿ ¯²¿Å
¾¯®
¼
w paper plane.
No
¿±ªÁ¯¿ ¯²¿Å
in paper plane.
Fig. 26: Each quark is described by a tangle family with an infinite number of tangles of two strands. The
six families define the six quark types and the three generations. The same classification arises for
anti-quarks, which are represented by the respective mirror tangles; they are not shown here. In the strand
tangle model, the number of generations is thus related to the structure of the Higgs braid, which is itself a
result of the three dimensions of space. (Figure improved from reference [121].)

63
ÇÈÉ ÊËÌÍÌÎ ÊÏ ÐÈÉ
Ñ ÒÉÓÐÊÎ ÍÉÎÉËÔÐÌÊÎÕ
Ö×Ø Ù
its neutrino
ele
ÚÖÛÜÝ Ù
its neutrino
ÞØÜÝ Ù
its neutrino
ßà
Ñ
additional
twirl
ßà
Ñ
additional
twirl
ÉÒá ÎÉâÐËÌÎÊ
tangle
ã×Þäåæ
tau
ÎÉâÐËÌÎÊ
tangle
ã×Þäåæ
muon
ÎÉâÐËÌÎÊ
tangle
ã×Þäåæ
ÉÒÉ
ctron
tangle
ã×Þäåæ
muon
tangle
ã×Þäåæ
tau
tangle
ã×Þäåæ
ÒÉÓÐ
ons with
ÊÎÉ ç
äèèé êÜéÜÝ
ëéì
c
ÜÝí éäÞ îå
ìéÖ
Ö×Ýèå
ìéï
lef
Öð ÝìØÖÛäÝÜé
ÛäèñÖð Úñ×Ûèìí
leptons
ÊÎÉ
additional
êÛ×
äí
ÒÉÓÐ
ons with
t
ò
o
ç
äèèé êÜéÜÝé
ëÖñ
äÛí éäÞîå
ìéÖ
Ö×Ýè
å
ìéï
ÊÎÉ
additional
êÛ×
äí
etc.
Fig. 27: Each lepton is described by a tangle family with an infinite number of tangles of three strands.
The simplest tangles for each lepton are shown at the top of the figure. The tangles due to one added Higgs
boson are shown further down. The six families define the six lepton types and the three generations.
Anti-leptons are represented by the corresponding mirror tangles; they are not shown.

64
31 Classifying tangles leads to the spectrum of elementary fermions
This and the following sections summarize how classifying rational strand tangles leads to the
observed spectrum of elementary fermions and bosons, with their observed quantum numbers, as
told in references [118–120]. As shown above, only rational 3d tangles describe the appearance of
antiparticles and realize the Dirac equation. In the strand tangle model, the rational 3d tangles of
elementary particles can consist of only one, two or three strands. More than three strands form
composite particles, such as protons, or even more complex systems, such as atoms or people.
The composition statement can be deduced from the overview of elementary fermion tangles
in Figure 25 and of the elementary boson tangles in Figure 34. No simpler rational 3d tangle is
missing. Any more complex rational 3d tangle with an added strand, falls into one of the following
two cases:
The more complex tangle can be a tangle already found in the two figures. For
example, a photon tangle that is extended with an additional strand can yield, depending
on the details, a graviton tangle, a quark tangle, or an (unbroken) Witangle. Similarly,
a strand added to a quark tangle can lead to a lepton tangle. These cases thus lead from
one elementary particle to another.
The more complex tangle can be taken apart into simpler tangles. For example, this
occurs if a strand is added to a gluon tangle or a W tangle. The resulting four-stranded
core can decay into several tangle cores. The same occurs if a strand is added to a
lepton; such a four-strand tangle can also be separated into simpler cores.
As a result, all elementary tangles have at most three strands. Specifically, elementary fermions
consist of either two or three strands. One-stranded rational particle tangles can neither have spin
1/2nor have mass, because the belt trick does not apply to them, as argued at the beginning of
this article. It turns out that
BTwo-stranded fermions are quarks. Three-stranded fermions are leptons.
The simplest rational 3d tangles for elementary fermions are shown in Figure 25. Simpler rational
3d tangles made of two or three strands are not possible. One notes that the cores of the heavier
charged leptons can be seen as combinations of electron cores and W cores, as is expected from
their decay behaviour. Likewise, the cores of the heavier quarks can be seen as combinations of
lighter quark cores and W cores. The six tails of the leptons blend into the vacuum strands and
allow them to appear as free particles. The four tails of the quarks prevent them from arising
as free particles; they need to form composites to appear as free particles. This yields the quark
model of hadrons, which was presented elsewhere [121].
All massive elementary particles also have additional tangles, as illustrated in Figure 26 for
quarks and in Figure 27 for leptons. Each massive elementary particle is described by an infinite
family of tangles that contains the simplest possible core, the simplest core plus one Higgs braid,
the simplest core plus two Higgs braids, etc. Thus, the strand tangle model reproduces Yukawa
coupling as an important aspect of particle mass.
The structure of the Higgs braid limits both quarks and leptons to three generations. Figure 26
shows how the infinite class of quark-like tangles is split into six infinite families, corresponding
to three generations with two quarks each. In other words, the infinitely many possible quark and
antiquark tangles consist of 6 + 6 separate infinite tangle families. Each family is an infinite series

65
of tangles. In a family, each tangle differs from the next by one Higgs braid. A similar structure
of six tangle families arises for the leptons. They are illustrated in Figure 27.
An important check for the tangle–particle assignments are the resulting quantum numbers. In
nature, quantum numbers lead to conservation laws that restrict particle reactions.
Baryon and lepton number count the tangle cores with two or three strands. (Tether braid-
ing models certain non-perturbative effects, including baryon number non-conservation and
sphalerons, that arise in the early universe.)
Spin was already discussed in Part I. Because elementary fermions are composed of only two
or three strands, one obtains:
Test 17: The strand tangle model implies that all elementary fermions are massive.
Test 18: Strands implies that all elementary fermions have spin 1/2.
The first statement confirms that neutrinos are massive Dirac fermions. It agrees with all ex-
periments so far. Also the second statement, which eliminates supersymmetry, agrees with all
experiments so far. All observed fermions with higher spin values, such as various nuclei, are
composed.
Flavour quantum numbers count the tangles of a specific flavour (or family). Quark tangles au-
tomatically provide the correct values by counting the respective cores. Flavour change is achieved
by tether braiding. This process only occurs in the weak interaction, because only the weak inter-
action moves tethers against each other. In the strand tangle model, only the weak interaction can
braid and unbraid tethers. The strand tangle model thus reproduces the observation that only the
weak interaction can change quark flavours and lead to quark mixing. Similarly, the strand tangle
model reproduces the observation that only the weak interaction leads to neutrino mixing. The
similarity between the tangle of the electron neutrino and any section of the vacuum tangle has
important consequences for dark energy. This topic will be explored elsewhere.
Charge parity C is the topological chirality of a tangle core. This assignment reproduces all
the observed C parities of the elementary particles.
Electric charge is the number of crossings involved in the topological chirality of a tangle.
Each crossing yields a charge e/3or −e/3, depending on the sign of the crossing [120]. This
assignment reproduces all observed electric charge values of the elementary particles. Because
of its topological definition based on chirality, electric charge does not depend on tangle shape,
i.e., does not depend on the state of a particle; electric charge can have two signs; it is additive;
antiparticles have opposite charge; electric charge interacts in a preferred way with photons of one
(geometric) chirality or helicity; and electric charge is automatically conserved. There does not
seem to be any other way to define electric charge. Magnetic charge cannot arise.
Parity P describes the behaviour of a tangle under spatial reflections. This assignment re-
produces all the observed P parities of the elementary particles. (Parity violation by the weak
interaction is discussed below.)
Time reversal T changes the spin and the motion direction of a tangle. This assignment repro-
duces the observed behaviour under time reversal of all elementary particles. The CPT theorem is
automatically satisfied, as observed.
Weak and strong charge are also explained for each fermion and boson. They are explored
in references [119] and [121]. The charges explain the interactions to which the corresponding
particles are subjected and reproduce all observations.
In the strand model, quantum numbers are topological invariants. The topological basis for

66
quantum numbers is the general reason that they are integers. The strand definitions explain why
quantum numbers are either additive or multiplicative, and why they are conserved.
Using the tangle model of the strong interaction, the quark–tangle assignments in Figure 25
fully reproduce the quark model of hadrons, as shown in references [119] and [121]. The allowed
and forbidden meson quantum numbers are explained. The meson tangle structures also provide
natural and correct retrodictions of which mesons violate CP symmetry. The hadron tangles also
reproduce all meson and baryon mass sequences.
No other elementary fermions appear topologically possible.
Test 19: Strands predict the lack of contradictions between tangle properties and observed parti-
cle properties, such as forbidden values of quantum numbers, new quantum numbers or
unexpected non-conservation. For example, millicharged particles, leptoquarks, Majo-
rana particles, or weakly charged particles without mass are predicted not to exist.
Test 20: Strands predict the lack of any unknown energy scale in high-energy physics.
Test 21: Strands predict the lack of any substructure in elementary particles that differs from
tangles of strands. This includes the preons and ribbons discussed in Appendix D, the
superstrings discussed in Appendix E, Möbius bands, prequarks, knots, tori, loops, and
any other localised or extended substructure [102–105,111,112].
Test 22: Strands predict the lack of any new elementary fermion of any kind. This includes the
lack of sterile neutrinos, magnetic monopoles [239], and supersymmetric partners.
Any experimental counter-example would falsify the strand tangle model. The last prediction will
be extended to a full prediction about dark matter below.
It should be noted that the tangle model of fermions is compatible with several particle models
in the research literature that are deduced from general relativity with and without torsion. In
models such as those of Popławski [240] or of Burinskii [241,242], an elementary fermion is
assumed to be a Planck-scale torus. In the tangle model, Figure 24 shows that an electron can be
seen – once the unobservable tethers are ignored and only crossings are kept – as three spinning
crossings. In practice, this yields a structure similar to a fuzzy torus that is always larger than a
few Planck lengths and that can be as large as a Compton wavelength. The same properties are
deduced in the cited papers on torus models.
In short, in the strand tangle model, the classification of elementary fermion tangles leads to the
three generations of leptons and quarks. No additional elementary fermion appears possible. As a
consequence, the fermions due to tangles cancel the anomalies of the standard model, as required
[243,244]. All observed quantum numbers are due to topological properties of the particle tangles.
In contrast, the fundamental constants – mass values, mixing angles and coupling constants – are
due to geometric properties, namely to the (average) shape of tangles. Without exception, the
statements deduced from the strand tangle model agree with observations.
32 Classifying tangle deformations leads to interactions and gauge groups
So far, only free particles have been described with tangles. Free particle motion and free La-
grangians arise when tangle cores rotate and move as rigid structures. In free particle motion,
only the tethers fluctuate by continuously changing their shape. The next step is to explore the in-
teractions of particles and to determine their properties. The present section provides a summary
of previous papers on the strand tangle explanation for the origin of the gauge groups, the origin

67
óôõö÷øùúûüýõ÷þö
interaction is twist transfer
f
õøúþùý
photon
ÿ
eak interaction is poke transfer
f
õøúþùý
weak
b
ù

ùý
Strong interaction is slide transfer
f
õøúþùý
gluon
v
ûö

ú
v
ûö

ú
v
ûö

ú
R
õþ
e
õúõþ

÷õø úù
v
õ

or twist
R
õþ
e
õúõþ

÷õø úù
v
õ

or poke
R
õþ
e
õúõþ

÷õø úù
v
õ

or slide
÷
t
þ

÷
 
û
v
e
one generator
that generates
U
pokes have
3 generators
that generate
SU
slides have
9 -  
generators
that generate
SU
Observation

þý

(
ûöõ

÷þúõ

photon
gluon
weak
b
ù

ùý
Observation

þý

(
ûöõ

÷þúõ

Observation

þý

(
ûöõ

÷þúõ

Fig. 28: In an interaction, a gauge boson rotates the bent strand segment enclosed by a dotted circle. The
observable deformations of tangle cores are classified by the three Reidemeister moves. The three
Reidemeister moves specify the generators of the three observed gauge interactions and determine the
generator algebras, as shown in the subsequent figures. The full gauge group arises when the rotations by
the angle πinduced by the generators are generalized to arbitrary angles. As a result, the three
Reidemeister moves determine the three gauge groups [114,118]. (Figure improved from reference [119].)
of quantum electrodynamics, the origin of quantum chromodynamics with the quark model, and
the origin of the standard model [114,115,118–121].
In line with the geometric effects of Hermitian operators, gauge interactions arise when the
tangle cores do not behave like rigid structures:
BGauge interactions are observable deformations of tangle cores.
This statement is of interest because all observable tangle core deformations are local and can be

68
K
eeping the encircled strand se
g 
xed, a double twist
c  g 
o a straight strand, thus
w
twist.
T 
, or
r


 

e,
generates a U(1) Lie group.
A 
g 
i!
"#$% g&g '
edom
  a )* + a $* , a $ '
or photons
The resulting model for the photon and



.
phase
motion
or
/
or
Fig. 29: The twist, the first Reidemeister move, generates the local Lie group U(1). When twists –
rotations of the region inside the dotted circle by π– are generalized to arbitrary angles, they yield all the
group elements of U(1). The twist also implies the strand tangle model for the photon.
classified. The classification is possible with the help of the moves published by Reidemeister in
1927 [144]. There are only three possible Reidemeister moves: twists,pokes, and slides. These
three moves are illustrated on the left side of Figure 28. Every observable tangle core deformation
at a given position is composed of the three Reidemeister moves. Exploring the three Reidemeister
moves yields a surprising result: the three moves generate the local gauge groups U(1), (unbroken)
SU(2), and SU(3) [114,115]. Each of the three Lie groups also corresponds to a local gauge
symmetry. In addition, each Reidemeister move generates the tangle for its gauge bosons.
Figure 28 shows how any approaching gauge boson core deforms a fermion core. The gauge
boson is absorbed and disappears, and the induced local fermion core deformation leads to a
local phase change in the fermion core. Exploring the details yields complete agreement with
experiments. In particular, modelling interactions as core deformations implies minimal coupling,
as the deformation depends on the density of arriving gauge bosons. Minimal coupling and local
gauge symmetry lead to the usual Lagrangians of QED, of QCD, of the weak interaction with its
symmetry breaking, and thus to the full Lagrangian of the standard model [118–121].
Test 23: Strands predict the lack of the tiniest deviation from minimal coupling, for any gauge
interaction, at any measurable energy or scale.
This prediction agrees with all observations [68]. Strands visualize the result.
Figure 29 shows that a twist, or first Reidemeister move, can be visualized as the local rotation
by πof a strand segment. In the figure, the segment is enclosed by a dotted circle that can be
imagined as a transparent plastic disc glued to it. Generalized twists can be seen as local rotations
by an arbitrary angle of the segment or disc. One notes directly that performing a double twist is
equivalent to no rotation. It is straightforward to check that the set of generalized twists also fulfils
all other axioms of the U(1) Lie group [114]. As a result, photons are permanently rotating twists.
The freedom in defining the twist phase yields local U(1) gauge symmetry. The twist model for
photons implies the photon propagator and the Lagrangian of the electromagnetic field.
Core twists reproduce quantum electrodynamics in all its details, as shown in reference [120].
Figure 30 and Figure 31 illustrate a selection of processes involving virtual particles. Perturbative
QED, including the running of the fine structure constant, appears automatically [120]. In the
limit of many photons, classical electrodynamics arises from strands. Electric fields are volume

69
t2
0123 45
erage
of crossing
switches
photon
electron
0123
F
3
y624
6 7
14
8:42
Fermion-antifermion annihilation
t1positron
electron
Q;<=
positron
Q;
>=
photon
photonphoton
positron electron
0123 45
erage
of crossing
switches
F
3
y624
6 7
14
8:42
Virtual particle-antiparticle pair
t3
photon
0123
t1
t2
photon
photon
photon
54?@@2
54?@@2
54?@@2
positron electron
Fig. 30: Two processes from quantum electrodynamics are shown in their strand representation. Top: the
appearance and disappearance of a virtual electron-positron pair. Bottom: the annihilation of an electron
and a positron. No difference to quantum electrodynamics occurs at any measurable energy.
densities of virtual photons, i.e., of bound twists. Magnetic fields are flow densities of (bound)
twists. In other words, photon exchange or twist exchange implies that the electromagnetic field
is defined by the space-time rotation rate it induces on an electric charge. This leads to minimal
coupling and visualizes the descriptions by Feynman [158], Hestenes [206–208], and Baylis [220,
221]. Electric charge results from topological tangle chirality.
Test 24: No massless, electrically charged elementary particle will be found.
Test 25: Not the slightest deviation from the local U(1) gauge group or from quantum electrody-
namics, at any measurable energy, will be observed.
Thus far, all observations have confirmed the strand tangle model of electromagnetism [120],

70
BCDEGHE IJHLMHG
photon
electron electron
NOPV
t2
t1
t
W
BCDEGHE IJHLMHG
photon
electron
t2
t1
t
W
positron
positron
electron
NOPV
t2
t1
t
W
electron
and photon
electron
positronelectron
positron
electron
electron
XYZ[\]^_ `_d fh]
tual photon
Int
Z]`[\h_j
ele
[\]^_ `_d k^lh\]^_
Fig. 31: Two further processes from quantum electrodynamics are shown in their strand representation.
Top: a free electron emitting and absorbing a virtual photon. Bottom: an electron and a positron interacting
via a virtual photon. No difference to quantum electrodynamics occurs at any measurable energy.
which fully reproduces quantum electrodynamics. For example, Furry’s theorem is automatically
realized by strands. First estimates of the fine structure constant agree with measurements [120].
In the tangle model, the gauge group SU(2) arises from pokes, that is, from the second Rei-
demeister move. Pokes can be seen as local rotations by the angle πof two strand segments, as
illustrated in Figure 32. Again, the three possible pokes are best visualized by imagining that
the dotted circle encloses a transparent plastic disc that contains, in this case, two parallel strand
segments. The plastic disc behaves like a belt buckle. Its three possible rotations by πform the
algebra of SU(2) [114,115,118]. Indeed, Figure 32 directly shows that the three pokes behave
under concatenation in the same way as itimes the Pauli spin matrices
τx=iσx=i 0 1
1 0!, τy=iσy=i 0−i
i0!, τz=iσz=i 1 0
0−1!(30)

71
The poke, or second R
mnompmnqsmu px
ve, on pairs of strands generates an SU(2)
Lie group, because the three rotations by generate the algebra of SU(2):
z
z
x
y
z
z
y
Fig. 32: The three types of pokes, the second Reidemeister moves, are visualized. Pokes can be modelled
by rotating by the angle πa region – inside the dotted circle – containing both strands. (Other
visualizations, deforming only one strand, are also possible.) The three pokes generate the Lie algebra of
SU(2) and imply a model for (unbroken) weak bosons. The Pauli matrices can be read off directly [114].
behave under multiplication [114]. Indeed, the matrices can be directly read off from the figure.
Generalized pokes are defined as local rotations by an arbitrary angle of plastic disc enclosing
the two strand segments. It is straightforward to verify that generalized pokes realize the SU(2)
Lie group axioms [114]. The freedom in defining phases yields the corresponding local gauge
symmetry. Together with the minimal coupling illustrated in Figure 28, this result yields the full
(unbroken) weak interaction Lagrangian.
Strands imply that only massive fermions can exchange weak bosons: only massive particles
interact weakly. Strands imply and predict massive neutrinos. Strands explain why Pauli matri-
ces appear in the context of spin, in equation (23), and in the context of the weak interaction, in
equation (30). As a consequence, due to the tangle structure of particles, maximal parity violation
arises: parity violation occurs because core rotations due to spin 1/2resemble the core deforma-
tions due to the group SU(2) of the weak interactions [118,119]. In addition, also SU(2) breaking
arises: a vacuum strand is included in the unbroken, massless weak bosons, and leads to the W
and Z boson tangles [118,119]. Also the Higgs mechanism is reproduced. In other words, W and
Z bosons are derived from permanently rotating pokes. The freedom in defining the poke phase
yields local SU(2) gauge symmetry. The models for W and Z imply their propagators and the
Lagrangian of the weak field. Particle mixing is a consequence of the strand tangle model of the
weak interaction: it arises from tether braiding, itself a specific type of poke deformation.
Test 26: Strands predict the lack of deviations from the weak interaction properties of the standard
model with massive mixing Dirac neutrinos, at any measurable energy.
So far, all observations confirm the strand tangle model of the weak interaction, which completely
reproduces the conventional description of the standard model of particle physics [68].
In the tangle model, the local gauge group SU(3) arises from slides, the third Reidemeister
moves. The set of the fundamental slides is illustrated in Figure 33. Slides are best visualized by
imagining that the dotted circle encloses a transparent plastic disc that contains, in this case, two

72
Slides, or third Reidemeister moves, acting on strand pairs in thre
{|}~ }~~{}
can
{
{{
ed to the generators of the
{
group SU(3).
iλ1
iλ4
iλ6



t
Fig. 33: The ten important deformations deduced from the slide, the third Reidemeister move, are
visualized. Slides can be modelled by rotating the region inside the dotted circle by the angle π. Each
triplet row defines an SU(2) subgroup. (Other visualizations are also possible, e.g., by deforming only one
strand.) With the definition of λ8, the eight slides iλ1, . . . , iλ8(thus without iλ9and iλ10) generate the
Lie group SU(3) and imply a model for gluons. Each of these eight deformations corresponds to itimes a
Gell-Mann matrix [114].

73
crossing strand segments. Exploring the algebra of the local disc rotations by πin the figure, one
observes that they behave like itimes the matrices [114]:
λ1=


010
100
000

, λ2=


0−i0
i0 0
000

, λ3=


100
0−1 0
0 0 0

,
λ4=


001
000
100

, λ5=


0 0 −i
0 0 0
i0 0 

, λ9=


−100
0 0 0
0 0 1

,
λ6=


000
001
010

, λ7=


0 0 0
0 0 −i
0i0

, λ10 =


0 0 0
0 1 0
0 0 −1

,
and λ8=1
√3


1 0 0
0 1 0
0 0 −2

.
In particular, the matrices can be directly read off from the figure. The slides 3, 9, and 10 are
linearly dependent; to get an orthonormal basis, it is conventional to use only slide 3 and the
additionally defined slide λ8= (λ10 −λ9)/√3. The threefold axis at the centre of the three-strand
configuration yields the factor 1/√3, which is due to the sine of the angle 2π/3. This yields
the orthonormal basis of the eight Gell-Mann matrices λ1, ..., λ8that realize the trace relations
tr λn= 0 and tr(λnλm) = 2δnm. As Figure 33 shows, only λ8is an actual slide, or third
Reidemeister move, in the original sense of the term. This might explain the late discovery of the
relation between the third Reidemeister move and the algebra of SU(3). The matrices λ9and λ10
are not Gell-Mann matrices. However, together with their corresponding slides, the two matrices
are useful for visualizing the three SU(2) subgroups of SU(3).
In other words, the slides 1 to 8 generate the Lie algebra of SU(3). Generalized slides, or
generalized third Reidemeister moves, are rotations by an arbitrary angle of the two crossing
strand segments enclosed by the circular plastic discs illustrated in Figure 33. These generalized
slides realize all the required axioms and yield the full, local SU(3) Lie group [114]. This is the
main result of previous publications [114,115,118,119]. The implied freedom of phase definition
yields local SU(3) gauge symmetry. As a result, gluons are permanently rotating slides. The model
for gluons implies the gluon propagator and the Lagrangian of the strong field.
As shown in reference [121], the strand tangle model for the strong interaction implies the
quark model of hadrons, yielding their correct quantum numbers. Colour fields are densities of
virtual gluons. The gluon tangles yield glueballs, with their observed quantum numbers. Mod-
elling the strong interaction as gluon exchange implies minimal coupling and asymptotic freedom.
The strand tangle model also implies that the strong interaction cannot produce CP violation. The
tangle model of hadrons yields the correct sequence of tangle complexity and thus yields the
correct mass sequences among hadrons. The colour charge of a quark tangle is given by the orien-
tation of its three-ended side in space. Due to their strand structure, gluons carry a double colour
charge. As a consequence, colour charge is conserved.
Test 27: Not the smallest deviation from quantum chromodynamics, at any measurable energy,
will ever be observed.

74
Thus far, the strand tangle model of the strong interaction, including the first estimate of the strong
coupling constant, agrees with quantum chromodynamics and with all observations [121].
Thus, strands yield an emergent model for electromagnetic fields, weak nuclear fields, and
strong nuclear fields. Gauge field intensities are densities of the corresponding gauge bosons,
which are strand configurations specified by the Reidemeister moves, as the next section shows.
Because only three Reidemeister moves exist, the strand tangle model implies:
Test 28: No other gauge group – such as SU(5), SO(10), E8, or any gauge supergroup – will be
observed, at any energy. Grand unification is ruled out.
This prediction agrees with all data [68]. The tangle model eliminates many unification attempts
and has consequences for the Yang-Mills millennium problem, as told in Appendix F. Due to their
rejection of other symmetries, strands respect the Coleman-Mandula theorem [245].
The visualization of the three gauge interactions with strands agrees with and extends the ideas
of Boudet [246]. He describes gauge theory using geometric concepts, in parallel to the approach
by Hestenes [206–208]. Strands can be seen as a way to simplify those descriptions.
In short, when gauge interactions are modelled as deformations of elementary particle tangle
cores, classifying these deformations yields only three possible types: each Reidemeister move
leads to a gauge interaction. The three possible Reidemeister moves – twists, pokes and slides –
determine the gauge groups U(1), SU(2) and SU(3), as implied by Figure 28 and as explored in
references [114,115,118–121]. The resulting charges, conserved quantities, symmetry breaking,
and other aspects agree with observations. In the strand tangle model, gauge fields are densities
of the corresponding gauge bosons and lead to minimal coupling. No other gauge groups are pre-
dicted to exist in nature. The calculations of the coupling constants that were started in references
[120,121] need to be improved.
33 Classifying tangles leads to the spectrum of elementary bosons
In the strand tangle model, elementary bosons consist of one, two, or three strands [114,118–
121]. More strands imply composite particles or groups of several bosons. The Reidemeister
moves, illustrated in Figure 28, directly suggest that one-stranded bosons correspond to photons,
three-stranded bosons to gluons, and two-stranded bosons to the weak unbroken W1,W2or W3
bosons. This holds before the breaking of SU(2) gauge symmetry; in the strand model, symmetry
breaking occurs when two-stranded weak boson tangles incorporate a vacuum strand, yielding the
three-stranded, massive W and Z bosons. A complete overview of all the possible elementary
boson tangles is given in Figure 34.
All (unbroken) elementary gauge boson tangles are trivial, that is, topologically untangled, as
Figure 34 shows. The tangles of vector bosons, i.e., of spin 1 bosons, can be described as one bent
strand among straight strands. The bent strand implies spin 1 for all gauge bosons. The bends in
the tangles can transfer or hop from one strand to the next, yielding boson behaviour. The size
of the bend defines the wavelength and corresponds to the region carrying energy. The region in
which the strand is curved is the region that allows observation and detection of the boson.
The W and Z bosons, also shown in Figure 34, form a special case. They are composed of
three strands that (asymptotically) lie along a line. They arise from the unbroken bosons by the
addition of a vacuum strand. This structure – which distinguishes them from electrons and electron
neutrinos – leads to spin 1, to boson behaviour, and to non-vanishing mass. No other elementary

75
 
eight gluons

W3

before
  

¡
Z boson
 
Higgs boson
wavelength
wavelength
¢ 
photon
W boson
Virtual bosons:
£
eak (real) vect
¤ 
¤¤
after

 

¡¥
thus massive
(
¤¦ § ¨¦ ©
¦ 

W1, W2 (before
  


¡
wavelength
wavelength
Elementary bosons are simple con
ª
gurations o
©¢
¥  ¤   §
 ¨¤¨
¡

¨

« ¢
« ¢
« ¢
« ¬
« ¢
« ¢
« ¢

graviton
wavelength
«
Real bosons:
Fig. 34: The proposed rational 3d tangles for each elementary boson are illustrated. The tangles are
composed of one, two, or three strands. For each gauge boson, the tangle structure determines the spin
value 1 because any curved strand can rotate by 2πand return to its original shape. The graviton has spin 2
because each of its curved strands can rotate by πand return to the original shape. All boson tangle cores
rotate during propagation. Both photons and gluons are massless and thus are described by a single tangle.
The W and Z tangles are asymptotically linear. The W, Z and Higgs are localizable and thus have mass;
therefore they have also more complex tangles, in addition to the simplest ones shown. No additional
elementary bosons are predicted to exist. The W tangle core is the only topologically chiral one and thus is
the only electrically charged elementary boson.

76
boson with spin 1 appears topologically possible. The W boson is the only elementary boson with
a topologically chiral core and thus is the only boson with an electric charge.
Interactions – absorption or emission of gauge bosons – are due to deformations of tangle
cores. The probability of such deformations determines the coupling constant uniquely [118–121].
The challenge of calculating the coupling constants with high precision remains open.
In the strand tangle model, the Higgs boson is a full braid of three strands with six crossings. As
shown in Figure 34, the Higgs braid is deduced from the Borromean rings. Random fluctuations
in its tangle do not lead to its rotation. For this reason, the Higgs tangle has spin 0. The tangle is
not chiral and has positive P parity. Among rational 3d tangles, only a full braid has spin 0 and
reproduces the observed positive parity values. Only a full braid has vanishing electric and strong
charge.
Test 29: Strands predict the lack of any additional Higgs boson.
No other elementary boson with spin 0 appears topologically possible. Observing an additional
Higgs boson would falsify the strand tangle model.
For every massive particle, Higgs braids can be added to the simplest tangle core. This does not
change any quantum number because the Higgs has the same quantum numbers as the vacuum.
Therefore, every massive particle – fermion or boson – is described by an infinite set of tangles.
The set contains the simplest possible core, the core plus one Higgs braid, the core plus two braids,
etc. Some of these options are illustrated in Figure 26 and Figure 27. The value of particle mass
is influenced by the addition of single or multiple Higgs braids. Only the Higgs braid reproduces
this mass-influencing property.
The processes illustrated in Figure 26 and Figure 27 imply that the Higgs boson couples to
itself. The self-coupling, illustrated in Figure 37, implies that the Higgs boson is massive. Both
properties are observed. In contrast, no Higgs braid can be added to cores of massless elementary
particles. Thus, massless elementary particles are described by a single tangle.
The graviton, the last elementary boson, has an unlocalized core that returns to itself after
a rotation by π; thus it has spin 2 and is a boson. Due to its non-localized structure, shown in
Figure 34, the graviton has a vanishing mass. During Dirac’s trick, a massive spinning particle
continuously emits and absorbs virtual gravitons. This mechanism for gravitational mass is as
expected. With the help of the graviton, strands describe general relativity with its field equations
and its Lagrangian, including the cosmological constant [116–118]. Also black hole thermody-
namics, the Unruh effect, and relativistic quantum gravity arise. Cosmology is also reproduced.
No other elementary boson with spin 2 appears topologically possible.
As mentioned, topologically trivial tangles imply that all elementary particles with integer spin
are bosons, and vice versa. Particle composition was explored in Section 9. Therefore,
Test 30: Strands predict that there is no elementary boson with spin 3 or larger.
This deduction agrees with observations: all known high-spin bosons are composed.
No additional elementary vector gauge boson appears possible: neither is a higher number of
strands possible in an elementary particle nor is a more complex tangling of strands with boson
properties possible. The boson tangles for photons, gluons and gravitons imply that these particles
are massless; their tangle cores can rotate unhindered by tethers, in contrast to the W, Z and Higgs
bosons, which cannot; they have to perform a kind of belt trick to rotate and therefore have mass.
Test 31: Strands predict the lack of any additional elementary vector gauge boson.

77
Once tangles are assigned to the gauge bosons, the corresponding charges can be defined from the
topological properties of their tangles. Electric charge is related to the topological chirality of a
tangle [120]. Colour charge is related to the threefold spatial symmetry of quark and gluon tethers
[121]. Weak charge is due to a combination of tangle topology and geometry. Thus,
Test 32: Strands predict conservation of electric and colour charge, but the lack of weak charge
conservation.
Test 33: Strands predict the lack of additional charge types.
Strands naturally imply the lack of localized composites made of photons. The possibility
of extremely short-lived ZZ-composites or W+-W−composites seems unrealistic. In contrast,
glueballs, composites of gluons, are allowed by the strand tangle model; so are many other boson
composites, from meson molecules to organic molecules. Strands forbid some boson composites:
Test 34: Strands predict the lack of ‘photonballs’.
Test 35: Strands predict the lack of glueballs with incorrect quantum numbers [121].
In contrast, strands reproduce the existence of entangled bosons, such as biphotons. These are
regularly observed [247].
The classification of elementary bosons implies
Test 36: Strands predict the lack of further elementary bosons, of any spin.
In particular, the strand tangle model predicts the absence of additional gauge bosons, supersym-
metric bosons, and additional Higgs bosons. In combination with the previous section on fermions,
the prediction becomes even more general:
Test 37: Strands predict the lack of any new elementary particle – including dilatons, inflatons,
anyons, axions [248], metaparticles [249], and any elementary dark matter particle.
Test 38: Dark matter, both galactic and cosmological, is predicted to consist of known matter:
black holes and known particles.
The predictions follow from the strand tangle model of elementary particles and from its classifi-
cation of the possible rational 3d tangles – assuming no mistakes or oversights.
In short, classifying elementary boson tangles reproduces the unbroken gauge bosons with
trivial tangles of one, two or three strands, reproduces the observed mass-producing Higgs boson
with a braid, yields W and Z mass, implies the twisted two-stranded spin-2 graviton, and leaves
no room for additional elementary particles beyond the standard model.
34 All measurements are electromagnetic
In nature, every measurement process and every measurement device makes use of electromag-
netism. For example, all human senses, including hearing or touch, are electromagnetic. All seven
base units of the International System of Units (SI) are defined and applied using electromagnetic
means of observation. Likewise, every comparison with a standard or a unit uses electromag-
netism. Examples are the reading of a length on a ruler, of the hand of a watch, of a balance, or of
a thermometer scale.
In the strand tangle model, the fundamental principle illustrated in Figure 7defines all ob-
servations, all measurements and all physical observables as consequences of crossing switches.
The details of the electromagnetic interaction – briefly summarized in Figure 19 as being due
to twists – imply that crossing switches are observable because they couple to electromagnetic

78
The fundamental, Planck
®¯°±² ³´µ¶¯µ³±²
o
· ¸¹º »¸¼½¾¿ ¸ ½¾ ÀÁº ÂÿºÁ ·
or a photon
t
t+ ∆t
W=~
v=c
S=kln 2
r=p~G/c3
Fig. 35: A photon can be used to formulate the fundamental principle of the strand tangle model, when the
strand radius ris the Planck length.
fields. Strands thus confirm at the most fundamental level that without electromagnetism, there
are no observations and no measurements.
Test 39: Performing any non-electromagnetic observation or measurement is impossible. Per-
forming one would falsify the strand tangle model.
A report by Chew [250] also makes this statement.
Strands thus explain why crossing switches are observable: crossing switches couple to elec-
tromagnetism. In contrast, single strand segments or simple strand deformations do not couple to
electromagnetism and are not observable. Therefore, strands explain the fundamental principle.
This allows reformulating the fundamental principle in the way illustrated in Figure 35.
The relation between crossing switches and electromagnetism also explains how the minimum
time p4~G/c5arises in the fundamental principle. A crossing switch could, in principle, take
an arbitrarily short time. However, such an ultra-rapid, trans-Planckian crossing switch cannot
couple to the electromagnetic field; a photon wavelength shorter than a (corrected) Planck length
is impossible. Such a crossing switch would have no physical effect and would be unobservable.
Test 40: Strands predict the lack of any measurable effect due to time or length intervals shorter
than the corrected Planck limits.
Observing any trans-Planckian effect would falsify the strand tangle model. Again, strands explain
the fundamental principle.
In short, the fundamental principle implies that all measurements are electromagnetic. Strands
also imply that only crossing switches that take longer than the corrected Planck time are observ-
able. No trans-Planckian effects are predicted to be measurable. These deductions agree with all
observations.
35 Strands make predictions about elementary particle mass values
In the strand tangle model, the mass value of every elementary particle is due to the frequency
of the belt trick that appears because of spontaneous strand fluctuations when a tangle moves
[118]. The mass value is thus emergent. In the strand tangle model, the mass is also influenced by
the Higgs mechanism. These connections imply several predictions that can be tested even before

79
calculating any mass value. As mentioned in Section 30, only localized particle tangles have mass.
This statement can be made more precise.
Test 41: Only localizable tangles have inertial mass. Strands predict that only localizable tangles
interact with the Higgs field, and thus have Yukawa couplings. In particular, in the strand
tangle model, only fermions, W, Z and the Higgs boson have positive mass.
Test 42: Because particle masses are due to the belt trick frequency, massive particles are sur-
rounded by a cloud of virtual gravitons. The gravitons are the two-stranded twists arising
in their tethers, as illustrated in Figure 2. Virtual gravitons have spin 2 and vanishing
mass. Therefore they induce a 1/r2dependency of gravity in flat space. Only localizable
tangles have gravitational mass.
Test 43: The mass values of all elementary particles are positive,fixed,unique and constant in
time and space, across the universe.
Test 44: Mass values for particles and antiparticles, i.e., for tangles and mirror tangles, are pre-
dicted to be equal.
All of these deductions from the tangle model agree with all observations [68].
The strand tangle model allows comparison of the inertial and gravitational mass values. On
the one hand, the belt trick generates a displacement and thus relates rotation and displacement.
This relation, illustrated in Figure 2, defines the inertial mass. On the other hand, the tether twists
generated by the belt trick correspond to virtual gravitons. Thus, the belt trick also defines and
generates gravitational mass.
Test 45: Strands predict that the inertial and gravitational mass of all particles are due to the same
process and thus are intrinsically equal, at all times, places and energies.
The deduction agrees with all observations. The prediction is of interest because some versions
of modified Newtonian dynamics (MOND) suggest a difference between the two mass values, at
least for large masses [251,252].
Strands imply that particle mass value depends on its tangle structure. The prediction that more
complex tangles have larger mass values (for the same number of tethers) is valid for all hadrons,
as shown in [118,119,121]. But the most important prediction concerns leptons.
Test 46: The neutrino tangle assignments explain their extremely small mass values. For the elec-
tron neutrino, the simplest tangle is similar to a vacuum configuration. Strands predict
that the electron neutrino mass arises only through the Higgs mechanism – no see-saw
mechanism – and thus is the smallest of all elementary particles.
Test 47: Strands imply and predict
mνe< mνµ< mντ< me< mµ< mτ.(31)
Test 48: Strands predict that neutrino masses and mixings can be calculated before they are mea-
sured with precision.
The prediction of the normal ordering of neutrino masses is expected to be testable in experiments
in the coming years.
In the strand tangle model, all particle tangle cores get flatter at higher energy – or higher
four-momentum. This change will influence the frequency of the belt trick. As a result,
Test 49: Strands imply the running of elementary particle masses with energy.

80
Qualitatively, the deductions from the tangle model agree with the observations and with standard
model calculations, such as those by Xing et al. [253] and by Huang and Zhou [254]. Performing
a quantitative test is a subject of research.
Thus far, only rough estimates of particle masses from belt trick probabilities have been
achieved [120,121]. No precise calculation method has been developed yet. It is expected that
exploring the behaviour of asymmetric bodies and their rotation in a viscous (Stokes) flow might
lead to better approximations [255]. This field of research is still in its infancy.
In short, calculating particle masses – and their running with energy – with the help of com-
puter simulations that determine the relation between the structure and shape of a tangle core, the
belt trick, and the motion of a tangle will allow independent tests of the strand tangle model. Even
before such calculations, additional tests of the strand tangle model are possible.
36 The principle of least action follows from strands
In nature, a higher value of action corresponds to a higher amount of change in a system. Accord-
ing to the fundamental principle of the strand tangle model, the action of a physical process is the
number of occurring crossing switches. Each crossing switch yields an action ~. Therefore,
BLeast action is the smallest number of crossing switches.
BEvery motion and every process in nature minimizes the number of crossing switches.
This equivalence is consistent with the fundamental principle because a crossing switch is less
probable than other strand deformations. The reason was given in Section 34: a crossing switch
couples to the electromagnetic field. Every crossing switch is a form of change; every crossing
switch requires an ‘effort’ or ‘cost’, whereas a strand shape deformation without a crossing switch
does not. Thus, strands confirm that that physical action is the measure of change.
In other words, strands suggest an underlying reason and explanation for the principle of least
action: the fundamental principle implies that nature minimizes crossing switches. This appears to
be the only explanation for the principle available in the literature. Therefore,
Test 50: Strands predict the lack of the slightest deviation from the least action principle and from
the description of motion with Lagrangians. Discovering such a deviation would falsify
the strand tangle model.
In particular, the result applies to motion in the quantum domain. Indeed, the Dirac Lagrangian
for a free particle arises by minimizing the number of crossing switches in spinors. The results of
the last sections confirm that the principle of least action also applies to gauge interactions.
In short, the strand tangle model implies the principle of least action for all processes in nature.
This result allows evaluation of the standard model as a whole.
37 Strands limit the possible interaction vertices of the standard model
This section summarizes the results on particle physics published in references [118–121]. In the
strand tangle model, gauge interactions are due to tangle core deformations that occur by transfer
of Reidemeister moves, as illustrated in Figure 28. The figure also shows that as a consequence,
particle reactions and decays are described by Feynman diagrams.

81
ÄÅÆÇ ÈÉ
erage
of crossing
switches
ÊËÌÍÎÏ
ed
Ð
Í
ÑÒÓÔÒ ÕÖÔ×
ÎÔÓ
Tangle model
ØÙÚÛ
c
ÜÝÞßà
á
âÜÝÞßà
âÜÝÞßà âÜÝÞßà
ã
t2
ØÙÚÛ
t1
âÜÝÞßà
âÜÝÞßà
âÜÝÞßà
á
c
ÜÝÞßà
ã
ØÙÚÛ Þä
erage
of crossing
switches
ÊËÌÍÎÏ
ed
Ð
Í
ÑÒÓÔÒ ÕÖÔ×Î
ÔÓ
ØÙÚÛ
ÚÝåæ æ
ÛÝØßÙæå
á
ÚÝ
åæ
ã
ÚÝ
åæ ÚÝåæ ã
ØÙÚÛ
ÚÝ
åæ
ÚÝ
åæ
ÚÝ
åæ
á
ÚÝåæ æ
ÛÝØßÙæå
ØÙÚÛ
á
á
ØÙÚÛ
ã
photon photon
á
á
ØÙÚÛ
á
á
á
á
ØÙÚÛ
photon photon
á
á
á á
ØÙÚÛ
ã
photon
á
á
ã
photon
ØÙÚÛ
á
á
ãã
ã
ã
çÅèéè ÅêëÅéÈÄ
e a pair of tethers
or a long tether
Tangle model
photon photon
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
á á
á á
á á
á á
ã
á á
á á
Fig. 36: The first interaction vertices allowed by fermion and boson tangle topologies imply the full
Lagrangian of the standard model and its renormalizability. (Figure improved from reference [119].)

82
ìíîï
ðñòóôõ
ed
öó
÷øùúø ûüúýô
úù
þí
ÿÿ
H
Z
Z
Z
Z
Z
þí
ÿÿ
H
þí
ÿÿ
H
ZZ
time
þ
í
ÿÿ
H
þí
ÿÿ
H
time
þí
ÿÿ
H
þí
ÿÿ
H
time
þí
ÿÿ
H
þ
í
ÿÿ
H
þí
ÿÿ
H
W W
time
þ
í
ÿÿ
H
time
þí
ÿÿ
H
þ
í
ÿÿ
H
þ
í
ÿÿ
H
þí
ÿÿ
H
þí
ÿÿ
H
þ
í
ÿÿ
H
þí
ÿÿ
H
þ
í
ÿÿ
H
þí
ÿÿ
H
time
time
q
gluon
q
time
q
gluon
q
time
ðñòóôõ
ed
öó
÷øùúø ûüúýô
úù
Tangle model
þí
ÿÿ
H
fermion
fermion
H q
H q
þí
ÿÿ
H
time average
of crossing
switches
Tangle model
vacuum
gluon gluon
gluon gluon
gluon gluon
þí
ÿÿ
H
W W
gluon
gluon
gluon gluon
gluon gluon
gluon gluon
t
2
t
1
t
2
t
1
t
2
t
1
t
2
t
1
t
2
t
1
ì
t
ï
q
core is rotated
by

/ 


ì
t
ï
t
í
h
ontal axis.
þí
ÿÿ
H
þí
ÿÿ
H
þí
ÿÿ
H
þí
ÿÿ
H
þí
ÿÿ
H
ìíîï

erage
of crossing
switches
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
D 
e a pair of tethers
or a long tether
Z
W W
W W
Fig. 37: The remaining interaction vertices allowed by fermion and boson tangle topologies imply the full
Lagrangian of the standard model and its renormalizability. (Figure improved from reference [119].)

83
The tangles of the elementary fermions and bosons given in Sections 31 and 33 allow only a
limited number of interaction vertices. The possible interaction vertices are listed in Figure 36
and Figure 37. These vertices cover all observed Feynman diagrams of the standard model with
massive Dirac neutrinos [256,257]. The vertices reproduce the conservation of charge, parity,
energy, momentum, and spin. No additional vertex arises and no observed vertex is missing [119].
The lack of additional interaction vertices is due to the structure of the rational 3d tangles for the
elementary particles. This vertex list agrees with observations [68]. In other words, strands restrict
the possible particle reactions and decays to the observed ones.
In the Lagrangian of the standard model, strand tangles imply, due to the Dirac equation and
the few allowed elementary rational 3d tangle structures, mass terms for the charged leptons, the
neutrinos and the quarks. Strand tangles imply, due to the Reidemeister moves, the terms for the
bosons of the three gauge interactions. Strands imply the existence of antiparticles. Due to the
interaction vertices and the resulting minimal coupling, tangles imply the dynamical or interac-
tion terms for all leptons. Strands imply, due to the Higgs tangle and its vertices, the Higgs mass
and dynamical terms. Strand tangles imply, due to the chiral fermion structures, maximal par-
ity violation. Strand tangles imply, due to tether movements, the Cabibbo-Kobayashi-Maskawa
mixing matrix and the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix. Strand tangles imply,
due to their average shape, all fundamental constants. As a result, the full standard model, in-
cluding massive mixing Dirac neutrinos, arises from rational 3d tangles of strands [258]. Strands
explain the origin of the standard model, solve the hierarchy problem, and answer the questions
raised by Gell-Mann et al. in 1983 [259]. Strands imply that no alternative or modification to the
standard model Lagrangian with massive neutrinos is possible. This conclusion agrees with all
experimental data [68].
Figure 36 and Figure 37 show that only triple and quadruple interaction vertices arise in the
tangle model, and never more than two involved fermions. This limitation is due to the limited
number of strands in the tangles of elementary particles. The limitation, together with the de-
tailed properties of the vertices in the figures, implies that QED,QCD, and the theory of the weak
interaction are renormalizable if they are extrapolated to point-like particles [260]. Because of
the minimum length, strands eliminate the Landau pole. These results are as expected, given that
tangles yield a finite description of quantum field theory.
In short, the fundamental principle restricts the rational 3d tangles for elementary particles to
the observed ones. The elementary tangles in turn restrict the interaction vertices of the standard
model to the observed ones. Strands also imply the renormalizability and finiteness of the standard
model, when formulated in the point particle approximation. In other words, the fundamental
principle implies quantum physics and particle physics. Therefore, the standard model of particle
physics with massive neutrinos is simple, finite, unique and elegant.
38 Strands imply experimental tests of the tangle model
Rational 3d tangles imply the standard model of particle physics with massive mixing Dirac neu-
trinos. The connection leaves no freedom of choice; it is unique.
Test 51: The tangle model predicts that the standard model with massive mixing Dirac neutrinos
is valid at all measurable energies, without any modification. Strands predict the lack of
measurable effects beyond the standard model.

84
This conclusion agrees with all observations [68]. The conclusion is equivalent to stating that
strands explain the origin of all interactions and all particles. In addition, the strand tangle model
predicts an answer to the open questions of particle physics:
Test 52: The strand tangle model predicts that the calculated values for the fundamental constants
– elementary particle masses, coupling constants, mixing angles and phases – agree with
experiment. The values can be calculated from the average tangle shapes of fluctuating
tangles. Values that differ from the observed ones are impossible.
So far, the estimates deduced from the tangle model agree with observations [118–121]. The tests
include the comparison with observations of calculated neutrino masses and mixing angles, lepton
masses, quark masses and mixing angles, the CP violating phases, the Higgs, W, and Z boson
masses, and the three coupling constants.
The strand tangle model implies the same perturbation expansions as conventional quantum
field theory [120,121]. The perturbation expansions are due to the complex tangles arising from
fluctuations at small scales.
Test 53: The tangle model predicts that the calculated runnings with four-momentum of the fun-
damental constants agree with experiment. Due to the lack of grand unification, super-
symmetry, and new energy scales, the running couplings do not need to converge.
These calculations are a topic for future research.
In the research literature, alternatives to the strand tangle model are rare. Only one other unified
model attempts to deduce the fundamental constants: the octonion model due to Singh [261]. So
far, the octonion model appears to predict additional particles and possibly additional interactions
[262]. These have not yet been observed.
Stated simply, the strand tangle model predicts the lack of breakthroughs in fundamental exper-
imental physics. Equivalently, strands predict that the standard model and general relativity limit
how humans can shape their environment and what we can achieve. Of course, this prediction
needs to be continuously challenged.
The strand tangle model implies a natural cut-off at the Planck scale and a well-defined con-
tinuum limit that arises when the strand radius is assumed to vanish.
Test 54: Strands predict that physical space has three dimensions at all measurable scales.
Test 55: Strands predict that there is only one vacuum state in nature. In particular, no domain
walls between “different” vacuum states exist.
So far, the predictions agree with observations. Additional tests of the strand tangle model appear
to be hard to find. The predicted lack of trans-Planckian effects includes a lower limit on the
effective size of elementary particles:
Test 56: No elementary particle has an intrinsic diameter smaller than the minimum length.
The lower size limit leads to a question: can deviations from locality be observed?
The predicted size of elementary particles could imply non-vanishing electric dipole moments.
Present measurements for the electron are closest to achieving Planck scale sensitivity [73,74]:
the experimental upper limit is the elementary charge multiplied by around 1000 Planck lengths.
However, the strand tangle model predicts no dipole moment for the electron. For all elementary
particles, tight tangles limit electric dipole moments to a few times the minimum length times
the elementary charge. Measuring the electric dipole moment of the W, Z and the Higgs seems
impossible. Likewise, measuring the dipole moment of a single quark to such a precision also
appears out of reach, even by measuring the dipole moment of the neutron.

85
Deviations from point-like behaviour can also be searched for in interactions and decays. How-
ever, collider experiments will never reach the required short distances, neither for the electromag-
netic nor for the weak interaction. For the nuclear interactions, the tangle structure could lead to
experimental effects in the non-perturbative domain. However, the possibility for detection ap-
pears remote [121]. Likewise, no decay process provides the required spatial accuracy.
A further deviation from point-like behaviour is entanglement. So far, all observations about
entanglement and decoherence agree with the tangle model. The possibility of detecting rela-
tivistic quantum gravity effects should be explored further. The relation between entanglement,
decoherence, and gravity is an extensive research field [263–265]. Mass superpositions provide
interesting experimental challenges. Strands predict no observable deviations from locality.
In other words, direct observation cannot confirm the details of strand tangles.
Test 57: Tangles predict the lack of measurable deviations from locality in nature.
Tangles imply that wave functions, mass, couplings, spin, charges, magnetic moments and all other
known particle properties completely describe elementary particles, including all the observable
deviations from local behaviour.
In short, strands predict the lack of new physics and the lack of trans-Planckian effects. Be-
cause stands are not observable, tangles imply that all particle properties are completely known
and predict that no additional observable particle properties can be found. Tangles can be tested by
calculating the fundamental constants – particle masses, coupling constants, and mixing angles –
and their running with four-momentum. Any direct experimental confirmation of particle tangles
appears highly unlikely.
39 Strands yield Galileo’s book of the universe
In 1623, Galileo published his book Il saggiatore. In chapter 6 he writes (translated by the author):
“Natural philosophy is written in this huge book that is continually open in front of our
eyes, I mean the universe, but one cannot understand it if one does not first learn to
understand the language and to know the characters in which it is written. It is written
in mathematical language, and the characters are triangles, circles and other geometric
figures, means without which it is impossible for humans to understand a word; without
these it is a wandering in vain through a dark labyrinth.”
The tangle model confirms that the book of nature is not written with equations, but is indeed
written in mathematical language that uses geometric figures. The fundamental principle with its
triangles in electron tangles and its circles in photon tangles yields quantum mechanics and the
standard model of particle physics. Additional fluctuating geometric shapes of strands [116,117]
yield space, general relativity and black hole horizons.
40 Particle tangles have open issues
The above presentation of the strand tangle model may contain inconsistencies or errors. First, it
could be that the tangle assignments for the quarks, leptons, or bosons require corrections. For
example, a smooth transition between quarks and antiquarks is difficult to visualize.

86
Test 58: Even if specific tangle corrections will be necessary, the strand tangle model is predicted
to agree with observations and thus remain valid.
A second open point is the completeness of the classification of the rational 3d tangles given
above. For example, when a braid is formed from three sets of two strands each, which set of
elementary particles corresponds to the resulting configuration? Similar questions regarding other
tangles are yet to be answered comprehensively.
A further point was not explored in this article. The propagation of a quantum particle does
not require that its strands are always the same ones. A propagating particle can take in a vacuum
strand and, in exchange, leave behind a strand from its tangle. This process is easiest to visualize
for photons; a twist can be imagined to ‘jump’ from one strand to the next. It is unlikely that such
processes lead to observable effects; however, the point requires further study.
In short, it is possible that future investigations lead to minor corrections to the tangle model.
41 Conclusion
Ciò che per l’universo si squaderna:
...
La forma universal di questo nodo
...
Dante, Paradiso, Canto XXXIII, 87, 91.
Starting from the tethers of Dirac, the ideas of Battey-Pratt and Racey, and the Planck limits,
fermions are modelled as tangles of unobservable fluctuating strands with Planck radius. Every
crossing switch yields a quantum of action ~. This fundamental principle contains all of physics.
As a consequence of the fundamental principle, propagating particles are modelled as advanc-
ing and rotating rational 3d tangles that fluctuate. Every particle is connected to the cosmological
horizon. The tangle crossing density defines the wave function. Crossing densities of fluctuating
rational 3d tangles reproduce the probabilistic behaviour of quantum theory in all details, including
superpositions, decoherence, and entanglement.
Only strands yield, using the belt trick, spin 1/2and fermion behaviour. Only strands yield
emergent wave functions, spinors, and the Dirac equation. Only strands yield, by classifying
strand deformations with the Reidemeister moves, the gauge interactions and the observed Lie
gauge groups. Only strands yield, by classifying the possible tangles, the three generations of
elementary fermions and all the elementary bosons, with their observed quantum numbers. Only
strands yield massive mixing Dirac neutrinos. Only strands yield, from the topology of the tangles,
the Higgs mechanism, the breaking of SU(2), lepton mixing, and maximal parity violation of the
weak interaction. Only strands yield, from their Planck radius and from the average shapes of
tangles, unique elementary particle mass values, coupling constants, and mixing angles.
Only strands yield, from the topology of the tangles, the observed Feynman vertices, particle
reactions, and decay modes. Only strands yield the quark model, glueballs, and the lack of CP
violation in the strong interaction. Only strands yield quantum electrodynamics, quantum chro-
modynamics, and the theory of the weak interaction. Only strands yield the Lagrangian of the
standard model with massive neutrinos, without any measurable modification.
From their tangling, strands yield the three dimensions of space. From their crossing switches,
strands yield maximum force, space curvature, Bekenstein’s entropy bound, the thermodynamics

87
of black holes, and Einstein’s field equations. Only strands yield both the Hilbert Lagrangian and
the standard model Lagrangian with massive neutrinos.
In short, the strand tangle model, every particle is connected to the cosmological horizon.
A simple fundamental principle implies all known observations and deduces numerous experi-
mental predictions. Every deduced consequence agrees with observations. Also, everybody can
understand quantum mechanics. Strands predict the lack of new physics and the possibility of
calculating the observed fundamental constants to high precision, including their running with
four-momentum. The strand tangle model proves that the standard model of particle physics –
with massive mixing Dirac neutrinos – is simple, unique, complete, and elegant.
42 Outlook
In the experimental domain, the strand tangle model derives, from its fundamental principle, more
than 50 precise predictions and tests. Strands predict the lack of measurable physics beyond the
standard model with massive mixing Dirac neutrinos, the lack of measurable corrections to general
relativity, the lack of measurable trans-Planckian effects, and the normal ordering of neutrino
masses. If only one of the predictions and tests disagrees with observations, the strand tangle
model is falsified. Despite intense experimental efforts, no such disagreement has appeared.
The main challenges for future research are the mathematical aspects and the numerical sim-
ulation of the strand processes presented above. Exploring the Heisenberg picture of quantum
theory will deepen understanding. Deducing Schwinger’s quantum action principle from strands
is appealing. Deducing the Dyson–Schwinger equations and the Ward–Takahashi identity from
strands of zero radius is attractive. Deducing axiomatic quantum field theory from tangles of
zero radius is worthwhile. The relation between crossings, qubits, entanglement and quantum
gravity should be explored. The consequences of strands for entanglement entropy should be
deepened. The relation of rotating tangles to Hestenes’ geometric algebra can be studied. Anima-
tions of particle propagators and Feynman vertices will be useful. Exploring perturbation theory
and renormalization in quantum field theory and quantum gravity is promising. An example is the
Kinoshita–Lee–Nauenberg theorem [266]. Ultraviolet issues and questions about anomalies will
become more accessible using strands. Non-perturbative effects will be clarified.
In cosmology, an estimate of the cosmological constant from strands should be derived, in-
cluding its time dependency or lack thereof. The strand tangle model for gravity should be used to
clarify the issue of galaxy rotation curves by deducing the existence or nonexistence of MOND.
In the domain of numerical simulation, precise calculation methods for the fundamental con-
stants and their running with four-momentum need to be developed. The study of the quantification
of the chirality of tangle cores should be deepened. Exploring the rotation of asymmetric and teth-
ered bodies in viscous flows will allow better approximations for the mass values of elementary
and composed particles. Numerical QCD is expected to be enhanced. Geometric ab initio approxi-
mations for quark and neutrino mixing, for the Higgs mechanism, and for the W and Z masses will
be developed. All calculations are predicted to agree with the measurements. A lack of agreement
would refute the strand tangle model. With a finite effort, neutrino masses and mixing angles can
be calculated before they are measured.

88
Acknowledgments and declarations
The author thanks Thomas Racey and Michel Talagrand for their valuable advice. The author
also thanks Bernd Thaller, Isabella Borgogelli Avveduti, Saverio Pascazio, Eric Rawdon, Luca
Bombelli, Volodimir Simulik, Sergei Fadeev, Lou Kauffman, Peter Schiller, Rodolfo Gambini and
Jorge Pullin for fruitful discussions. Part of this work was supported by a grant from the Klaus
Tschira Foundation. The author declares that he has no conflicts of interest and no competing
interests. No additional data are associated with this work.
Appendix
A General relativity and all of nature deduced from strands
In the strand tangle model, everything observed in the universe – space, horizons, and particles –
is composed of strands. The corresponding strand structures are illustrated in Figure 38. Thus
BThe whole universe is expected to be one long closed strand – i.e., a strand with the
topology of a ring – that crisscrosses all of nature and forms space, particles and hori-
zons. This strand is becoming increasingly tangled over time.
Figure 39 illustrates the strand tangle model of cosmology. Strands realize Heraclitus’ idea that
everything in nature is connected to everything else: nature has no exact parts.
In the strand tangle model, flat empty space is an aggregate composed of fluctuating untan-
gled strands, packed as densely as possible. The lack of tangles implies its emptiness. Strands
fluctuations imply Lorentz invariance, and avoid the obstacle that static finite models cannot do so
[267]. Curved space is an irregular aggregate of strands, in which some strand pairs are twisted,
representing real or virtual gravitons. Black hole horizons are weaves of strands. Because the
Planck limits are part of the fundamental principle of the strand tangle model, strands imply the
maximum force c4/4G. As shown elsewhere [116–118], strands imply the usual expressions for
black hole mass, the moment of inertia of black holes, black hole entropy, and black hole temper-
ature. The usual logarithmic correction to black hole entropy also arises – but it is unmeasurably
small. Despite their countless tethers, black holes can still rotate, making use of the belt trick.
In the strand model of black holes, for an observer at spatial infinity, mass is distributed over
the horizon. This mass distribution explains the non-vanishing moment of inertia of black holes
deduced by Ha [268]. Together, these properties imply Einstein’s field equations and the Hilbert
Lagrangian, without any measurable deviation, as shown in [8,10,17,116,117]. This derivation
is an example of how a fundamental inequality, here F6c4/4G, leads to precise equations of
motion. All deduced properties of gravitation agree with all experiments, as shown by Will [47].
Also all LIGO observations about gravitational waves confirm general relativity. In particular, no
black hole merger exceeds the power limit P6c5/4G[45,46], and thus none of the other black
hole limits.
All particles are rational 3d tangles of strands. Fields are particle densities. The present
article focuses on elementary particles and fields. Therefore, only particle strands are drawn in
the illustrations, without vacuum strands. For gravitons, the usual Feynman diagrams arise [269].
Every elementary particle is effectively a defect in space. This old dream has been explored by

89
Observation
:
N
(
or long
oo
times)
Vir
!"# $"%&
(for short
oo
times)
The
'
at
v")!!*

a homogeneous strand aggregate
time average
of crossing
switches
Observation
:
C
urv
e+ &$")
e
w
,o-
trivial metric
C
urv
e+ v")!!*

an irregular strand aggregate
A .#")0 #e %3
on

a weave of strands
Observation after time average of crossing swit
c,
:
a horizon
4
i.e.
4  
, 5
,c6 c6o78 4
with
m4
moment of iner
4
entrop
94
and temperature.
time average
of crossing
switches
A
par
)#e

a tangle of strands
time average
of crossing
switches
Observation
:
P%.".#;
+e&;
B6c<
dots show
the midpoints
of crossings.
Fig. 38: Strands can form aggregates (space), weaves (horizons), and tangles (particles). The present text
concentrates on particles. For the exploration of space, curvature and black hole horizons, see references
[116] and [117].
many researchers, including Hehl and Kröner [270] or Kleinert [271]. Exploring all possible defect
structures of the strand vacuum yields horizons and particles, and no other possibility. Nature has
no domain walls, line defects, singularities, white holes, space-time foam, or wormholes [116,
117]. Nevertheless, strands solve all problems of gravitation collapse [272].
The strand tangle model of cosmology is illustrated in Figure 39. Strands predict the existence
of a cosmological horizon. The tangle model defines a preferred reference frame, namely the
frame in which the cosmological horizon and the cosmic background radiation are isotropic. This
is also the natural reference frame to observe the maximum possible elementary particle energy,
p~c5/4G, were the value achievable. This description with Planck limits resembles the approach
by Amelino-Camelia [273]. In highly boosted reference frames, the background formed by hori-
zon strands would not be isotropic and the derivation of the Dirac equation in Part III would not
apply. However, at all realistic boost values, deviations from the Dirac equation are unmeasurable.
In addition, strands confirm that the idea of ‘wave function of the universe’ makes no sense.
In short, in the strand tangle model, all systems found in nature – particles of radiation and

90
Strand tangle model:
=>?@
c
EF?
EGEI
>JKG
LEM
>O
on
TQR RSUVX
expanding universe
Observations:
=>?@ K Y
erage
of crossing
switches
none
Cosmological
horizon (dark)
Physical space or
cosmic vacuum (whit
@[\
made o
f ]@^ F
@G
_
packed
u^=K
^IG
@] F
=MK^]
F\
E`F
@MYK`G
@a
locally
JuMY
ed, fl
K
=E
^KY
erage
Background
space
d
gre
_[
g
^
E=nL
_F
>JKG
a
^
E=E`F
@MYK`G
@
i
osmological
tethers
The present universe
Particle
tangle;
made of
tangled
strands
Fig. 39: The whole universe is expected to consist of a single strand, without ends. The figure illustrates
its early history (top) and its present state (bottom).
matter, flat and curved space, cosmological and black hole horizons – consist of strands. Strands
imply that black hole horizons have the usual entropy. Space curvature and gravitation are de-
scribed by Einstein’s field equations, without any measurable deviation, in agreement with all
experiments. Strands also yield a model of cosmology in agreement with observations.
B Strand crossings visualize qubits and Urs
Qubits, a contraction of ‘quantum bits’, play an important role in two research fields: quantum
computing and relativistic quantum gravity. In modern quantum computing research, physical
qubits are commonly realized as trapped ions, quantum dots, nitrogen-vacancy diamonds, or su-
perconducting loops. The state of a qubit can be visualized with the Bloch sphere, which illustrates
the SU(2) structure of the qubit state space [274–276]. In quantum computing, quantum gates are

91
represented by unitary operators. The different types of quantum gates can be represented as
different operations on the Bloch sphere.
The strand tangle model for fermions – as illustrated in Figure 20, Figure 21 and Figure 22 – is
useful for visualizing qubits. In the strand tangle model, the Bloch sphere describes the orientation
of the tangle core and the twist state of its tethers. In the strand tangle model, unitary operators are
represented by strand deformations that keep the average shape of the tangle core untouched, but
change its orientation in space. In this way, a strand tangle visualizes a single-qubit state; many
tangles visualize multi-qubit states.
Research questions about qubits are also explored in the context of relativistic quantum gravity.
How can one imagine a region of space-time that includes vacuum, matter, and particles in terms
of qubits? In particular, do qubits form lattices, random structures, or other types of structures? Is
the number of qubits finite, countably infinite or uncountably infinite? These questions are about
the microscopic, Planck-scale constituents of nature as a whole [277]. In 2001, Zizzi formulated
the aim of this quest with the expression ‘it from qubit’ [278].
Given the possibility of describing quantum theory, gravity, and quantum field theory with
strand tangles [116,118–120], relations between tangles and qubits arise almost naturally. Given
that space, curvature, and particles are due to strands and their tangling, one can interpret the
fundamental principle of Figure 7as a maximally simplified qubit. As long as the strand segments
are kept close to each other in a crossed configuration, the two strands can be seen as a simplified
belt buckle realizing SU(2). In other words, a strand crossing is a qubit.
The fluctuations of the strands continuously change the structure and the interdependencies
of the qubits found in nature. Strands clarify that the number of their crossings – the number of
qubits – is countable in principle, but is not fixed. With this interpretation of tangle model, strands
confirm that nature is built of qubits.
The quest to construct nature from qubits is not new. A ‘qubit’ is the modern expression for the
‘Ur’, the fundamental and continuous alternative, described by SU(2), that Weizsäcker introduced
in the 1970s [279,280]. Weizsäcker dreamt of deducing all laws of nature by building them from
Urs [281]. (The correct German plural is ‘Ure’.) The research approach was explored by Kober
[282] and by Lyre [283,284]. The fundamental principle of Figure 7, which generates SU(2), can
be seen as the simplest possible realization of an Ur. Thus, the strand tangle model can be said to
realize Weizsäcker’s wish: nature is built of Urs.
An approach to qubits similar to that of strands arose recently. In 2016, Freedman and Head-
rick introduced bit threads to describe entanglement, holography and the Ryu-Takayanagi entropy
[285]. Central to the idea of bit threads is the hope that space can be seen as a consequence of the
entanglement of filiform entities with Planck-sized diameter [286]. Various authors explored the
topic [287–294]. It can be expected that research on bit threads will follow that on strands.
In short, strand tangles are useful tools for teaching and visualizing qubits in quantum comput-
ing. In the field of relativistic quantum gravity, the fundamental principle can be seen as a specific
version of building ‘it from qubit’ and ‘nature and her laws from Urs’.
C Strands define quantum fields
Traditionally, quantum fields are introduced as part of the axioms of quantum field theory [295].
In particular, quantum field theory defines fields assuming point-like particles and using local

92
commutator relations and operator-valued local densities. In the description of nature provided by
quantum field theory, particle masses, couplings, and mixing angles must be added by hand and
cannot be explained. Strands propose a different view.
BIn the strand tangle model, quantum fields emerge from fluctuating rational 3d tangles
made of strands with Planck radius.
The minimum length forbids using local operators or distributions as fundamental concepts. The
traditional, axiomatic quantum field concept is an approximation. All concepts of axiomatic field
theory, from C∗-algebras to fibre bundles, are approximations. Quantum field theory only arises
when strands are approximated to have zero radius.
The strand tangle model is not based on approximations. As a result, nothing has to be added
by hand and all constants are explained. In the strand tangle model, particles are rational 3d
tangles of unobservable strands with Planck radius. Strand shape fluctuations lead to fluctuations
in the strand crossings. Averaged over time, strand crossings in massive particles yield complex
wave functions. Crossing switches define probability densities, matter fields and all observables.
Gauge fields arise from the strand crossing switches of the related gauge bosons. Thus, all fields
emerge from jiggling strands. Like traditional quantum field theory, the tangle model treats all
fields equally. And strands lead to axiomatic quantum fields and local operators in the limit that
the strand radius goes to zero.
In the strand tangle model, identical particles are countable and indistinguishable; their spin
and their statistics are closely related. Particles are (approximately) localized excitations of the
(untangled) vacuum. Excitations are realized by the tangling of the untangled strands of the vac-
uum. Rational tangles allow for virtual particles, antiparticles, particle transformations, and parti-
cle reactions. Thus, the (rational) tangle model reproduces particle creation by tangling, particle
annihilation by untangling, particle absorption by tangle combination, and particle emission by
tangle separation. The strand description of operators as deformations of tangle cores given in Sec-
tion 12 yields the description of particle creation and annihilation as operators. Both operators are
almost local, realizing the observational constraints and theoretical demands. Both operators en-
sure an integer number of particles. The tangle model eliminates the need for superselection rules.
Strands eliminate the need for ‘second quantization’ – and for any other type of quantization. Nev-
ertheless, the strand description of operators as deformations of tangle cores automatically yields
the usual result that quantum fields are operator-valued distributions [267].
Vacuum excitation results in both particles and antiparticles. The strand tangle topology de-
termines the particle type and its quantum numbers. The (rational) tangle model also implies that
(certain) particles can interact: different tangles can be combined, or a single tangle can be sep-
arated into two or three tangles. The (rational) tangle model also implies that (certain) particles
can transform into each other: tether braiding changes particle type. The strand tangle model re-
produces the three generations and the particle spectrum as the result of tangle classification. The
strand tangle model reproduces particle quantum numbers, their mass values, and their mixing
angles as the consequence of topological and geometric tangle properties. The minimum length
built into the strand tangle model resolves the issues of renormalization.
The strand tangle model thus reproduces all effects traditionally attributed to quantum field
theory. As a result, in the strand tangle model, an expression such as ‘an electron is an excitation
of the electron field’ is either misleading or wrong. More drastically, concepts such as ‘electron
field’ or ‘neutrino field’ make no sense.

93
Strands simplify quantum field theory in other ways. Whenever space is assumed to be de-
scribed by a lattice, the Nielsen–Ninomiya theorem implies the necessity of doubling the number
of chiral fermions [296–298]. This theorem has generated numerous studies on its extensions and
limitations. The theorem is also the reason for various difficulties in numerical simulations of
elementary particles. In the strand tangle model, because of the minimum length, space is neither
discrete nor continuous. In particular, space is not a lattice and thus has no Brillouin zone. There-
fore, the strand tangle model does not realize the conditions for the doubling of fermions and the
no-go theorem published by Nielsen and Ninomiya.
In 1955, Haag proved that an interacting quantum field theory is mathematically inconsis-
tent. The proof is based on continuous space, on the properties of vacuum polarization, and on
its behaviour under renormalization [299,300]. However, given that nature is not characterized
by continuous space, nor by continuous observables, nor local operators, the conditions for the
theorem are not satisfied. Again, strands simplify quantum field theory.
In the tangle model, the issue of ultraviolet completeness loses its importance and is effectively
dissolved. The same happens to renormalization, even in the case of quantum gravity. Anomalies,
briefly addressed above, lose their importance.
In short, jiggling tangles and their crossing densities reproduce, visualize and simplify both
wave functions and quantum fields. Both concepts emerge from the fundamental principle of the
strand tangle model. Strands predict that there is only one possible quantum field theory that
describes particle properties and is compatible with gravity: the standard model with massive
Dirac neutrinos and with the observed fundamental masses, couplings and mixing angles is the
only possible option. A few aspects of this uniqueness are presented in the following appendices.
D On preons
In 1979, both Harari [301] and Shupe [302] proposed models of composite quarks and leptons.
In both models, there are only two fundamental particles, which Harari called rishons and Shupe
quips. Nowadays, such hypothetical particles are generally called preons. Both approaches as-
sumed that the preons had a charge e/3, anti-preons −e/3and that leptons were composed of
three preons, as already proposed before [303]. Shupe explored states with fewer and more preons
as models for gauge bosons and gravitons, deduced the vanishing mass of preons, and tried to
model Feynman diagrams.
Many explorations followed and continue to this day, too numerous to mention all [304–310].
Approaches that tried to add interactions or dynamics had no success. In retrospect, this lack of
success resulted from the insistence to find a new Lagrangian and new symmetries.
If one calls a strand crossing a ‘preon’ and its mirror inverse an ‘anti-preon’, one finds several
similarities: fractional electric charge ±e/3, masslessness, and a description of both fermions and
bosons. However, preons do not explain the gauge groups, whereas strands do. Likewise, preons
do not explain particle masses, gauge coupling constants, or general relativity, whereas strands do.
Similar limitations also apply to the fascinating preon approach by Bilson-Thompson based
on braided ribbons [106–110], which has generated numerous subsequent studies. Despite their
similarity to strands, ribbons do not seem to explain wave functions, their dynamics, the gauge
groups, the fundamental constants, or general relativity. However, replacing each edge of a ribbon
with a strand yields an approach similar to the tangle model and could yield new insights.

94
In short, point-like preons seem to be more similar to fundamental quantum numbers and to
strand crossings than to fundamental particles. Ribbon-like preons do not seem to provide a unified
description. It appears that tangles of strands solve the problems that preons left open: strands
explain the gauge groups, the fundamental constants, general relativity, and wave functions.
E On superstrings, supermembranes, and strands
At first sight, strands differ from superstrings and supermembranes in several respects. Super-
strings are characterized by Planck tension, have a Lagrangian, live in 10 dimensions, realize du-
alities, carry supersymmetric superfields, and possibly imply grand unification [311,312]. Super-
strings have been generalized to various types of supermembranes, such as D-branes. In contrast,
strands have no tension or Lagrangian, generate and populate the usual 3 + 1 dimensions, carry no
fields nor any other observables, and exclude supersymmetry and grand unification. There is only
one type of strand. All strands have Planck radius. The entire universe is one closed strand.
At second sight, the differences between strands and superstrings might be much smaller. If
one adds two ‘strand surface coordinates’ for each strand, a higher number of dimensions can
be imagined to arise. Elementary particles with three strands, with two surface dimensions each,
would give a ten-dimensional space-time. Even Calabi-Yau manifolds might arise in this way.
Also the Planck tension might disappear in this way.
Strands include non-commutativity. An aggregate of crossing strands resembles a fermionic
space. Supersymmetry might be seen as a consequence of ubiquitous crossing switches.
In the strand tangle model, crossings yield observable Lagrangians. The original superstring
Lagrangian might only be a bookkeeping tool, not a real observable Lagrangian. The fundamental
supersymmetry that is assumed in superstring theory might not be observable, and neither might
its super-Lie group be. On the other hand, experiments imply that the idea of grand unification
that is often assumed to exist in superstring theory [311,312] must be abandoned.
Certain supermembranes respect the Planck limits, like strands do. Like strands, certain types
of supermembranes arise in finite numbers in finite volumes of three-dimensional space. If one
imagines that the superstring tension is responsible for forming strands, the resulting strands might
nevertheless be wobbly, as assumed in the strand tangle model. Strands might thus be seen as
tubular supermembranes.
Finally, strands can be thought of as realizing specific forms of gauge/gravity duality, of
AdS/CFT duality, and of holography, provided that these superstring concepts are modified by
taking Planck limits into account. Strands can be considered a way of realizing the freedom from
anomalies that started superstring theory.
In short, there might exist a mapping between certain supermembranes and strands. Testing any
hypothesis connecting supermembranes and strands will require research focusing on a concrete
model for elementary particles. Such a model would have to go beyond the statement that ‘particles
are vibrating superstrings’ and might well be achievable in the coming years.

95
F On the Yang-Mills millennium problem
In the year 2000, the Clay Mathematics Institute asked, in its list of millennium problems, to prove
the existence of a non-trivial quantum field theory for every (non-Abelian) compact simple gauge
group in continuous flat space-time. The problem, formulated by Witten and Jaffe [313], also asks
to prove that each such quantum field theory leads to a finite mass gap. In the millennium problem,
a quantum field theory is defined as a structure realizing the axioms by Streater and Wightman as
well as those by Osterwalder and Schrader. All these axioms are based on perfect locality and
continuous quantum fields. This millennium problem has two aspects: a physical aspect about
nature, and a mathematical aspect about axiomatic quantum field theory.
The physical aspect of the Yang-Mills millennium problem is answered negatively both by
nature and by the strand tangle model. In nature, continuous space-time does not exist, due to the
minimum length. In nature, additional gauge groups do not exist: only the two nuclear interactions
are observed to be non-Abelian gauge theories. Strands confirm the observations by excluding
every imaginable alternative to the standard model (with massive Dirac neutrinos): due to the
existence of just three Reidemeister moves and of the minimum length, additional gauge groups
and additional gauge particles are impossible, as told in the main section of this article. Strands
clarify the relation between the three dimensions of space, the three possible gauge groups, and
the three fermion generations. In particular, strands imply that the emergence of three-dimensional
space and the lack of higher gauge groups are two sides of the same coin. Strands thus answer the
challenge that motivated the millennium problem negatively.
The physical Yang-Mills millennium problem has no relation to nature for four reasons. First,
the Yang-Mills millennium problem explicitly assumes that continuous space, continuous fields,
perfect locality, perfectly local operators and perfect point particles exist. Thus, the millennium
problem explicitly disregards the minimum length and all the other Planck limits of relativis-
tic quantum gravity. Secondly, the millennium problem disregards the way that spatial three-
dimensionality arises in nature. Thirdly, the millennium problem disregards the way that particles,
quantum fields, and gauge groups arise in nature. Fourthly, the millennium problem disregards the
way that particle masses and couplings arise in nature.
All four reasons for the lack of any relation between the Yang-Mills millennium problem and
nature are independent of the strand tangle model. Whatever the unified description of nature
may be, it does not comply with the assumptions of spatial continuity and of locality in the Yang-
Mills millennium problem. Whatever the unified description may be, it will, by definition, explain
why three-dimensionality arises. Whatever the unified description may be, it will, by definition,
explain why only the observed gauge interactions exist. Whatever the unified description may be,
it will, by definition, explain why only the observed particle masses and couplings occur. Thus,
the statement of the Yang-Mills millennium problem contradicts nature and observations.
In summary, if the Yang-Mills millennium problem is restricted to the physical aspect, namely
the existence of additional quantum field theories in nature, the answer is that such theories do not
exist. This has been known experimentally for roughly a century. Strands confirm and derive this
observational result, possibly for the first time. The best one can do is to restrict the Yang-Mills
millennium problem to proving the existence of a finite mass gap in the special case of the gauge
group SU(3) of the strong nuclear interaction (in vacuum). Only this case has a positive answer,
due to the existence of glueballs – both in observations [68] and in the strand tangle model [121].

96
The mathematical aspect of the millennium problem has further issues. Continuous space is
in contrast with the minimum length in nature. In addition, continuous flat space has no units of
length and time. Units of length and time are needed to discover and determine the quantum of
action ~. Continuous space is in contrast with quantum field theory. But there is more.
In mathematics, one can choose to ignore nature and physics and assume both continuity and
quantum field theory at the same time. This leads to a further issue. The statement of the millen-
nium problem defines a quantum field theory as a structure realizing specific axioms. The axioms
assume that the quantum of action ~and the speed of light care finite. Now, exact space continuity
implies that the Planck length is zero. This implies that the gravitational constant Gmust vanish.
A vanishing Gis in contrast to observations. It also implies that mass units cannot be defined and
mass values cannot be measured.
In mathematics, one can choose to ignore nature and physics and assume continuity, quantum
field theory, and non-vanishing G. However, if one insists that Gis finite after all, then one has a
non-vanishing minimum length. The minimum length implies that there are no points, no sets and
no axioms, as deduced in detail in Appendix G. Quantum field theory and finite G, taken together,
imply that there is no axiomatic quantum field theory.
In mathematics, one can choose to ignore nature and physics further and assume both that
continuity and axiomatic quantum field theory are valid at the same time. The two assumptions,
taken together, imply the description of particles as exact point particles. This purely mathematical
result contradicts general relativity: no object can be smaller than the Schwarzschild radius deter-
mined by its mass. Thus the concepts of ‘particle’ and ‘quantum field’ assumed in the millennium
problem contradict general relativity and have no relation to actual particles in nature or actual
quantum fields. The two mathematical assumptions have a further unfortunate consequence. Dif-
ferent elementary particles differ at least at the Planck scale. Only Planck scale differences explain
the existence of different types of quarks and the existence of electrons, muons and tau leptons.
Eliminating the Planck scale by assuming point particles prevents deriving the existence of dif-
ferent types of elementary particles. Assuming point particles prevents deriving the three particle
generations. This implies that the existence of different elementary particles cannot be derived but
must be assumed. In particular, it is impossible to deduce whether additional gauge bosons exist.
Therefore, it is impossible to derive whether additional gauge interactions with additional gauge
groups exist. It is equally impossible to derive whether additional particles, such as generalizations
of glueballs for additional gauge interaction, exist. Thus, it is impossible to deduce the existence
of any mass parameter or any mass gap.
In short, the three limits of nature – the speed limit c, the quantum of action ~, and the max-
imum force c4/4G– imply a minimum length, the lack of continuity of space, and the lack of
perfect locality. These properties are essential for the appearance of wave functions, quantum field
theory, gauge theories, elementary particles, measurement units, mass, physical observables, and
the three dimensions. In contrast, the Yang-Mills millennium problem starts with other assump-
tions: continuity of space, perfect locality, point-like particles, axiomatic quantum field theory,
gravity in flat space, perfect three-dimensionality, and predetermined gauge bosons and mass gen-
eration mechanisms. Due to the logical contradictions in the assumptions, a mathematical proof of
the existence of additional Yang-Mills theories and their mass gaps is impossible. The only known
explanation for gauge theories and gauge groups, the strand tangle model, implies the lack of any
Yang-Mills theory beyond SU(3) and the existence of a finite mass gap for SU(3) [121].

97
G On physics’ axioms and Hilbert’s sixth problem
The following arguments do not rely on strands. However, strands allow visualizing them.
Combining maximum speed c, the quantum of action ~, and maximum force c4/4Gyields the
existence of a minimum length in nature, given by twice the Planck length. The minimum length
is also the smallest possible length measurement error. The same applies to time. Therefore,
measurement results without errors are impossible – for every physical observable.
The lack of exact measurement results for all physical observables implies the lack of sharp
boundaries in nature. There is no exact way to distinguish two values or even two objects. In
nature, there is no way to separate space into separate regions. At the Planck scale, nature is
inherently fuzzy. Nature has no parts that can be distinguished exactly. Nature has no parts.
The lack of sharp boundaries implies the lack of sets in nature. The lack of exact distinctions
and of exact parts implies the impossibility of defining elements in nature.
In mathematics, all axioms are based on distinctions, elements, sets and boundaries. However,
all these concepts do not exist in nature, because of the smallest length and time measurement
errors. The minimum length implies the lack of axioms for nature: physics as a whole cannot be
based on axioms. Hilbert’s dream [314], most clearly stated in his sixth problem [315,316], in
which he asked for an axiomatic system for all of physics, cannot be realized whenever gravity
and quantum theory are combined. (The minimum length also avoids Gödel’s problems [317].)
In contrast, branches of physics can be based on axioms. Axioms are possible in quantum field
theory without gravity, or in relativistic gravity without quantum theory – because in these domains
the minimum length does not arise. In branches of physics, length and time measurements without
errors and infinitely precise can be assumed. Only the combination of quantum theory, relativity,
and gravity introduces minimum errors and eliminates the possibility of defining axioms.
Despite the lack of axioms for unification, a complete and unified description of nature remains
possible: it just has to be logically circular. Strands provide such a logical circular description:
(background) space and time are part of the fundamental principle while, at the same time, (physi-
cal) space and time arise from averaging strands. Even though space and time are approximations,
they must be used to talk about nature. There is no way to describe nature without space and time.
The fundamental constants c,~,Gand k– and thus all measured quantities, directly or indirectly
– have the metre and the second in their units and use space and time in their fundamental im-
plementations. Space, time and boundaries are necessary to think, to define information, and to
communicate. Even the act of counting requires space and time. A description of nature without a
space-time background is impossible.
Because of the minimum measurement errors in relativistic quantum gravity, as argued in Sec-
tion 2, no equations exist in this domain. A statistical description of nature is necessary.
The first mathematical proposal that eliminates exact parts has been given by Kauffman [318,
319]. In his proposal, all sets and elements are approximate concepts that result from the folding
of a unique fundamental entity. Kauffman’s proposal is realized by the fundamental principle of
Figure 7. In the strand tangle model, nature consists of a single closed strand and all parts are
approximate. The strand tangle model confirms the unity of nature and the lack of exact parts in
nature. All ‘parts’ of nature – points, instants, particles – are low-energy approximations. An early
proponent of this approach was Dante: in 1320, in the last canto of his Commedia, he described
the universe as a knot spread out all over nature.

98
In short, every unified description of motion incorporating relativistic quantum gravity im-
plies measurement errors for length and time. Therefore, precise distinctions are impossible at the
Planck scale – and so is an axiomatic theory of relativistic quantum gravity. The lack of axioms
requires that a unified description is logically circular, statistical, and contains no parts. Although
relativistic quantum gravity implies that points and instants do not exist, emergent space and time
made of points must be used as communication tools to describe nature. Fluctuating strands with
Planck radius whose crossing switches define the quantum of action incorporate these require-
ments. No other option is known or appears possible.
[1] M. Planck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der königlich-preußischen
Akademie der Wissenschaften zu Berlin 5, 440 (1899).
[2] R. Penrose and W. Rindler, Spinors and space-time: Volume 1, Two-spinor calculus and relativistic
fields, Vol. 1 (Cambridge University Press, 1984).
[3] N. Bohr, Atomtheorie und Naturbeschreibung (Springer, 1931).
[4] A. Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik 4(1905).
[5] E. A. Rauscher, The Minkowski metric for a multidimensional geometry, Lett. Nuovo Cim. 7S2, 361
(1973).
[6] G. W. Gibbons, The Maximum Tension Principle in General Relativity, Found. Phys. 32, 1891
(2002),arXiv:hep-th/0210109.
[7] C. Schiller, Maximum force and minimum distance: physics in limit statements (2003),
arXiv:physics/0309118.
[8] C. Schiller, General Relativity and Cosmology Derived From Principle of Maximum Power or Force,
Int. J. Theor. Phys. 44, 1629 (2005),arXiv:physics/0607090.
[9] C. Schiller, Comment on “Maximum force and cosmic censorship”, Phys. Rev. D 104, 068501
(2021),arXiv:2109.07700 [gr-qc].
[10] C. Schiller, Tests for maximum force and maximum power, Phys. Rev. D 104, 124079 (2021),
arXiv:2112.15418 [gr-qc].
[11] K. S. Thorne, Nonspherical gravitational collapse – a short review, in Magic without Magic. John
Archibald Wheeler. A collection of essays in honor of his sixtieth birthday, edited by J. Klauder
(Freeman, 1972) p. 231–258.
[12] H. J. Treder, The planckions as largest elementary particles and as smallest test bodies, Found. Phys.
15, 161 (1985).
[13] R. J. Heaston, Identification of a Superforce in the Einstein Field Equations, Journal of the Washing-
ton Academy of Sciences 80, 25 (1990).
[14] V. de Sabbata and C. Sivaram, On limiting field strengths in gravitation, Found. Phys. Lett. 6, 561
(1993).
[15] C. Massa, Does the gravitational constant increase?, Astrophys. Space Sci. 232, 143 (1995).
[16] L. Kostro and B. Lange, Is c**4/G the greatest possible force in nature?, Phys. Essays 12, 182 (1999).
[17] C. Schiller, Simple Derivation of Minimum Length, Minimum Dipole Moment and Lack of Space-
time Continuity, Int. J. Theor. Phys. 45, 221 (2006).
[18] J. Barrow and G. Gibbons, Maximum tension: with and without a cosmological constant, Mon. Not.
Roy. Astron. Soc. 446, 3874 (2014).
[19] M. P. Dabrowski and H. Gohar, Abolishing the maximum tension principle, Phys. Lett. B 748, 428
(2015),arXiv:1504.01547 [gr-qc].
[20] M. R. R. Good and Y. C. Ong, Are black holes springlike?, Phys. Rev. D 91, 044031 (2015),
arXiv:1412.5432 [gr-qc].
[21] Y. L. Bolotin, V. A. Cherkaskiy, A. V. Tur, and V. V. Yanovsky, An ideal quantum clock and Principle
of maximum force, arxiv (2016), arXiv:1604.01945 [gr-qc].

99
[22] V. Cardoso, T. Ikeda, C. J. Moore, and C.-M. Yoo, Remarks on the maximum luminosity, Phys. Rev.
D97, 084013 (2018),arXiv:1803.03271 [gr-qc].
[23] Y. C. Ong, GUP-corrected black hole thermodynamics and the maximum force conjecture, Phys.
Lett. B 785, 217 (2018),arXiv:1809.00442 [gr-qc].
[24] J. D. Barrow, Non-Euclidean Newtonian cosmology, Class. Quant. Grav. 37, 125007 (2020),
arXiv:2002.10155 [gr-qc].
[25] J. D. Barrow, Maximum force and naked singularities in higher dimensions, Int. J. Mod. Phys. D 29,
2043008 (2020),arXiv:2005.06809 [gr-qc].
[26] J. D. Barrow and N. Dadhich, Maximum force in modified gravity theories, Phys. Rev. D 102, 064018
(2020),arXiv:2006.07338 [gr-qc].
[27] K. Atazadeh, Maximum force conjecture in Kiselev, 4D-EGB and Barrow corrected-entropy black
holes, Phys. Lett. B 820, 136590 (2021).
[28] A. Jowsey and M. Visser, Counterexamples to the maximum force conjecture, Universe 7(2021),
arXiv:2102.01831 [gr-qc].
[29] V. Faraoni, Maximum force and cosmic censorship, Phys. Rev. D 103, 124010 (2021),
arXiv:2105.07929 [gr-qc].
[30] V. Faraoni, Reply to “Comment on ‘Maximum force and cosmic censorship”’, Phys. Rev. D 104,
068502 (2021).
[31] L.-M. Cao, L.-Y. Li, and L.-B. Wu, Bound on the rate of Bondi mass loss, Phys. Rev. D 104, 124017
(2021),arXiv:2109.05973 [gr-qc].
[32] N. Dadhich, Maximum Force for Black Holes and Buchdahl Stars (2022), arXiv:2201.10381 [gr-qc].
[33] C. J. Hogan, Energy Flow in the Universe, NATO Sci. Ser. C 565, 283 (2001),arXiv:astro-
ph/9912110.
[34] V. G. Gurzadyan and A. Stepanian, Hubble tension and absolute constraints on the local Hubble
parameter, arxiv (2021), arXiv:2108.07407 [astro-ph.CO].
[35] S. Di Gennaro, M. R. R. Good, and Y. C. Ong, Black hole Hookean law and thermodynamic frag-
mentation: Insights from the maximum force conjecture and Ruppeiner geometry, Phys. Rev. Res.
4, 023031 (2022),arXiv:2108.13435 [gr-qc].
[36] G. E. Volovik, Negative Newton constant may destroy some conjectures, Mod. Phys. Lett. A 37,
2250034 (2022),arXiv:2202.12743 [gr-qc].
[37] A. Jowsey and M. Visser, Reconsidering maximum luminosity, Int. J. Mod. Phys. D 30, 2142026
(2021),arXiv:2105.06650 [gr-qc].
[38] D. W. Sciama, Gravitational radiation and general relativity, Phys. Bull. 24, 657 (1973).
[39] C. Sivaram, A general upper limit on the mass and entropy production of a cluster of supermassive
objects, Astrophysics and Space Science 86, 501 (1982).
[40] L. Kostro, The quantity c**5/G interpreted as the greatest possible power in nature, Phys. Essays 13,
143 (2000).
[41] V. Cardoso, T. Ikeda, C. J. Moore, and C.-M. Yoo, Remarks on the maximum luminosity, Phys. Rev.
D97, 084013 (2018),arXiv:1803.03271 [gr-qc].
[42] A. Loeb, Five novel observational tests of general relativity (2022), arXiv:2205.02746.
[43] C. Schiller, From maximum force via the hoop conjecture to inverse square gravity, Gravitation and
Cosmology 28, 304 (2022).
[44] A. Kenath, C. Schiller, and C. Sivaram, From maximum force to the field equations of general
relativity - and implications, Int. J. Mod. Phys. D 31, 2242019 (2022),arXiv:2205.06302 [gr-qc].
[45] B. P. Abbott et al. (LIGO Scientific, Virgo), GWTC-1: A Gravitational-Wave Transient Catalog of
Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs,
Phys. Rev. X 9, 031040 (2019),arXiv:1811.12907 [astro-ph.HE].
[46] R. Abbott et al. (LIGO Scientific, Virgo), Properties and Astrophysical Implications of the 150
MBinary Black Hole Merger GW190521, Astrophys. J. Lett. 900, L13 (2020),arXiv:2009.01190
[astro-ph.HE].
[47] C. M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Rel. 17, 4
(2014),arXiv:1403.7377 [gr-qc].

100
[48] H. W. Zimmermann, Particle Entropies and Entropy Quanta. I. The Ideal Gas, Z. Phys. Chem. 214,
187 (2000).
[49] H. W. Zimmermann, Particle Entropies and Entropy Quanta II. The Photon Gas, Z. Phys. Chem. 214,
347 (2000).
[50] H. W. Zimmermann, Particle Entropies and Entropy Quanta. III. The van der Waals Gas, Z. Phys.
Chem. 216, 615 (2002).
[51] H. W. Zimmermann, Particle Entropies and Entropy Quanta: IV. The Ideal Gas, the Second Law of
Thermodynamics, and the P–t Uncertainty Relation, Z. Phys. Chem. 217, 55 (2003).
[52] H. W. Zimmermann, Particle Entropies and Entropy Quanta V. The P–t Uncertainty Relation, Z.
Phys. Chem. 217, 1097 (2003).
[53] U. Hohm and C. Schiller, Testing the minimum system entropy and the quantum of entropy, Entropy
25,10.3390/e25111511 (2023).
[54] C. A. Mead, Possible connection between gravitation and fundamental length, Physical Review 135,
B849 (1964).
[55] F. Karolyhazy, Gravitation and quantum mechanics of macroscopic objects, Il Nuovo Cimento A
(1965-1970) 42, 390 (1966).
[56] L. J. Garay, Quantum gravity and minimum length, Int. J. Mod. Phys. A 10, 145 (1995),arXiv:gr-
qc/9403008.
[57] M. Maggiore, A generalized uncertainty principle in quantum gravity, Physics Letters B 304, 65
(1993).
[58] A. Kempf, G. Mangano, and R. B. Mann, Hilbert space representation of the minimal length uncer-
tainty relation, Physical Review D 52, 1108 (1995).
[59] X. Calmet, M. Graesser, and S. D. Hsu, Minimum length from quantum mechanics and classical
general relativity, Physical Review Letters 93, 211101 (2004).
[60] X. Calmet, M. L. Graesser, and S. D. Hsu, Minimum length from first principles, International Jour-
nal of Modern Physics D 14, 2195 (2005).
[61] X. Calmet, On the precision of a length measurement, The European Physical Journal C 54, 501
(2008).
[62] S. Hossenfelder, Can we measure structures to a precision better than the Planck length?, Classical
and Quantum Gravity 29, 115011 (2012).
[63] S. Hossenfelder, Minimal length scale scenarios for quantum gravity, Living Reviews in Relativity
16, 2 (2013).
[64] A. N. Tawfik and A. M. Diab, A review of the generalized uncertainty principle, Reports on Progress
in Physics 78, 126001 (2015).
[65] T. P. Singh, Quantum gravity, minimum length and holography, Pramana 95, 1 (2021).
[66] C. Schiller, From maximum force to physics in 9 lines towards relativistic quantum gravity,
Zeitschrift für Naturforschung A (2022),arXiv:2208.01038 [gr-qc].
[67] P. Antonini, M. Okhapkin, E. Goklu, and S. Schiller, Test of constancy of speed of light with rotating
cryogenic optical resonators, Phys. Rev. A 71, 050101(R) (2005),arXiv:gr-qc/0504109.
[68] R. Workman et al. (Particle Data Group), Review of Particle Physics, Prog. Theor. Exp. Phys. 2022,
083C01 (2022).
[69] A. Addazi, J. Alvarez-Muniz, R. A. Batista, G. Amelino-Camelia, V. Antonelli, M. Arzano,
M. Asorey, J.-L. Atteia, S. Bahamonde, F. Bajardi, et al., Quantum gravity phenomenology at the
dawn of the multi-messenger era—a review, Progress in Particle and Nuclear Physics 125, 103948
(2022).
[70] V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeßler, F. J. Rueß, T. Stöferle,
A. Westphal, A. M. Gagarski, G. A. Petrov, et al., Quantum states of neutrons in the earth’s gravita-
tional field, Nature 415, 297 (2002).
[71] H. Abele, S. Baeßler, and A. Westphal, Quantum States of Neutrons in the Gravitational Field and
Limits for Non-Newtonian Interaction in the Range between 1 µm and 10 µm, Quantum Gravity:
From Theory to Experimental Search , 355 (2003).
[72] V. Nesvizhevsky and K. Protasov, Quantum states of neutrons in the earth’s gravitational field: state

101
of the art, applications, perspectives, Trends in Quantum Gravity Research , 65 (2006).
[73] V. Andreev, D. G. Ang, D. DeMille, J. M. Doyle, G. Gabrielse, J. Haefner, N. R. Hutzler, Z. Lasner,
C. Meisenhelder, B. R. O’Leary, C. D. Panda, A. D. West, E. P. West, X. Wu, and A. Collaboration,
Improved limit on the electric dipole moment of the electron, Nature 562, 355 (2018).
[74] T. S. Roussy, L. Caldwell, T. Wright, W. B. Cairncross, Y. Shagam, K. B. Ng, N. Schlossberger, S. Y.
Park, A. Wang, J. Ye, et al., An improved bound on the electron’s electric dipole moment, Science
381, 46 (2023).
[75] D. J. Bird, S. Corbato, H. Dai, J. Elbert, K. Green, M. Huang, D. Kieda, S. Ko, C. Larsen, E. Loh,
et al., Detection of a cosmic ray with measured energy well beyond the expected spectral cutoff due
to cosmic microwave radiation, Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 441, no. 1, p.
144-150 441, 144 (1995).
[76] J. Bekenstein, Black holes and the second law, Lettere al Nuovo Cimento 4, 737 (1972).
[77] S. W. Hawking, Particle creation by black holes, Communications in mathematical physics 43, 199
(1975).
[78] H. Nicolai, Quantum gravity: The view from particle physics, in General Relativity, Cosmology
and Astrophysics: Perspectives 100 years after Einstein’s stay in Prague, edited by J. Biˇ
cák and
T. Ledvinka (Springer International Publishing, Cham, 2014) pp. 369–387.
[79] G. Weber, Thermodynamics at boundaries, Nature 365, 792 (1993).
[80] E. P. Verlinde and M. R. Visser, Black hole entropy and long strings, International Journal of Modern
Physics D 31, 2242006 (2022).
[81] S. Carlip, Dimension and Dimensional Reduction in Quantum Gravity, Classical and Quantum Grav-
ity 34, 193001 (2017),arXiv:1705.05417 [gr-qc].
[82] M. Botta Cantcheff, Spacetime geometry as statistic ensemble of strings (2011), arXiv:1105.3658.
[83] L. Bombelli and B. B. Pilgrim, A tale of two actions a variational principle for two-dimensional
causal sets, Classical and Quantum Gravity 38, 085014 (2021).
[84] A. F. Jercher, D. Oriti, and A. G. A. Pithis, Complete Barrett-Crane model and its causal structure,
Phys. Rev. D 106, 066019 (2022),arXiv:2206.15442 [gr-qc].
[85] T. Asselmeyer-Maluga and H. Rosé, On the geometrization of matter by exotic smoothness, General
Relativity and Gravitation 44, 2825 (2012).
[86] T. Asselmeyer-Maluga, Braids, 3-manifolds, elementary particles: Number theory and symmetry in
particle physics, Symmetry 11,10.3390/sym11101298 (2019).
[87] M. Gardner, Riddles of the Sphinx and Other Mathematical Puzzle Tales (Mathematical Association
of America, 1987) p. 47.
[88] J. Hise, Belt trick 2, animation at youtube.com/watch?v=DHFdBWU36eY (2017).
[89] A. Martos, Dirac’s belt trick for spin 1/2 particle, animation at vimeo.com/62228139 (2013).
[90] A. Martos, Belt trick for the exchange of two fermions, animation at vimeo.com/62143283 (2013).
[91] J. Hise, Anti-twister (revised), animation at youtube.com/watch?v=LLw3BaliDUQ (2016).
[92] J. Hise, Antitwister 4K, animation at youtube.com/watch?v=eR9ZCwYPhhU (2020).
[93] J. Baez and M. Weiss, Photons, schmotons (2000).
[94] E. P. Battey-Pratt and T. J. Racey, Geometric Model for Fundamental Particles, Int. J. Theor. Phys.
19, 437 (1980).
[95] L. H. Kauffman, New invariants in the theory of knots, The American Mathematical Monthly 95,
195 (1988).
[96] L. H. Kauffman, On knots (Princeton University Press, 1987) Chap. 6.
[97] J. R. Goldman and L. H. Kauffman, Rational tangles, Advances in Applied Mathematics 18, 300
(1997).
[98] H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen
entsprechen, J. Reine Angew. Math. 55, 25 (1858).
[99] L. Kelvin, On vortex atoms, in Proc. R. Soc. Edin, Vol. 6 (1867) pp. 94–105.
[100] W. Thomson, II. On Vortex Atoms, The London, Edinburgh, and Dublin Philosophical Magazine
and Journal of Science 34, 15 (1867).
[101] W. Thomson, 4. On Vortex Atoms, Proceedings of the Royal Society of Edinburgh 6, 94 (1869).

102
[102] L. Faddeev and A. J. Niemi, Knots and particles (1996), arXiv:hep-th/9610193.
[103] R. J. Finkelstein, The elementary particles as quantum knots in electroweak theory, International
Journal of Modern Physics A 22, 4467 (2007).
[104] R. J. Finkelstein, Knots and preons, International Journal of Modern Physics A 24, 2307 (2009).
[105] J. S. Avrin, Knots on a Torus: A model of the elementary particles, Symmetry 4, 39 (2012).
[106] S. O. Bilson-Thompson, A topological model of composite preons (2005), arXiv:hep-ph/0503213.
[107] S. O. Bilson-Thompson, F. Markopoulou, and L. Smolin, Quantum gravity and the standard model,
Classical and Quantum Gravity 24, 3975 (2007).
[108] S. Bilson-Thompson, J. Hackett, L. Kauffman, and L. Smolin, Particle identifications from symme-
tries of braided ribbon network invariants (2008), arXiv:0804.0037.
[109] S. Bilson-Thompson, J. Hackett, and L. H. Kauffman, Particle topology, braids, and braided belts,
Journal of Mathematical Physics 50, 113505 (2009).
[110] S. Bilson-Thompson, J. Hackett, L. Kauffman, Y. Wan, et al., Emergent braided matter of quantum
geometry, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 8, 014 (2012).
[111] S. K. Ng, On a knot model of mesons and baryons (2002), arXiv:hep-th/0210024.
[112] S. K. Ng, Quantum knots and new quantum field theory (2000), arXiv:math/0004151.
[113] S. Wolfram, The Wolfram physics project (2021).
[114] C. Schiller, On the relation between the three Reidemeister moves and the three gauge groups, Inter-
national Journal of Geometrical Methods in Modern Physics 21, 2450057 (2024).
[115] C. Schiller, Deducing the three gauge interactions from the three Reidemeister moves (2009),
arXiv:0905.3905.
[116] C. Schiller, Testing a conjecture on the origin of space, gravity and mass, Indian Journal of Physics
96, 3047 (2022).
[117] C. Schiller, Testing a microscopic model for black holes deduced from maximum force, in A Guide
to Black Holes, edited by A. Kenath (Nova Science, 2023) Chap. 5.
[118] C. Schiller, A Conjecture on Deducing General Relativity and the Standard Model with Its Funda-
mental Constants from Rational Tangles of Strands, Phys. Part. Nucl. 50, 259 (2019).
[119] C. Schiller, Testing a conjecture on the origin of the standard model, Eur. Phys. J. Plus 136, 79
(2021).
[120] C. Schiller, Testing a conjecture on quantum electrodynamics, J. Geom. Phys. 178, 104551 (2022).
[121] C. Schiller, Testing a conjecture on quantum chromodynamics, International Journal of Geometrical
Methods in Modern Physics 20, 2350095 (2023).
[122] S. L. Adler, Quantum Theory as an Emergent Phenomenon: The Statistical Mechanics of Matrix
Models as the Precursor of Quantum Field Theory (Cambridge University Press, 2004).
[123] L. de la Peña, A. Cetto, and A. Hernández, The Emerging Quantum: The Physics Behind Quantum
Mechanics (Springer International Publishing, 2014).
[124] A. Zeilinger, A foundational principle for quantum mechanics, Foundations of Physics 29, 631
(1999).
[125] L. Hardy, Quantum theory from five reasonable axioms (2001).
[126] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Informational derivation of quantum theory, Phys.
Rev. A 84, 012311 (2011).
[127] W. Sulis, A process algebra model of QED, in Journal of Physics: Conference Series, Vol. 701 (IOP
Publishing, 2016) p. 012032.
[128] W. Sulis, A Process Algebra Approach to Quantum Electrodynamics, International Journal of Theo-
retical Physics 56, 3869 (2017).
[129] W. Sulis, The classical-quantum dichotomy from the perspective of the process algebra, Entropy 24,
184 (2022).
[130] I. Licata, Quantum Mechanics Interpretation on Planck Scale, Ukrainian Journal of Physics 65, 17
(2020).
[131] G. Grössing, Sub-quantum thermodynamics as a basis of emergent quantum mechanics, Entropy 12,
1975 (2010).
[132] G. ’t Hooft, Emergent quantum mechanics and emergent symmetries, AIP Conference Proceedings

103
957, 154 (2007).
[133] D. Acosta, P. Fernandez de Cordoba, J. Isidro, and J. G. Santander, An entropic picture of emergent
quantum mechanics, International Journal of Geometric Methods in Modern Physics 9, 1250048
(2012).
[134] D. Minic and S. Pajevic, Emergent “quantum” theory in complex adaptive systems, Modern Physics
Letters B 30, 1650201 (2016).
[135] D. Bedingham and P. Pearle, Continuous-spontaneous-localization scalar-field relativistic collapse
model, Phys. Rev. Research 1, 033040 (2019).
[136] B. Dakic and C. Brukner, Quantum theory and beyond: Is entanglement special? (2009),
arXiv:0911.0695.
[137] A. Grinbaum, Quantum theory as a critical regime of language dynamics, Foundations of Physics
45, 1341 (2015).
[138] J. Walleczek, G. Grössing, P. Pylkkänen, and B. Hiley, Emergent Quantum Mechanics: David Bohm
Centennial Perspectives (MDPI, 2019).
[139] O. Krötenheerdt and S. Veit, Zur Theorie massiver Knoten, Beiträge zur Algebra und Geometrie 5,
61 (1976).
[140] J. Cantarella, R. B. Kusner, and J. M. Sullivan, On the minimum ropelength of knots and links,
Inventiones mathematicae 150, 257 (2002).
[141] W. B. R. Lickorish, Prime knots and tangles, Transactions of the American Mathematical Society
267, 321 (1981).
[142] C. C. Adams, The knot book (American Mathematical Soc., 1994).
[143] J. Cantarella, A. LaPointe, and E. J. Rawdon, Shapes of tight composite knots, Journal of Physics A:
Mathematical and Theoretical 45, 225202 (2012).
[144] K. Reidemeister, Elementare Begründung der Knotentheorie, Abhandlungen aus dem Mathematis-
chen Seminar der Universität Hamburg 5, 24 (1927).
[145] A. Krugly, Statistical physics and thermodynamics of processes in general relativity, Gravitation and
Cosmology 26, 162 (2020).
[146] S. Powers and D. Stojkovic, An event centric approach to modeling quantum systems (2023),
arXiv:2306.14922 [physics.gen-ph].
[147] V. Giovannetti, S. Lloyd, and L. Maccone, Geometric event-based quantum mechanics, New Journal
of Physics 25, 023027 (2023).
[148] J. A. Wheeler, How Come the Quantum?, Annals of the New York Academy of Sciences 480, 304
(1986).
[149] G. K. Francis and L. H. Kauffman, Air on the Dirac strings, Contemporary Mathematics 169, 261
(1994).
[150] H. C. Ohanian, What is spin?, American Journal of Physics 54, 500 (1986).
[151] W. Pauli, The connection between spin and statistics, Physical Review 58, 716 (1940).
[152] A. Blum, From the necessary to the possible: the genesis of the spin-statistics theorem, The European
Physical Journal H 39, 543 (2014).
[153] B. Thaller, Quantenmechanik visualisiert und animiert, animation at https://gams.uni-graz.at (2022).
[154] B. Thaller, Visual quantum mechanics: selected topics with computer-generated animations of
quantum-mechanical phenomena (Springer, 2007).
[155] B. Thaller, Advanced visual quantum mechanics (Springer, 2005).
[156] D. Bohm, R. Schiller, and J. Tiomno, A causal interpretation of the Pauli equation (A), Il Nuovo
Cimento (1955-1965) 1, 48 (1955).
[157] D. Bohm and R. Schiller, A causal interpretation of the Pauli equation (B), Il Nuovo Cimento (1955-
1965) 1, 67 (1955).
[158] R. Feynman, QED: The Strange Theory of Light and Matter (Princeton University Press, 2019).
[159] M. Born and P. Jordan, Zur Quantenmechanik, Zeitschrift für Physik 34, 858 (1925).
[160] J. A. Heras, Can Maxwell’s equations be obtained from the continuity equation?, American Journal
of Physics 75, 652 (2007).
[161] E. Kapuscik, Comment on “Can Maxwell’s equations be obtained from the continuity equation?” by

104
JA Heras [Am. J. Phys. 75, 652–657 (2007)], American Journal of Physics 77, 754 (2009).
[162] J. A. Heras, Reply to “Comment on ‘Can Maxwell’s equations be obtained from the continuity
equation?,’” by E. Kapuscik [Am. J. Phys. 77, 754 (2009)], American Journal of Physics 77, 755
(2009).
[163] J. A. Heras, How to obtain the covariant form of Maxwell’s equations from the continuity equation,
European Journal of Physics 30, 845 (2009).
[164] J. A. Heras, An axiomatic approach to Maxwell’s equations, European Journal of Physics 37, 055204
(2016).
[165] L. Burns, Maxwell’s equations are universal for locally conserved quantities, Advances in Applied
Clifford Algebras 29, 62 (2019).
[166] P. A. M. Dirac, The Principles of Quantum Mechanics, 27 (Oxford University Press, 1981).
[167] M. Namiki, S. Pascazio, and C. Schiller, What is wave-function collapse by measurement?, Physics
Letters A 187, 17 (1994).
[168] E. Joos and H. D. Zeh, The emergence of classical properties through interaction with the environ-
ment, Zeitschrift für Physik B Condensed Matter 59, 223 (1985).
[169] W. Zurek, Decoherence and the transition from quantum to classical, Physics Today 44(10), 36
(1991).
[170] B. Drossel and G. Ellis, Contextual wavefunction collapse: an integrated theory of quantum mea-
surement, New Journal of Physics 20, 113025 (2018).
[171] S. L. Adler, Why decoherence has not solved the measurement problem: a response to PW Anderson,
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern
Physics 34, 135 (2003).
[172] P. Facchi, D. Lidar, and S. Pascazio, Unification of dynamical decoupling and the quantum zeno
effect, Physical Review A 69, 032314 (2004).
[173] M. Namiki, S. Pascazio, and H. Nakazato, Decoherence and quantum measurements (World Scien-
tific, 1997).
[174] H. Ammann, R. Gray, I. Shvarchuck, and N. Christensen, Quantum delta-kicked rotor: Experimental
observation of decoherence, Physical Review Letters 80, 4111 (1998).
[175] J.-M. Raimond, M. Brune, and S. Haroche, Manipulating quantum entanglement with atoms and
photons in a cavity, Reviews of Modern Physics 73, 565 (2001).
[176] A. J. Leggett, Testing the limits of quantum mechanics: motivation, state of play, prospects, Journal
of Physics: Condensed Matter 14, R415 (2002).
[177] M. Arndt, K. Hornberger, and A. Zeilinger, Probing the limits of the quantum world, Physics World
18, 35 (2005).
[178] I. Takagi, Y. Kayanuma, and K. G. Nakamura, Ultrafast quantum path interferometry to determine
the electronic decoherence time of the electron-phonon coupled system in n-type gallium arsenide,
Physical Review B 107, 184305 (2023).
[179] M. Tegmark, Apparent wave function collapse caused by scattering, Foundations of Physics Letters
6, 571 (1993).
[180] A. Caldeira and A. J. Leggett, Influence of damping on quantum interference: An exactly soluble
model, Physical Review A 31, 1059 (1985).
[181] R. Gambini and J. Pullin, The Montevideo interpretation of quantum mechanics: a short review,
Entropy 20, 413 (2018).
[182] R. Gambini and J. Pullin, The Montevideo interpretation: how the inclusion of a quantum gravita-
tional notion of time solves the measurement problem, Universe 6, 236 (2020).
[183] D. Bohm and Y. Aharonov, Discussion of Experimental Proof for the Paradox of Einstein, Rosen,
and Podolsky, Phys. Rev. 108, 1070 (1957).
[184] E. S. Fry, Bell inequalities and two experimental tests with mercury, Quantum and Semiclassical
Optics: Journal of the European Optical Society Part B 7, 229 (1995).
[185] P. Aravind, Borromean entanglement of the GHZ state, in Potentiality, entanglement and passion-
at-a-distance (Springer, 1997) pp. 53–59.
[186] L. H. Kauffman and S. J. Lomonaco, Quantum entanglement and topological entanglement, New

105
Journal of Physics 4, 73 (2002).
[187] S. Mironov, Topological entanglement and knots, Universe 5(2019).
[188] D. Melnikov, A. Mironov, S. Mironov, A. Morozov, and A. Morozov, From topological to quantum
entanglement, Journal of High Energy Physics 2019, 116 (2019).
[189] G. M. Quinta and R. André, Classifying quantum entanglement through topological links, Physical
Review A 97, 042307 (2018).
[190] L. H. Kauffman and E. Mehrotra, Topological aspects of quantum entanglement, Quantum Informa-
tion Processing 18, 1 (2019).
[191] S. Heusler, P. Schlummer, and M. S. Ubben, The topological origin of quantum randomness, Sym-
metry 13 (2021).
[192] P. M. Tam, M. Claassen, and C. L. Kane, Topological Multipartite Entanglement in a Fermi Liquid,
Phys. Rev. X 12, 031022 (2022).
[193] N. Bao, N. Cheng, S. Hernández-Cuenca, and V. P. Su, Topological Link Models of Multipartite
Entanglement, Quantum 6, 741 (2022).
[194] N. D. Mermin, Bringing home the atomic world: Quantum mysteries for anybody, American Journal
of Physics 49, 940 (1981).
[195] S. Filatov and M. Auzinsh, Towards Two Bloch Sphere Representation of Pure Two Qubit States and
Unitaries (2024), arXiv:2403.10587 [quant-ph].
[196] S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, in The logico-
algebraic approach to quantum mechanics (Springer, 1975) pp. 293–328.
[197] S. Weinberg, Testing quantum mechanics, Annals of Physics 194, 336 (1989).
[198] N. Gisin, Weinberg’s non-linear quantum mechanics and supraluminal communications, Physics
Letters A 143, 1 (1990).
[199] P. Majumder, B. Venema, S. Lamoreaux, B. Heckel, and E. Fortson, Test of the linearity of quantum
mechanics in optically pumped hg 201, Physical Review Letters 65, 2931 (1990).
[200] S. Kochen, A reconstruction of quantum mechanics, Foundations of Physics 45, 557 (2015).
[201] R. Colbeck and R. Renner, No extension of quantum theory can have improved predictive power,
Nature Communications 2, 411 (2011).
[202] R. Colbeck and R. Renner, Is a system’s wave function in one-to-one correspondence with its ele-
ments of reality?, Physical Review Letters 108, 150402 (2012).
[203] R. Colbeck and R. Renner, The completeness of quantum theory for predicting measurement out-
comes, in Quantum Theory: Informational Foundations and Foils, edited by G. Chiribella and R. W.
Spekkens (Springer Netherlands, Dordrecht, 2016) pp. 497–528.
[204] M. F. Pusey, J. Barrett, and T. Rudolph, On the reality of the quantum state, Nature Physics 8, 475
(2012).
[205] E. Schrödinger, Die Wellenmechanik (E. Battenberg, 1963).
[206] D. Hestenes, Zitterbewegung in quantum mechanics, Foundations of Physics 40, 1 (2010).
[207] D. Hestenes, Quantum mechanics of the electron particle-clock (2019), arXiv:1910.10478.
[208] D. Hestenes, Zitterbewegung structure in electrons and photons (2019), arXiv:1910.11085.
[209] A. M. Steane, Relativity Made Relatively Easy Volume 2: General Relativity and Cosmology (Oxford
University Press, 2021) Chap. 27.
[210] A. M. Steane, An introduction to spinors (2013), arXiv:1312.3824.
[211] W. Payne, Elementary spinor theory, American Journal of Physics 20, 253 (1952).
[212] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique quantique, Vol. I–II (Hermann Paris, 1973).
[213] N. Dombey and A. Calogeracos, Seventy years of the Klein paradox, Physics Reports 315, 41 (1999).
[214] V. Fock and D. Iwanenko, Über eine mögliche geometrische Deutung der relativistischen Quanten-
theorie, Zeitschrift für Physik 54, 798 (1929).
[215] V. Simulik and I. Krivsky, Once more on the derivation of the Dirac equation (2013),
arXiv:1309.0573.
[216] I. Krivsky, V. Simulik, I. Lamer, and Z. Taras, The Dirac equation as the consequence of the quantum-
mechanical spin 1/2 doublet model, TWMS Journal of Applied and Engineering Mathematics 3, 62
(2013).
[217] V. Simulik, Relativistic quantum mechanics and field theory of arbitrary spin (Nova Science Pub-

106
lishers, 2020) Chap. 2.
[218] A. Loinger and A. Sparzani, Dirac equation without Dirac matrices, Il Nuovo Cimento (1955-1965)
39, 1140 (1965).
[219] L. Lerner, Derivation of the Dirac equation from a relativistic representation of spin, European Jour-
nal of Physics 17, 172 (1996).
[220] W. E. Baylis, Surprising symmetries in relativistic charge dynamics (2004), arXiv:physics/0410197.
[221] W. Baylis, R. Cabrera, and J. Keselica, Quantum/classical interface: classical geometric origin of
fermion spin, Advances in applied Clifford algebras 20, 517 (2010).
[222] S. L. Adler, Quantum theory as an emergent phenomenon: Foundations and phenomenology, Journal
of Physics: Conference Series 361, 012002 (2012).
[223] S. L. Adler and A. C. Millard, Generalized quantum dynamics as pre-quantum mechanics, Nuclear
Physics B 473, 199 (1996).
[224] G. ’t Hooft, Emergent Quantum Mechanics and Emergent Symmetries, AIP Conference Proceedings
957, 154 (2007),https://pubs.aip.org/aip/acp/article-pdf/957/1/154/11584702/154_1_online.pdf.
[225] H.-T. Elze, Does quantum mechanics tell an atomistic spacetime?, Journal of Physics: Conference
Series 174, 012009 (2009).
[226] L. De la Peña, A. Valdés-Hernández, and A. Cetto, Quantum mechanics as an emergent property of
ergodic systems embedded in the zero-point radiation field, Foundations of Physics 39, 1240 (2009).
[227] M. Blasone, P. Jizba, and F. Scardigli, Can quantum mechanics be an emergent phenomenon?, Jour-
nal of Physics: Conference Series 174, 012034 (2009).
[228] G. Grössing, Sub-quantum thermodynamics as a basis of emergent quantum mechanics, Entropy 12,
1975 (2010).
[229] G. Grössing, J. Mesa Pascasio, and H. Schwabl, A classical explanation of quantization, Foundations
of Physics 41, 1437 (2011).
[230] T. J. Hollowood, The emergent Copenhagen interpretation of quantum mechanics, Journal of Physics
A: Mathematical and Theoretical 47, 185301 (2014).
[231] R. G. Torromé, Emergent quantum mechanics and the origin of quantum non-local correlations,
International Journal of Theoretical Physics 56, 3323 (2017).
[232] D. Deutsch, The beginning of infinity: Explanations that transform the world (Penguin UK, 2011).
[233] G. P. Galdi, On the motion of a rigid body in a viscous fluid: a mathematical analysis with applica-
tions, Handbook of Mathematical Fluid Mechanics 1, 653 (2002).
[234] O. Gonzalez, A. Graf, and J. Maddocks, Dynamics of a rigid body in a Stokes fluid, Journal of Fluid
Mechanics 519, 133 (2004).
[235] M. Rahmani and A. Wachs, Free falling and rising of spherical and angular particles, Physics of
Fluids 26, 083301 (2014).
[236] P. Ern, F. Risso, D. Fabre, and J. Magnaudet, Wake-induced oscillatory paths of bodies freely rising
or falling in fluids, Annual Review of Fluid Mechanics 44, 97 (2012).
[237] M. Khlopov, What comes after the standard model?, Progress in Particle and Nuclear Physics 116,
103824 (2021).
[238] V. Berezin, Markov’s maximon and quantum black holes, Physics of Particles and Nuclei 29, 274
(1998).
[239] N. E. Mavromatos and V. A. Mitsou, Magnetic monopoles revisited: Models and searches at colliders
and in the cosmos, International Journal of Modern Physics A 35, 2030012 (2020).
[240] N. J. Popławski, Nonsingular Dirac particles in spacetime with torsion, Physics Letters B 690, 73
(2010).
[241] A. Burinskii, The Dirac-Kerr-Newman electron, Gravitation and Cosmology 14, 109 (2008).
[242] A. Burinskii, Emergence of the Dirac equation in the solitonic source of the Kerr spinning particle,
Gravitation and Cosmology 21, 28 (2015).
[243] C. Bouchiat, J. Iliopoulos, and P. Meyer, An anomaly-free version of Weinberg’s model, Physics
Letters B 38, 519 (1972).
[244] C. Geng and R. E. Marshak, Uniqueness of quark and lepton representations in the standard model
from the anomalies viewpoint, Physical Review D 39, 693 (1989).

107
[245] S. Coleman and J. Mandula, All possible symmetries of the S matrix, Physical Review 159, 1251
(1967).
[246] R. Boudet, Quantum Mechanics in the Geometry of Space-time: Elementary Theory (Springer,
2011).
[247] K. G. Katamadze, A. V. Pashchenko, A. V. Romanova, and S. P. Kulik, Generation and application
of broadband biphoton fields (brief review), JETP Letters 115, 581 (2022).
[248] F. Chadha-Day, J. Ellis, and D. J. Marsh, Axion dark matter: What is it and why now?, Science
Advances 8, eabj3618 (2022).
[249] L. Freidel, J. Kowalski-Glikman, R. G. Leigh, and D. Minic, Theory of metaparticles, Physical
Review D 99, 066011 (2019).
[250] G. F. Chew, Quantum cosmology based on discrete Feynman paths, Tech. Rep. (Lawrence Berkeley
National Lab.(LBNL), Berkeley, CA (United States), 2002).
[251] M. Milgrom, MOND–particularly as modified inertia, Acta Physica Polonica B 42 (2011).
[252] M. Milgrom, Models of modified-inertia formulation of MOND (2022).
[253] Z.-Z. Xing, H. Zhang, and S. Zhou, Impacts of the Higgs mass on vacuum stability, running fermion
masses, and two-body Higgs decays, Physical Review D 86, 013013 (2012).
[254] G.-Y. Huang and S. Zhou, Precise values of running quark and lepton masses in the standard model,
Physical Review D 103, 016010 (2021).
[255] G. Dietler, R. Kusner, W. Kusner, E. Rawdon, and P. Szymczak, Chirality for crooked curves (2020),
arXiv:2004.10338.
[256] M. Veltman, Diagrammatica: the path to Feynman diagrams (Cambridge University Press, 1994)
pp. 249–272.
[257] J. C. Romao and J. P. Silva, A resource for signs and Feynman diagrams of the Standard Model,
International Journal of Modern Physics A 27, 1230025 (2012).
[258] W. N. Cottingham and D. A. Greenwood, An introduction to the standard model of particle physics
(Cambridge University Press, 2007).
[259] R. Jackiw, N. N. Khuri, S. S. Schweber, and E. Witten, Shelter Island II: Proceedings of the 1983
Shelter Island conference on quantum field theory and the fundamental problems of physics (Courier
Dover Publications, 2016).
[260] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (CRC Press, 2018).
[261] T. P. Singh, Quantum theory without classical time: octonions, and a theoretical derivation of the
fine structure constant 1/137, International Journal of Modern Physics D 30, 2142010 (2021).
[262] T. P. Singh, Gravitation, and quantum theory, as emergent phenomena, Journal of Physics: Confer-
ence Series 2533, 012013 (2023).
[263] G. C. Ghirardi, A. Rimini, and T. Weber, Unified dynamics for microscopic and macroscopic sys-
tems, Phys. Rev. D 34, 470 (1986).
[264] D. L. Danielson, G. Satishchandran, and R. M. Wald, Gravitationally mediated entanglement: New-
tonian field versus gravitons, Physical Review D 105, 086001 (2022).
[265] M. Arzano, V. D’Esposito, and G. Gubitosi, Fundamental decoherence from quantum spacetime,
Communications Physics 6, 242 (2023).
[266] A. Khalil and W. A. Horowitz, Initial state factorization and the kinoshita-lee-nauenberg theorem,
Journal of Physics: Conference Series 889, 012002 (2017).
[267] M. Talagrand, What Is a Quantum Field Theory? (Cambridge University Press, 2022).
[268] Y. K. Ha, External energy paradigm for black holes, International Journal of Modern Physics A 33,
1844025 (2018).
[269] F. A. Berends and R. Gastmans, Quantum gravity and the electron and muon anomalous magnetic
moments, Physics Letters B 55, 311 (1975).
[270] F. Hehl and E. Kröner, Über den Spin in der allgemeinen Relativitätstheorie: Eine notwendige Er-
weiterung der Einsteinschen Feldgleichungen, Zeitschrift für Physik 187, 478 (1965).
[271] H. Kleinert, Gravity and crystalline defects, in The Twelfth Marcel Grossmann Meeting: On Re-
cent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic
Field Theories (In 3 Volumes) (World Scientific, 2012) pp. 1195–1197.

108
[272] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973)
Chap. 44, “Beyond the end of time”.
[273] G. Amelino-Camelia, Testable scenario for relativity with minimum length, Physics Letters B 510,
255 (2001).
[274] M. Le Bellac, Introduction à l’information quantique, Tech. Rep. (Université de Nice, France, 2006).
[275] P. Zizzi and E. Pessa, From SU(2) gauge theory to qubits on the fuzzy sphere, International Journal
of Theoretical Physics 53, 257 (2014).
[276] S. Heusler, P. Schlummer, and M. S. Ubben, A Knot Theoretic Extension of the Bloch Sphere Rep-
resentation for Qubits in Hilbert Space and Its Application to Contextuality and Many-Worlds The-
ories, Symmetry 12, 1135 (2020).
[277] E. Witten, Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens
To The Algebra of Observables In The Thermodynamic Limit? (2021), arXiv:2112.11614.
[278] P. Zizzi, Quantum computation toward quantum gravity, General Relativity and Gravitation 33, 1305
(2001).
[279] C. von Weizsäcker, Die Einheit der Natur: Studien, Wissenschaftliche Reihe (Carl Hanser, 1971).
[280] C. von Weizsäcker, Aufbau der Physik (Carl Hanser, 1985).
[281] W. Heisenberg, Der Teil und das Ganze: Gespräche im Umkreis der Atomphysik (Piper Verlag, 2013)
Chap. 21.
[282] M. Kober, Über die Beziehung der begrifflichen Grundlagen der Quantentheorie und der Allge-
meinen Relativitätstheorie (2015), arXiv:1510.01196.
[283] H. Lyre, CF von Weizsäcker’s reconstruction of physics: Yesterday, today, tomorrow, in Time, Quan-
tum and Information (Springer, 2003) pp. 373–383.
[284] H. Lyre, Quantentheorie der Information: zur Naturphilosophie der Theorie der Ur-Alternativen und
einer abstrakten Theorie der Information, in Quantentheorie der Information (Brill mentis, 2004).
[285] M. Freedman and M. Headrick, Bit threads and holographic entanglement, Communications in
Mathematical Physics 352, 407 (2017).
[286] A. Rolph, Quantum bit threads, SciPost Physics 14, 097 (2023).
[287] N. Bao and H. Ooguri, Distinguishability of black hole microstates, Physical Review D 96, 066017
(2017).
[288] J. Harper, M. Headrick, and A. Rolph, Bit threads in higher-curvature gravity, Journal of High Energy
Physics 2018, 1 (2018).
[289] H. Yan et al., Geodesic string condensation from symmetric tensor gauge theory: A unifying frame-
work of holographic toy models, Physical Review B 102, 161119 (2020).
[290] C. A. Agón, E. Cáceres, and J. F. Pedraza, Bit threads, Einstein’s equations and bulk locality, Journal
of High Energy Physics 2021, 1 (2021).
[291] J. F. Pedraza, A. Russo, A. Svesko, and Z. Weller-Davies, Sewing spacetime with Lorentzian threads:
complexity and the emergence of time in quantum gravity, Journal of High Energy Physics 2022, 1
(2022).
[292] M. Headrick and V. E. Hubeny, Covariant bit threads, Journal of High Energy Physics 2023, 1 (2023).
[293] Y.-Y. Lin and J.-C. Jin, Thread/state correspondence: from bit threads to qubit threads, Journal of
High Energy Physics 2023, 1 (2023).
[294] J. Lin, Y. Lu, and Q. Wen, Geometrizing the partial entanglement entropy: from PEE threads to bit
threads, Journal of High Energy Physics 2024, 1 (2024).
[295] K. Fredenhagen, K.-H. Rehren, and E. Seiler, Quantum field theory: Where we are, Approaches to
Fundamental Physics: An Assessment of Current Theoretical Ideas 721, 61 (2007).
[296] H. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (I). Proof by homotopy theory,
Nuclear Physics B 185, 20 (1981).
[297] H. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (II). Intuitive topological proof,
Nuclear Physics B 193, 173 (1981).
[298] H. B. Nielsen and M. Ninomiya, No-go theorum for regularizing chiral fermions, Tech. Rep. (Science
Research Council, 1981).
[299] A. Maiezza and J. C. Vasquez, On Haag’s theorem and renormalization ambiguities, Foundations of

109
Physics 51, 1 (2021).
[300] D. Freeborn, M. Gilton, and C. Mitsch, How Haag-tied is QFT, really? (2022), arXiv:2212.06977.
[301] H. Harari, A schematic model of quarks and leptons, Physics Letters B 86, 83 (1979).
[302] M. A. Shupe, A composite model of leptons and quarks, Physics Letters B 86, 87 (1979).
[303] O. Greenberg and C. Nelson, Composite models of leptons, Physical Review D 10, 2567 (1974).
[304] H. Harari and N. Seiberg, A dynamical theory for the rishon model, Physics Letters B 98, 269 (1981).
[305] H. Harari and N. Seiberg, The rishon model, Nuclear Physics B 204, 141 (1982).
[306] P. ˙
Zenczykowski, The Harari–Shupe preon model and nonrelativistic quantum phase space, Physics
Letters B 660, 567 (2008).
[307] R. Raitio, Supersymmetric preons and the standard model, Nuclear Physics B 931, 283 (2018).
[308] P. ˙
Zenczykowski, Quarks, hadrons, and emergent spacetime, Foundations of Science 24, 287 (2019).
[309] R. J. Finkelstein, Are dyons the preons of the knot model?, International Journal of Modern Physics
A34, 1941009 (2019).
[310] N. G. Gresnigt, Topological preons from algebraic spinors, The European Physical Journal C 81, 506
(2021).
[311] J. G. Polchinski, String Theory, Volume I: An Introduction to the Bosonic String (Cambridge Univer-
sity Press, 1998).
[312] J. G. Polchinski, String Theory, Volume II: Superstring Theory and Beyond (Cambridge University
Press, 1998).
[313] A. Jaffe and E. Witten, Quantum Yang-Mills theory, The millennium prize problems 1, 129 (2006).
[314] D. Hilbert, Axiomatisches Denken, Mathematische Annalen 78, 405 (1918).
[315] D. Hilbert, Mathematical problems, Bulletin of the American Mathematical Society 37, 407 (2000).
[316] A. N. Gorban, Hilbert’s sixth problem: the endless road to rigour, Philosophical Transactions of the
Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20170238 (2018).
[317] C. Kiefer, Gödel’s undecidability theorems and the search for a theory of everything, International
Journal of Theoretical Physics 63, 52 (2024).
[318] L. H. Kauffman, Knot logic, in Knots and applications (World Scientific, 1995) pp. 1–110.
[319] L. H. Kauffman, Reflexivity and eigenform: The shape of process, Constructivist Foundations 4
(2009).