Quantum Holonomy Theory

Abstract

We present quantum holonomy theory, which is a non-perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a C*-algebra that involves holonomy-diffeomorphisms on a 3-dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi-classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi-classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi-classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost-commutative algebra emerges from the holonomy-diffeomorphism algebra in the same limit.

Full text

Quantum Holonomy Theory
Johannes Aastrup1& Jesper Møller Grimstrup 2
Abstract
We present quantum holonomy theory, which is a non-perturbative
theory of quantum gravity coupled to fermionic degrees of freedom.
The theory is based on a C∗-algebra that involves holonomy-diffeo-
morphisms on a 3-dimensional manifold and which encodes the canon-
ical commutation relations of canonical quantum gravity formulated in
terms of Ashtekar variables. Employing a Dirac type operator on the
configuration space of Ashtekar connections we obtain a semi-classical
state and a kinematical Hilbert space via its GNS construction. We
use the Dirac type operator, which provides a metric structure over
the space of Ashtekar connections, to define a scalar curvature op-
erator, from which we obtain a candidate for a Hamilton operator.
We show that the classical Hamilton constraint of general relativity
emerges from this in a semi-classical limit and we then compute the
operator constraint algebra. Also, we find states in the kinematical
Hilbert space on which the expectation value of the Dirac type operator
gives the Dirac Hamiltonian in a semi-classical limit and thus provides
a connection to fermionic quantum field theory. Finally, an almost-
commutative algebra emerges from the holonomy-diffeomorphism al-
gebra in the same limit.
1email: aastrup@math.uni-hannover.de
2email: jesper.grimstrup@gmail.com
1

Contents
1 Introduction 3
1.1 Outline of the central idea . . . . . . . . . . . . . . . . . . . . . 5
2 The Quantum Holonomy-Diffeomorphism algebra 8
2.1 The Holonomy-Diffeomorphism algebra . . . . . . . . . . . . . 8
2.2 The Quantum Holonomy-Diffeomorphism algebra . . . . . . . 10
2.3 The infinitesimal QHD(M)algebra . . . . . . . . . . . . . . . 10
2.4 The canonical commutation relations . . . . . . . . . . . . . . . 12
2.5 The spectrum of HD(M)...................... 14
3 Semi-classical states and a kinematical Hilbert space 15
3.1 Lattice formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 The continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 A semi-classical state on HD(M)................. 20
3.4 A Dirac type operator and the graded HD(M)algebra . . . . 21
3.5 Unbounded operators and the dQHD∗(M)algebra . . . . . 24
3.6 Constructing the state . . . . . . . . . . . . . . . . . . . . . . . . 27
3.7 The spectrum of the semi-classical states . . . . . . . . . . . . 28
3.8 A bimodule over volume-preserving diffeomorphisms . . . . . 31
4 Emerging elements of fermionic QFT 32
4.1 Infinitesimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 A two-point lattice state . . . . . . . . . . . . . . . . . . . . . . 33
4.3 An emergent one-particle state . . . . . . . . . . . . . . . . . . . 35
5 On a dynamical principle 37
5.1 The Hamilton constraint operator . . . . . . . . . . . . . . . . . 38
5.2 The operator constraint algebra . . . . . . . . . . . . . . . . . . 42
6 Emergence of an almost-commutative algebra 47
7 Background independency and action of the diffeomorphism
group 49
7.1 Invariance properties of Dand dQHD∗(M). . . . . . . . . . 49
7.2 Unitary equivalence . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3 Action of the diffeomorphism group . . . . . . . . . . . . . . . . 52
7.4 A lattice-independent formulation of Dand dQHD∗(M). . 53
8 The overlap function 55
2

9 The complex Ashtekar connection 57
10 Interpreting Din terms of the volume of M58
11 Summary and discussion 60
A The operator constraint algebra 64
A.1 The commutator [H(N),H(N′)] ................. 64
A.2 The commutator [D(¯
N),D(¯
N′)] ................. 69
1 Introduction
The search for fundamental principles in Nature is the leitmotiv of modern
physics. The aim is to explain rather than describe and what constitutes
an explanation is the invocation of a principle. The dream is to uncover an
ultimate principle behind it all – the fundamental theory – which will end
the reductionist ladder of descent and leave no door to peek behind.
The framework of quantum holonomy theory proposes such a first prin-
ciple. The theory is built over an algebra that encodes how diffeomorphisms
act on spinors. Thus, the fundamental building blocks are ”moving stuff in
space” and as such seem immune to further reduction: the question ”what
are diffeomorphisms made of?” makes little sense.
What we find is a non-perturbative and background independent quan-
tum mechanics of diffeomorphisms, where we on the one hand have an alge-
bra generated by holonomy-diffeomorphisms on a three-dimensional mani-
fold Mand on the other hand conjugate operators given by canonical trans-
lations on a configuration space of connections, over which the holonomy-
diffeomorphisms form a non-commutative algebra of functions. The interac-
tion between the algebra of holonomy-diffeomorphisms, which we denote by
HD(M), and the translation operators encodes the canonical commutation
relations of canonical quantum gravity formulated in terms of Ashtekar vari-
ables [1, 2]. This means that we establish a direct link between an algebra
generated by holonomy-diffeomorphisms and canonical quantum gravity.
This construction comes with a very high degree of canonicity: once the
number of spatial dimensions is chosen it only depends on a choice of gauge
group, where SU (2)matches our choice of three spatial dimensions via the
rotation group.
Once we have defined the algebra generated by holonomy-diffeomorphisms
and translation operators, denoted QHD(M), the question arises whether
states exist on this algebra. This issue is central to this paper. To address
3

it we first combine the conjugate translation operators into a Dirac type
operator. This constitutes a canonical metric over a configuration space of
Ashtekar connections, i.e. a geometry over a configuration space of spatial
geometries. The Dirac type operator entails a flow-dependent version of the
QHD(M)algebra and it is on this algebra that we find a semi-classical state
that provides us with a kinematical Hilbert space via its GNS construction.
Thus, each classical point gives rise to a different kinematical Hilbert space.
Furthermore, we find evidence that the overlap function, which measures
the transition from one semi-classical approximation to another, vanishes.
This suggest that there will be no quantum interference between different
semi-classical approximations. However, on certain operators in the GNS
construction of a semi-classical state the expectation value can look like a
different classical geometry.
The introduction of the Dirac-type operator opens up further avenues of
analysis. We obtain a tentative candidate for a Hamilton constraint opera-
tor by constructing a scalar curvature operator and show that the classical
Hamilton constraint emerges in a semi-classical limit from this operator.
This shows that the framework of quantum holonomy theory produces gen-
eral relativity in a semi-classical limit.
A central test for any theory of quantum gravity is to check whether the
constraint algebra closes off-shell. This determines to what extend general
covariance is maintained in the quantum theory and is usually understood
as a necessary requirement for the theory to be internally consistent. With a
candidate for a Hamilton constraint operator we are in a position where we
can perform this test. We first compute the commutator between two Hamil-
ton constraint operators – characterized by different choices of lapse fields
– and find that it reproduces off-shell the structure of the corresponding
classical Poisson bracket up to an anomalous term, which vanishes at all fi-
nite orders in perturbation theory. This result matches the observation that
the action of the diffeomorphism group on the flow-dependent QHD(M)
algebra is not strongly continuous and thus cannot have infinitesimal gen-
erators. The exception to this is exactly the perturbative regime, where
we thus find that the constraint algebra does close. Thus, we find that
the ’Hamilton-Hamilton’ sector of the quantum constraint algebra is free of
anomalies, off-shell, in exactly the regime where the constraint algebra is a
meaningful object.
This computation provides us with a candidate for a diffeomorphism
constraint operator and we continue to compute the commutator between
two different diffeomorphism constraint operators. As these computations
are very involved we perform them with a simplified version of the Hamilton
4

constraint operator. We believe that this simplification is the explanation
for the emergence of non-physical anomalous terms in this computation.
Nevertheless, the fact that our framework permits this type of computations
is, in our opinion, very encouraging and the result calls for a computation,
that involves the full Hamilton constraint operator derived from the scalar
curvature operator.
In another direction of analysis we show that the construction of the
Dirac type operator leads to a class of states on which its expectation value
gives a spatial Dirac operator in a semi-classical limit. A certain transforma-
tion, that introduces the lapse and shift fields, entails from these expectation
values the principle part of the Dirac Hamiltonian. This indicates that the
theory also harbors quantized matter degrees of freedom and that these are
canonical. Also, we find that the HD(M)algebra descends, in a semi-
classical limit, to a tensor product that involves an almost-commutative
algebra. Almost-commutative algebras form the cornerstone in the formula-
tion of the standard model of particle physics coupled to gravity in terms of
non-commutative geometry [3, 4], and the emergence of such an algebra – in
company with a spatial Dirac operator – opens up the possibility for emer-
gent gauge degrees of freedom and a connection to the mathematics of the
standard model itself. All together we find strong indications that a theory
of quantum gravity based on the QHD(M)algebra naturally involves both
additional fermionic and bosonic degrees of freedom.
The construction of the semi-classical state is carried out with a for-
mulation of the HD(M)algebra in terms of infinite sequences of lattice
approximations. At the present level of analysis we are unable to determine
which type of diffemorphisms this formulation is in fact capable of describing
and whether we are forced to restrict ourselves to analytic diffeomorphisms.
Also, more analysis is needed to describe the precise mathematical nature
of the continuum limit, on which the lattice formulation of the HD(M)
algebra depends. To be more precise, we know that the limit exist but the
exact details of the limit are not completely known.
1.1 Outline of the central idea
The cornerstone in the theory is the algebra HD(M), which is generated by
holonomy-diffeomorphisms on a 3-dimensional manifold M. A holonomy-
diffeomorphism maps a connection ∇into an operator acting on spinors on
MA∋∇→eX(∇)
5

where Ais the space of all smooth connections in a certain bundle and
where eX(∇)denotes a holonomy-diffeomorphisms along a vector-field X.
Thus, the algebra HD(M)depends on the manifold Mas well as a choice of
gauge group corresponding to the bundle over M. We choose a 3-dimensional
manifold and the group SU (2)since this corresponds to canonical quantum
gravity formulated in terms of Ashtekar variables1[1, 2]. Furthermore, we
choose the trivial bundle.
Once we have the algebra HD(M)it is natural to consider translations
on the space Aof connections
Uωξ(∇)=ξ(∇−ω)
where ωis a one-form with values in the Lie-algebra of SU(2). We find the
relation (UωeXU−1
ω)(∇)=eX(∇+ω)
which is in fact an integrated version of the canonical commutation rela-
tions of quantum gravity formulated in terms of Ashtekar variables. Thus,
we find that the algebra of holonomy-diffeomorphisms leads us directly into
the realm of canonical quantum gravity.
Note that the algebra HD(M)is manifestly non-commutative. This
means that an approach to quantum gravity based on this algebra also plays
into the heartland of non-commutative geometry.
In order to approach the problem of finding Hilbert space representations
of these algebraic structures we first introduce operators that correspond
to infinitesimal translations on A. Using infinite sequences of lattice ap-
proximations we construct a Dirac-type operator Dfrom these infinitesimal
translation operators and consider the algebra generated by HD(M)and
commutators with D. This algebra, which also encodes the canonical com-
mutation relations of quantum gravity, has a state that exist independently
of the lattice approximations. We apply the GNS construction on this state
to form a kinematical Hilbert space.
With a Dirac-type operator we then consider one-forms, which in the lan-
guage of non-commutative geometry have the form B=a[D, b], where aand
1To be precise, the choice of SU (2)corresponds to general relativity with an Euclidian
signature. The Lorentzian signature corresponds to a connection, which takes values in the
self-dual section of sl(2,C), the Lie-algebra of SL(2,C). We comment on this in section
9.
6

bare elements of the HD(M)algebra. We also consider the corresponding
curvature operators FB=[D, B]+1
2[B, B ],
which are curvature operators over the configuration space of Ashtekar con-
nections, and find that the densitized Hamiltonian constraint of general rel-
ativity formulated in terms of Ashtekar variables emerges in a semi-classical
limit from a scalar curvature operator built from operators FBin a way that
involves the orientation of the manifold M.
The paper is organized as follows: In section 2 we first introduce the
HD(M),QHD(M)and dQHD(M)algebras and show that the latter two
encode the canonical commutation relations of canonical quantum gravity
formulated in terms of Ashtekar variables. In section 3 we then introduce
a lattice formulation of the holonomy-diffeomorphism algebra and consider
states on QHD(M)and dQHD(M)and reach the conclusion that such
are unlikely to exist. We define the dQHD∗(M)algebra via the Dirac type
operator and show that semi-classical states exist hereon. Subsequently we
obtain a kinematical Hilbert space via the GNS construction. In section 4
we show that a spatial Dirac operator as well as the principal part of the
Dirac Hamiltonian emerges in a semi-classical limit from the expectation
value of the Dirac type operator on certain states in the kinematical Hilbert
space. We then consider the dynamics of general relativity in section 5 and
show how a candidate for a Hamiltonian constraint operator is obtained
from a scalar curvature operator. We then compute the constraint algebra.
In section 6 we show that an almost-commutative algebra emerges from the
HD(M)algebra in a semi-classical limit. The use of lattice approxima-
tions naturally entails the question of background independency, which is
discuss in section 7, where we also work out an abstract formulation of the
dQHD∗(M)algebra. Then we present in section 8 a tentative analysis of
the overlap function, which suggest that it vanishes, and in section 9 we con-
sider a complex Ashtekar connection. Finally, in section 10 we show that
the Dirac operator has an interpretation in terms of the quantized volume
of the manifold M. We conclude in section 11 and add an appendix, which
contains computations on the operator constraint algebra.
7

ξ(m)Hol(γ, ∇)ξ(m)
γ
Figure 1: An element in HD will parallel transport a vector on Malong the
flow of a diffeomorphism.
2 The Quantum Holonomy-Diffeomorphism alge-
bra
The first task is to introduce the Holonomy-Diffeomorphism algebra. This
algebra was first described in [5, 6], where its spectrum was analyzed. The
extension to the Quantum Holonomy-Diffeomorphism algebra then follows
canonically.
2.1 The Holonomy-Diffeomorphism algebra
Let Mbe a connected 3-dimensional manifold. We consider the two dimen-
sional trivial vector bundle S=M×C2over M, and we consider the space
of SU (2)connections acting on the bundle. Given a metric gon Mwe get
the Hilbert space L2(M , S, dg), where we equip Swith the standard inner
produkt. Given a diffeomorphism φ∶M→Mwe get a unitary operator φ∗
on L2(M, S, dg )via
(φ∗(ξ))(φ(m))=(∆φ)(m)ξ(m),
where ∆φ(m)is the volume of the volume element in φ(m)induced by a
unit volume element in munder φ.
Let Xbe a vectorfield on M, which can be exponentiated, and let ∇be
aSU (2)-connection acting on S. Denote by t→expt(X)the corresponding
flow. Given m∈Mlet γbe the curve
γ(t)=expt(X)(m)
8

running from mto exp1(X)(m). We define the operator
eX
∇∶L2(M, S, dg )→L2(M, S, dg)
in the following way: we consider an element ξ∈L2(M, S, dg )as a C2-valued
function, and define
(eX
∇ξ)(exp1(X)(m))=((∆ exp1)(m))Hol(γ , ∇)ξ(m).
Here Hol(γ, ∇)denotes the holonomy of ∇along γ. Again, the factor
(∆ exp1)(m)is accounting for the change in volumes, rendering eX
∇uni-
tary.
Let Abe the space of SU(2)-connections. We have an operator valued
function on Adefined via A∋∇→eX
∇.
We denote this function eX.
Denote by F(A,B(L2(M, S, dg))) the bounded operator valued func-
tions over A. This forms a C∗-algebra with the norm
Ψ=sup
∇∈A{Ψ(∇)},Ψ∈F(A,B(L2(M, S, dg )))
For a function f∈C∞
c(M)we get another operator valued function feX
on A.
Definition 2.1.1 Let
C=span{feXf∈C∞
c(M), X exponentiable vectorfield }.
The holonomy-diffeomorphism algebra HD(M , S, A)is defined to be the C∗-
subalgebra of F(A,B(L2(M , S, dg))) generated by C.
We will often denote HD(M , S, A)by HD or HD(M)when it is clear
which M,S,Ais meant. We will by HD(M , S, A)denote the ∗-algebra
generated by C.
It was shown in [6] that HD(M, S, A)is independent of the chosen metric
g.
9

2.2 The Quantum Holonomy-Diffeomorphism algebra
Let su(2)be the Lie-algebra of SU(2). It is well-known that two con-
nections in Adiffers by an element in Ω1(M, su(2)), and that for ∇∈A
and ω∈Ω1(M, su(2)),∇ +ωdefines a connection in A. Thus a section
ω∈Ω1(M, su(2)) induces a transformation of A, and therefore an operator
Uωon F(A,B(L2(M, S, g ))) via
Uω(ξ)(∇)=ξ(∇−ω).
Note that U−1
ω=U−ω.
Definition 2.2.1 Let us denote by QHD(M, S, A)the sub-algebra of
F(A,B(L2(M, S )))generated by HD(M , S, A)and all the operators Uω,ω∈
Ω1(M, su(2)). We will often denote QHD(M , S, A)by QHD or QHD(M)
when it is clear, which M,S,Ais meant. We call QHD the Quantum-Flow
algebra or the Quantum Holononomy-Diffeomorphism algebra.
We note that we have the relation
(UωfeXU−1
ω)(∇)=feX(∇+ω),(1)
where f∈C∞
c(M). However QHD(M)is not a cross product of HD(M)
with the additive group Ω1(M, su(2)), since the function of operators given
by ∇→eX
∇+ωneed not be in HD(M).
2.3 The infinitesimal QHD(M)algebra
To get closer to the formulation of the holonomy-flux-algebra2and canonical
quantization of gravity (see [8] for setup and notions) we need the infinites-
imal version of Utω. We simply do this by formally defining
Eω=d
dtUtωt=0.
Due to the relation (1) we get
[Eω, eX
∇]=d
dteX
∇+tωt=0.(2)
2By the holonomy-flux algebra we refer to the algebra used in loop quantum gravity,
see [7].
10

Thus the infinitesimal version of the Quantum Holonomy-Diffeomorphism
algebra is generated by the flows and the variables {Eω}ω∈Ω(M,T ). We de-
note this algebra by dQHD(M).
We note, that
Eω1+ω2=Eω1+Eω2.
This follows since the map Ω1(M, su(2)) ∋ω→Uωis a group homomor-
phism, i.e. U(ω1+ω2)=Uω1Uω2.
To see the connection to the holonomy-flux algebra let us analyze the
righthand side of (2). First we introduce local coordinates (x1, x2, x3). We
decompose ω:ω=ωi
µσidxµ. Due to the additive property of Eωand that
the action of C∞
c(M)commutes with Uωwe only have to analyze an ωof
the form σidxµ. For a given point p∈Mchoose the points
p0=p, p1=e1
nX(p),..., pn=en
nX(p)
on the path
t→etX (p), t ∈[0,1].
We write the vectorfield X=Xν∂ν. We have
eX
∇+tω
=lim
n→∞(1+1
n(A(X(p0)+tσiXµ(p0)))(1+1
n(A(X(p1))+tσiXµ(p1)))
⋯(1+1
n(A(X(pn)+tσiXµ(pn)),(3)
where ∇=d+A, and therefore
d
dteX
∇+tωt=0
=lim
n→∞1
nσiXµ(p0)(1+1
nA(X(p1)))⋯(1+1
nA(X(pn)))
+(1+1
nA(X(p0)))1
nσiXµ(p1)(1+1
nA(X(p2)))⋯(1+1
nA(X(pn)))
+ ⋮
+(1+1
nA(X(p0))))(1+1
nA(X(p2)))⋯(1+1
nA(X(pn−1))1
nσiXµ(pn)
(4)
11

Figure 2: The operator Eσidxµwill, when it is commuted with a flow, insert
Pauli matrices continuously along the course of the flow. This means that
it acts as a sum of flux operators with surfaces, which intersect the flow at
the points of insertion.
We see that before taking the limit limn→∞this is just the commutator of the
sum of the flux operators ∑k
1
nXµ(pk)FSk
i, where Skis the plane orthogonal
to the xµ-axis intersecting pk, and the holonomy operator of the path
t→etX (p), t ∈[0,1],
see figure 2.
It follows that Eσidxµis a series of flux-operators FS
isitting along the
path
t→etX (p), t ∈[0,1],
where the surfaces Sare just the planes othogonal to the xµdirection. But
since there are infinitely many of them, they have been weighted with the
infinitesimal length, i.e. with a dxµ, see figure 2. We can formally write
Eω=FS
µdxµ.
Note that the phenomenon from the holonomy-flux-algebra, that a path p
running inside a surface S, has zero commutator with the corresponding flux
operator is encoded in the quantum-flow-algebra, since the tangent vectors
of pwill be annihilated by the differential form dxµ.
2.4 The canonical commutation relations
We can also make the holonomies infinitesimal in order to see the canonical
commutation relations of general relativity. In the following we scale the
12

translation operators Uω→Uκω , where κis the quantization parameter.
Let us again first introduce a coordinate system (x1, x2, x3). We have the
vectorfields ∂µ, and we consider the operator
d
dses∂µs=0.
We note that if the local connection one form of ∇is Ai
µσidxµwe have
(d
dses∂µs=0)(ξ(x, ∇))=σiAi
µ(x)ξ(x, ∇).
We therefore consider the operator δ(x)d
ds es∂µs=0, where δ(x)is the delta
function located in x, as the operator σiAi
µ(x)located in x.
On the other hand consider ω=σldxν. We get
[Eσldxν, σiAi
µ(x)](∇)
=δ(x)d
ds[Eσldxν, es∂µ]s=0(∇)=κδ(x)d
ds
d
dtes∂µ
∇+tσldxνt=0s=0
=κδ(x)d
ds
d
dt(1+s(Ai
µσi+tσlδν
µ))t=0s=0=κσlδν
µδ(x).
If
fy(x)=1x=y
0x=y
we can therefore consider the operator fyEσldxνas ˆ
Eν
l(y)since then
[ˆ
Eν
l(y), σiAi
µ(x)]=κσlδν
µδ(x−y),(5)
which is the quantized canonical commutation relation of general relativity
formulated in terms as Ashtekar variables.
All together these results show that the algebra QHD(M)is intimately
related to canonical quantum gravity since it is simply the algebra from
which the infinitesimal operators forming the canonical commutation rela-
tions originate.
Note, however, that with the choice of SU(2)as the gauge group we
are not, a priori, dealing with a construction based on the original Ashtekar
connection, which takes values in the Lie-algebra of complexified SU(2)–
the self-dual sector of SL(2,C)–, but instead with an Ashtekar connec-
tion related to general relativity with a Euclidian signature3. We could, of
3The issue is a little more complicated than this, since it depends on the form of
the Hamiltionian as well. In the framework of this paper a real SU (2)connection does
imply a Euclidian signature since the Hamiltonian, which we eventually derive from our
construction, has the simple algebraic structure ’EEF ’, where Fis the field strength
tensor of the Ashtekar connection. See for instance [7].
13

course, try to work with SL(2,C)instead of SU(2), but it is our belief that
the complexification, which takes us from SU (2)to SL(2,C)should arise
naturally from the construction. See section 9 and 11.
2.5 The spectrum of HD(M)
In [6] we analyzed the spectrum of the algebra HD(M), which is defined
as the irreducible representations of HD(M)modulo unitary equivalence.
There we obtained two main results concerning the spectrum of HD(M),
which we will now review. Before we do so we first need to introduce the
concepts of a measurable connection and of a generalized connection. For
details and proofs of this section we refer to [6].
Definition 2.5.1 Let Fbe the group generated by flow operators eX. A
measurable U(n)-connection, n=1,...,∞, is a map ∇from Fto the group
of measurable maps from Mto U(n)satisfying
1. ∇(1)=1.
2. ∇(F1○F2)(m)=∇(F1)(m)○∇(F2)(F−1
1(m))
3. If F1and F2are the same up to local reparametrization over some set
U⊂M, then ∇(F1)U=∇(F2)U.
Next, let lbe a piece-wise analytic path in M. We identify l⋅l−1with
the trivial path starting and ending at the start point of l. Furthermore we
identify two paths that differ by a reparameterization.
Definition 2.5.2 Let Gbe a connected Lie-group. A generalized connection
is an assignment ∇(l)∈Gto each piece-wise analytic path l, such that
∇(l1⋅l2)=∇(l1)∇(l2).
With this we can now state the two main results from [6] on the spectrum
of HD(M):
Theorem 2.5.3 Any separable, irreducible representation of HD(M)is
unitarily equivalent to a representation of the form ϕ∇, where ∇is a mea-
surable U(2)-connection4.
4Here we disregard the special case where the representation decomposes into two U(1)
measurable connections.
14

Theorem 2.5.4 A generalized connection together with the counting mea-
sure on Mdoes not render a representation of HD(M).
Theorem 2.5.3 holds in more general settings – manifolds of arbitrary
dimensions and arbitrary vector bundles. The theorem is, however, partic-
ularly interesting in the case where Mis a three-dimensional manifold and
Sis a two-spinor bundle over Mwith SU(2)connections, since in this case
one can interpret the spectrum as the completion of a configuration space
of Ashtekar connections.
Theorem 2.5.4 basically states that the bulk of the spectrum found in
loop quantum gravity [7], which is given by generalized connections with
support on finite graphs, is excluded from the spectrum of HD(M).
It is an open question what the non-separable part of the spectrum of
HD(M)contains. The fact that we are for now unable to prove that the
entire spectrum is given by measurable connections may indicate that we
need to change the definition of HD(M). In particular, one may speculate
that the topology of HD(M), which is the C∗-topology, is not the right one
for our purpose. For further discussion of this point as well as other related
issues see [5].
Note finally that there is an open question as to how we get SU (2)
connections instead of U(2)connections. Of course, we can simply put
them in by hand – which is what we do in this paper – but a more natural
solution would be to introduce a real structure and a conjugate action of
the algebra, which would kill the U(1)factor.
3 Semi-classical states and a kinematical Hilbert
space
With the formulation of the Quantum Holonomy-Diffeomorphism algebra
completed the first task is to determine whether states exist on this algebra.
3.1 Lattice formulation
The question concerning states on QHD(M)and dQHD(M)is best ad-
dressed in a setting where everything is reformulated in terms of lattice
approximations. Aside from being a technical tool this also represents a co-
ordinate dependent formulation since an infinite sequence of nested lattices
corresponds to a coordinate system. The following is based on techniques
introduced in [5].
15

...
Figure 3: Ω is an infinite sequence of cubic lattices, where each lattice Γnin
Ωis a symmetric subdivision of the previous lattice Γn−1and where the union
of all lattices Γnis dense in M. Thus Ωrepresents a coordinate system in
M.
Before we continue let is first note that at the present level of analysis we
are unable to determine whether a lattice formulation exist for QHD(M)
and dQHD(M)or if it is necessary to restrict HD(M)to analytic flows.
We discuss this issue in section 3.2 and in the final discussion in section 11.
In the following we therefore leave open whether the lattice approximations
of HD(M),QHD(M)and dQHD(M)are constructed from diffeomor-
phisms or analytic diffeomorphisms.
In the lattice formulation an element in HD(M)is represented by an
infinite family of operators acting in increasingly accurate lattice approxi-
mations associated to an infinite system {Γn}of nested cubic lattices, see
figure 3. Thus, we begin with a single finite cubic lattice Γnwith vertices
and edges denoted by {vi}and {lj}. We assign to each edge via copy of
SU (2)∇(lj)=gj∈S U (2)(6)
and obtain the space An=(SU (2))∣ln∣
where lnis the number of edges in Γn.Anis an approximation of the
space of SU (2)-connections Aand the map ∇should be understood as an
approximation of a connection in A.
An element feXin HD(M)is at the level of a lattice Γnapproximated
by a finite family of oriented, weighted paths in Γn, denoted by n. Here
ndenotes a family {pi}of paths in Γn, where each piis a sequence of
adjacent edges
pi={li1,...,lin},
connecting two vertices in Γn. By we denote a corresponding set of weights
16

assigned to each edge in n. There is a natural product between such lattice
approximations given by composition of paths and a natural involution given
by reversal of the paths (see [5] for details). The lattice approximation of
HD(M)is denoted by HDn. We shall give a more precise definition of
HDnin the next section, where we describe the continuum limit.
With the space Anwe automatically have the Hilbert space L2(An)via
the Haar measure on SU (2). We will, however, need more structure in order
to construct a representation of HDnand therefore introduce the Hilbert
space Hn=L2(An, M2(C))×M∣vn∣(C)
where vnis the number of vertices in Γn. A representation of an element
nin HDnacts by multiplying the M2(C)factor in Hnwith the parallel
transports ∇(pi)=∇(li1)⋅...∇(lin), pi∈n,
and by acting on the M∣vn∣(C)factor with nas an vn×vn-matrix in the
sense that each path in nshifts lattice points according to its start and
end-points. Thus, an example of a representation of an element nin HDn
as an operator in Hncould be:






















1... 0
⋮⋱⋮
0... 1⋱0∇(p1)∇(p2)
0 0 ∇(p3)
0 0 0 ⋱1... 0
⋮⋱⋮
0... 1






















(7)
where ninvolves5the three paths {p1, p2, p3}.
We also need to define the operators Uωand Eωon the lattice. We start
with the latter since, if this is successfully constructed, a lattice approxima-
tion of the former is obtained via exponentiation hereof. Let sui(2)be the
Lie algebra of the i’th copy of SU (2)and choose an orthonormal basis {ea
i}
thereon. We also denote by {ea
i}the corresponding right translated vector
5In order to ease the notation we have not assigned weights to the paths in this example.
If there had been weights these would have appeared as numbers multiplied the ∇(pi)’s.
17

Figure 4: The action of a path in lattice on L2(M, S ). It transports the
content of the first square to the content of the second square and multiplies
with the holonomy of the connection ∇along the path.
fields and by Lea
ithe derivation with respect to the trivialization given by
{ea
i}. There is then a natural candidate for a lattice approximation of Eω
given by
Eω,n =2−nκ
i,a
ωa
iLea
i⊗1∣vn∣
where ωa
ihere denotes the value of ωevaluated at the start-point of the
edge liand where the sum runs over all edges in the lattice and over all
Lie-algebra indices. Here we have introduced a quantization parameter κ,
which corresponds to the transformation Uω→Uκω.
3.2 The continuum limit
Consider now a sequence {n}of lattice approximations as described above.
In order to give meaning to the notion that the sequence approximates an
element feXin HD(M)we need to define an action of nas an operator
in L2(M, S ). This is done in the following manner (see also section 5 in
[5]). First, we subdivide Minto cubes {ci}, where a cube ciis assigned
to each vertex vi∈Γn, and in such a way that the cubes fill out M. Let
us assume that nincludes a path pwhich connects two vertices viand
vj. Then nacts on a spinor ψ∈L2(M , S)by shifting each value of ψin
the cube cito the same relative location in the cube cjwhile multiplying
18

it with the holonomy transform ∇(p)of the connection along p, see figure
4. We denote this representation by ϕ(n). Furthermore, nhas the
same representation, denoted ϕ(n), in L2(M, S )as nexcept that each
holonomy with respect to ∇of a paths in the family is also multiplied with
the weight associated.
We say that a sequence {n}converges to feX∈HD(M)in the repre-
sentation ϕif
lim
n→∞(ϕ(n)−ϕ(feX)ξ)=0,(8)
for all ξ∈L2(M, S )and for all smooth connections, where ϕ(feX)denotes
the representation of feXin L2(M, S ). Keep in mind that the representation
ϕdepends on a chosen SU (2)connection and that (8) must hold for any
such choice. There is a natural equivalence relation between the sequence
{n}and {′
n}given by
{n}∼{′
n}iff lim
n→∞(ϕ(n)−ϕ(′
n))ξ=0 (9)
for all ξ∈L2(M, S )and all smooth connections. Note that this equivalence
relation is just saying that if {n}∼{′
n}then {n}and {′
n}con-
verge to the same element in HD(M), if they converge. If a family {n}
converges to feXin ϕwe shall say that ϕ(n)approximates feX.
This notion of convergence can also be defined for diffeomorphisms and
volume-preserving diffeomorphisms, see [5].
In a next step we need to describe how the algebra HD(M)is repre-
sented by sequences of lattice approximations. Let a, b, c be elements in
HD(M)and let {an},{bn}and {cn}be sequences of lattice approxima-
tions converging to a, b, c.{an},{bn}and {cn}are elements in equivalence
classes according to (9) and we choose these elements such that if a=bc
then the sequences {an}and {bncn}are identical. We call this choice of
lattice approximations of elements in HD(M)for a consistent set of lattice
approximations. The lattice approximation of HD(M)that arises in this
way at the level of Γnis denoted HDn.
The reason why it is necessary to require a consistent set of lattice ap-
proximations is that this is the only way one can preserve the algebra struc-
ture in lattice approximations of the flow-dependent version of dQHD(M),
which is the subject of the next section. It is, however, not clear to us if
it is in fact possible to obtain a consistent set of lattice approximations of
HD(M). One possibility is to restrict HD(M)to analytic flows, in which
case the requirement can be met, but we do not know if this restriction is
necessary. We shall return to this issue in the final discussion.
19

Finally, we also need a notion of convergence of states on HD(M)from
states on HDn. Let again {n}converge to feX∈HD(M)and let ξbe a
state on HD(M), i.e. a positive linear functional on HD(M). Let {ξn}be
a sequence of states with ξn∈Hn. We will say that ξnapproximates ξif
lim
n→∞nξnξn=ξ(feX)
for all elements in the equivalence class {n}. We say that two sequences
{ξn}and {ξ′
n}are equivalent if they both approximate the same state ξ. We
will, when discussing states and their approximations always assume that
we are dealing with such equivalence classes.
3.3 A semi-classical state on HD(M)
We are now going to write down necessary conditions for a state to exist on
HD(M)and dQHD(M), respectively. Let {ξ(n,i)(gi)}be a set of functions
on SU (2)associated to edges in Γn. To define a candidate for a state on
HD(M)and dQHD(M)we first write down the state ξnin Hn
ξn=cnΠ∣l∣
i=1ξ(n,i)⊗12⊗1∣vn∣,(10)
where cnis a normalization constant, and consider the continuum limit
n→∞of ξn, which we denote by ξc.
The necessary conditions for ξcto be a state on HD(M)and dQHD(M),
respectively, are the following6:
1) ξn∇(li)ξn=1−O(dx)
2)ξnLei1Lei2...Leimξn=O(dx2m),∀m, in∈{1,2,3}(11)
where dx =2−n, as well as:
3) the expectation values of ∇(li)and the expectation val-
ues of all powers of the Lei’s must converge to smooth
functions on Min the limit n→∞. i.e. the coefficients in
front of O(dx)and O(dx2m)must depend smoothly on the
points on the manifold. (12)
6to be precise, it might be possible to exchange smoothness in condition 3) with some
weaker requirement such as C1.
20

Conditions 1)and 3)ensure the the expectation value of a parallel trans-
port ∇(p)will converge in the limit n→∞. These conditions are easily sat-
isfied and we may therefore use the GNS construction to obtain the Hilbert
space Hξnand its limit Hξc. Furthermore, we may write
ξn∇(li)ξn=TrM21+dx A(n,i)−tB(n,i) (13)
where A(n,i)is the skew-self-adjoint component and where B(n,i)is the re-
mainder, which is a strictly positive operator, and where tis a formal pa-
rameter. If we let tplay the role of a quantization parameter we see that
the states on HD(M)constructed this way will always have a semi-classical
structure with Abeing the classical point and Bthe quantum correction.
Condition 2)ensures that the expectation value of the translation op-
erators Eω,n and all polynomials thereof will converge in the limit n→∞.
However, we strongly expect that this condition can never be satisfied si-
multaneous with condition 1), since this condition ensures that the state is
peaked around the identity of the group, whereas the second condition more
or less gives that the state is constant. These two requirements are mutually
exclusive.
Finally, condition 3)also ensures that the commutator between Eω,n
and elements in HDnexist in the large nlimit as well as the elements of
HD(M)themselves.
To sum up, we make the following conclusions:
1. There is a state ξcon HD(M)and by the GNS construction a Hilbert
space H(ξc,HD).
2. Within the framework of lattice approximations it appears that no
states on dQHD(M)exist. The infinitesimal translation operators
Eωare not well defined operators in H(ξc,HD).
3.4 A Dirac type operator and the graded HD(M)algebra
Our failure to find a state on dQHD(M)leads us to consider an alternative
approach, which involves a Dirac-type operator. At the end of our analysis
we shall see that this Dirac-type operator is, at this point, a somewhat su-
perfluous construction, which can be omitted. In the light of results of later
sections in this paper, where this Dirac-type operator plays an important
role, we choose however to include it in our analysis here too.
21

We return to the lattice approximation Γnand the operator Eω ,n. We
first add an additional factor to the Hilbert space Hn
H′
n=L2(An, Cl(T∗An)×M2(C))×M∣vn∣(C)
where Cl(T∗An)is the Clifford algebra over the co-tangent space over An.
This enables us to turn Eω,n into a Dirac-type operator over the space An
by substituting ωin Eω,n with an element of the Clifford algebra:
Eω,n
ω→e
Ð→Dn=2−nκ
i,a
ea
i⋅Lea
i⊗1∣vn∣(14)
where we by ea
ialso denote the element of Cl(T∗An)that corresponds to the
right-translated vector field ea
iand where ’⋅’ denotes Clifford multiplication.
One could think of the Clifford element ea
ias a lattice approximation of a
Grassmann valued element of Ω1(M , su(2)). The operator Dnis essentially
the Dirac type operator, which we studied in the papers [8]-[17] . We denote
by Dthe n→∞limit of Dn.
Here, however, we shall require the Clifford elements to form the non-
standard Clifford algebra
ea
i,eb
j=−2nδabδij .(15)
This means that the anti-commutator, in the large nlimit, will approach
a one-dimensional delta function. Of course, if fa
idenotes the standard
generators, we get the above generators via ea
i=2n
2fa
i.
Next, define the operation δn, which consist of taking the commutator7
with Dn
δnan∶=[an, Dn], an∈B(H′
n).
One can check that for an∈HDnδnanwill be a sum over insertions in an
of the form
2−nκσaea
j
where jnow labels the point of insertion in anand where σaare the skew-
adjoint Pauli matrices satisfying [σa, σb]=−2abcσc.
Definition 3.4.1 We define the graded HDnalgebra, which we denote by
HD∗
n, as the algebra generated by elements of HDnand elements δnan,
an∈HDn. Furthermore, we define the graded HD(M)algebra, which we
denote by HD∗(M), as the n→∞limit of HD∗
n.
7All operator brackets in this paper are graded. We shall occasionally write ’{⋅,⋅}’ for
the anti-commutator but it is understood that the bracket ’[⋅,⋅]’ is graded.
22

The n→∞limit is again as described in [5]. Also, we denote the n→∞
limit of δnby δ.
Note that the emphasis put here and in the following on the commutator
of the Dirac type operator Dinstead of Ditself is quite natural since the
central relation (1) describes the conjugation of a flow with the operator Uω,
the infinitesimal of which is of course the commutator. Thus, it does not
seem necessary that the operator Eω, nor its graded version D, should be
well defined in the lattice formulation as long as its commutators are.
We shall now consider the GNS construction of the algebra HD∗
non the
state ξnwhich we wrote down in (10) and which we now require to satisfy
condition 1) in (11). Since ξndoes not take values in the Clifford algebra
we immediately see that
ξnan(δnbn)cnξn=0,∀{an, bn, cn}∈HDn
and we check that
lim
n→∞ξn(δnan)∗δnanξn<∞, an∈HDn,(16)
where the sequence {an}is a representation of an element in HD(M). This
relation holds since δnanis a sum of insertions of the form 2−nκσaea
jand
since trCl(ea
ieb
j)=2nδabδij collapses the two sums in (16) to a single sum
with the right factor of 2−n. In fact, if we compute (16) for a single path
the result is simply the length Lof that path, computed with respect to the
metric provided by the lattice.
We readily generalize (16) to
lim
n→∞ξnan(δnbn)∗cn(δndn)enξn<∞,{an, bn, cn, dn, en}∈HDn,
where we again assume that the sequences {an},{bn},{cn},{dn}and {en}
are representations of elements in HD(M). Here, due to the Clifford algebra
the result can only be non-zero if {bn}and {dn}are representations of the
same element in HD(M). Finally, one checks that this result holds for all
polynomials:
lim
n→∞ξnPm(δnan1,...,δnani, am1,...,amj)ξn<∞,{ani, amj}∈HDn,
where Pmis an arbitrary polynomial of degree mand where the sequences
{ani},{amj}represent elements in HD(M). We therefore conclude that the
23

state ξnprovides us, in the limit n→∞, with a state on HD∗(M). We de-
note the corresponding Hilbert space, obtained from the GNS construction,
by H(ξc,HD∗).
3.5 Unbounded operators and the dQHD∗(M)algebra
The next step is to analyze whether the operator δitself can be understood
in terms of a GNS construction over a suitable algebra. To this end we start
by introducing a graded and flow-dependent version of the dQHDnalgebras
as well as the dQHD(M)algebra.
Definition 3.5.1 We define the flow-dependent dQHDnalgebra, which we
denote by dQHD∗
n, as the algebra generated by elements in HDnand by
elements of the form
[Dn,[Dn,...[Dn, a]...]] , a ∈HDn.
Furthermore, we define the flow-dependent dQHD(M)algebra, which we
denote by dQHD∗(M), as the n→∞limit of dQHD∗
n.
We are now going to show that the state ξcon HD∗(M)can be gener-
alized to a state on dQHD∗(M)as well. To ease the notation we shall in
the following write δnas δ.
First, consider the m’th commutator, with meven
ξnδmanξn, an∈HDn.(17)
One checks that the msums in (17) are collapsed by the trace over the
Clifford elements to m2 sums with the correct factor of dxm/2=2−m/2,
where each sum indexed by iruns over an insertion of the form
2−nκ2σaLea
i+2−2nκ2(σaea
i)(σbeb
i)(18)
where sums over the Lie-algebra indices aand bare implied. Thus, in the
limit this converges to m2 line integrals. The convergence of (17) in the
limit n→∞depends therefore entirely on how the expectation values of
the Lea
i’s behave. Therefore we now require the states ξnto satisfy both
condition 1) in (11) and condition 3) in (12), where the latter here simply
implies that the expectation values of powers of the vector-fields are required
24

to depend smoothly on the points in M. We thus conclude that (17) will
converge and be finite:
lim
n→∞ξnδmanξn<∞, an∈HDn, m even.(19)
Next, consider also the the m’th commutator, with muneven. In this
case the entire expression will be non-trivial with respect to the Clifford
algebra and thus its expectation value vanish
lim
n→∞ξnδmanξn=0, an∈HDn, m uneven.(20)
However, in the case where mis uneven one may consider the expression
(δma)(δka∗),
where kis uneven as well. This is an unbounded operator of order (m+k)2,
which involves (m+k)2 sums over insertions of the form (18). Again, since
the expectation values of each Lea
iis a smooth function on Mwe conclude
that this expression will have a finite expectation value
lim
n→∞ξn(δma)(δka∗)ξn<∞, an∈HDn, m, k uneven.(21)
Also, this expectation value will vanish if we instead choose m+kto be
uneven.
Next, we consider an expression of the form
(δm1a1,n)(δm2a2,n )...(δmlal,n), ai,n ∈HDn
If the sum ∑l
i=1miis uneven, then the expectation of this expression will
again vanish since it is non-trivial with respect to the Clifford algebra. If
the sum is even, then it depends whether each element ai,n in HDnequals
the inverse of one of the other elements ai,n in HDn(or, possible, equals the
inverse of a section of one of the other elements). If this is not the case then
the expression will again be non-trivial with respect to the Clifford algebra
and the expectation value will vanish. If this is the case, then it will involve
(∑l
i=1mi)2 sums over insertions (18), which is again well defined since the
right factor of dx appears, and we conclude:
lim
n→∞ξn(δm1a1,n)(δm2a2,n )...(δmlal,n)ξn<∞, ai,n ∈HDn.(22)
Finally, none of the above results are changed by insertion of an element
of HDn, for example (δman)bn(δmcn)with an, bn, cn∈HDn, and we may
therefore conclude that
lim
n→∞ξnPm(a1,n, δ a2,n, δ2a2,n,...,δmam,n)ξn<∞, ai,n ∈HDn,(23)
25

where Pmis a polynomial of order m. This, in turn, implies that ξnconverges
to a state ξcon dQHD∗(M)in the limit n→∞.
Note that the operator δis not a well defined operator in the GNS
construction around the state ξcsince the expectation value of δ2diverges
on ξc. This is in accord with our general philosophy that only operators
where Dis commuted with elements in HD(M)– and thus not the vacuum
state ξc– are well defined due to their one-dimensional smearing.
We denote by H(ξn,dQHD∗
n)the Hilbert space emerging from this GNS
construction over dQHD∗
nand by H(ξc,dQHD∗)its n→∞limit. The
Hilbert space H(ξc,dQHD∗)can – and will in the following – be understood
as a kinematical Hilbert space for quantum gravity since the operators in
dQHD∗(M)encode information about the canonical commutation relations
of canonical quantum gravity formulated in terms of Ashtekar variables. The
state ξcwill play the role of a vacuum state in our construction.
Note that we could have carried out the above construction without
reference to a Dirac type operator, but simply by inserting the right-invariant
vector fields into the flows by hand. As we pointed out above, we believe
that the Dirac type operator represents a significant structure and therefore
we have included it.
Note also that there exist an alternative approach, where instead of
the algebra dQHD∗(M)we consider an algebra generated by elements of
HD(M)and by second commutators between the Dirac type operator and
elements of HD(M)only:
a, {D, [D, b]} a, b ∈HD(M).
This algebra will not involve polynomials of vector-fields associated to the
same position and will therefore in some respect be better behaved. Since
this algebra also encodes the kinematics of canonical quantum gravity it
could be an interesting alternative to consider.
Let us finally point out that it is in fact an open technical question
exactly what algebra the GNS construction is built over. The uncertainty
has to do with the norm for the C∗-algebra that arises in the continuum
limit. Since the vector fields in the dQHD∗(M)algebra may have a non-
measurable effect on the spectrum of the holonomy-diffeomorphism algebra
(see section 3.7 for a discussion hereof) it may be that the relevant algebra is
built with respect to the counting measure instead of the Lebesque measure.
We shall return to this question in the final discussion.
26

3.6 Constructing the state
So far we have identified conditions for a state on dQHD∗(M)to exist. The
conditions were that the expectation value of ∇(li)must be infinitesimally
close to the identity and that expectation values of ∇(li)and of all powers
of the vector fields should be smooth with respect to M, see (12).
Let us now for a moment concern ourselves with the actual construction
of this state. In particular, let us consider a construction, which relies on
coherent states on SU (2)as have been considered by Hall [18, 19] and others
[20, 21, 22]. Thus, we pick a phase-space point (A(x), E(x)) of Ashtekar
variables and consider first the function
ξ(n,i)(g)=φκ
(A,E,i)(g)
where φκ
(A,E,i)(g)is the coherent state on a copy of SU (2)assigned to the
i’th edge in Γn, where κis a quantization parameter and which satisfies the
properties
lim
κ→0φκ
(A,E,i)κLea
iφκ
(A,E,i)=iEµ
a(¯
li),(24)
and
lim
κ→0φκ
(A,E,i)⊗v∇(li)φκ
(A,E,i)⊗v=(v, Hol(li, A)v),(25)
where v∈C2, and (,)denotes the inner product hereon. The index µdenotes
the spatial orientation of the ledge li. The construction of the coherent state
φκ
(A,E,i)is described in [20, 21, 22] and relies on a so-called complexifier,
which determines the exact localization properties of φκ
(A,E,i).
Consider first
Mκ
li=φκ
(A,E,i)∇(li)φκ
(A,E,i)SU(2) ,
which is the SU (2)-valued expectation value of a parallel transport on the
edge li.Mκ
liwill be of the general form
Mκ
li=gi(1−κ(ci+σjcj
i))
where ci, cj
iare positive parameters, which depend on the exact form of the
coherent state and on κ, and where gi=Hol(li, A). The matrix (1−κ(ci+
σjcj
i)) will have operator norm strictly smaller than 1, which implies that
the first condition in (12) cannot be satisfied. If, however, we scale κin the
coherent states φκ
(A,E,i)with a factor dx
φκ
(A,E,i)→φs
(A,E,i), s =κdx (26)
27

where dx in the lattice approximation equals dx =2−n, then we obtain
Ms
li=(1+dxA)(1−s(ci+σjcj
i))∶=1+dx(A+κB)(27)
where Bis a quantity that depends on both (A, E),κand the specific form
of the coherent state φs
(A,E,i). Thus, φs
(A,E,i)satisfies the first condition in
(12).
In the passing we note that Bin (27) cannot be a connection since
Ms
liMs
l−1
i=1.
The scaling of κhas, however, implications for the peakedness of φs
(A,E,i)
over the point Eµ
a(¯
li)in (24) since this is now shifted with a factor of dx−1.
Thus, we must adjust this with a second scaling
φs
(A,E,i)→φs
(A,dxE,i)
and we thus reach the result that the state built from
ξ(n,i)(g)=φs
(A,dxE,i)(g)
satisfies the requirements for the expectation values on it. Since Aand E
are smooth on Mall expectation values will be smooth as required.
Let us for later reference introduce the notation ξκ
(A,E)for a state on
dQHD∗(M), which is constructed from Hall’s coherent states on SU (2)in
this way.
3.7 The spectrum of the semi-classical states
We will now look at how the algebra affects the spectrum of the coher-
ent state, i.e. which kind of translations on the space of connections that
appears.
In order to see more clearly what happens to the spectrum, we will in
this section consider operators, which are a bit different from those in the
previous sections. Also we will consider a slightly different state. At the end
we will comment on the difference.
We first need a little bit of notation. The system af Graphs {Γn}induces
a coordinate system. It therefore also induces a one-norm
(x1, x2, x3)1=x1+x2+x3,
28

and a length function of path, i.e. if γ∶[a, b]→Mis a path, the length is
defined via
L(γ)=b
a˙γ(t)1dt.
Let pn={l1,⋯, lk}be a path in Γn, and let us assume that pnapproximates
pwhen n→∞, i.e. where pis a smooth path in M. Again we denote the
operator associated to pnwith ∇(pn)=∇(l1)∇(l2)...∇(lk). The double
commutator with the Dirac operator is given by
{Dn,[Dn,∇(pn)]} =
2−n(D1∇(pn)+∇(l1)D2∇(l2)⋯∇(lk)+
...+∇(l1)⋯∇(lk−1)Dk∇(lk)),(28)
where Didenotes here the Dirac operator on the copy of Gassociated to li,
i.e. the operator σaLeaon S U (2).
Instead of coupling the Leato σa, we can couple it to an ω∈Ω1(M, su(2))
in the following way: we consider lias a tangent vector in the starting point
piof li. In this way we can replace σaLeaon the i’th copy of Gwith
ωa
li(pi)Lea
i,
and insert this in (28) instead of Di. Since the length of liis 2−nwe propose
the operator
E(ω,p,n)=ωa
l1(p1)Lea
1∇(pn)+∇(l1)ωa
l2(p2)Lea
2∇(l2)⋯∇(lk)+
...+∇(l1)⋯∇(lk−1)ωa
lk(pk)Lea
k∇(lk).
Note, that this is nothing but the derivative with respect to tin t=0 of
U(tω,p,n)=exp(tωa
l1(p1)Lea
1)∇(l1)exp(tωa
l2(p2)Lea
2)∇(l2)
...exp(tωa
lk(pk)Lea
k)∇(lk).(29)
If we therefore compare to the formula (3) and if we neglect the ∇(li)and
the moving of the starting point of pto the endpoint of p, then we see
that in the continuum limit (n→∞) the operator U(ω,p,n)corresponds to a
translation of Aover pby ω, i.e.
(B(p,ω))X(m)=AX(m),(m, X )∉p
AX(m)+ωX(m),(m, X)∈p,
where (m, X)∈T M . Note that this is a ”singular” connection since it is
given by a smooth connection which is deformed over the path p.
29

Let us now look at what happens if we act on a semi-classical state.
We will here consider a semi-classical state, which is associated to a single
vertex. Let ψbe a spinor over M. With the notation of section 3.3 we
consider
ξ(n,A,E)(v)=ψ(v)∣l∣

i=1
ξ(n,i),
where vdenote a vertex in Γnand where (A, E)denotes a semi-classical
phase-space point. The state ξ(A,E)is the continuum limit of ξ(n,A,E ). Fur-
thermore we also consider
η(n,A,E)=∣l∣

i=1
ξ(n,i)
and η(A,E)as the continuum limit thereof. Let F=feXbe a flow. This is
approximated in the lattice Γnby a family of paths. Let Fmbe the path
ending in mgenerated by F, and let U(ω,F )be the operator ∑m∈MU(ω,Fm),
where U(ω,Fm)is the continuum limit of U(ω,Fm,n). From the above we see
that8
(U(ω,F )(ξ(A,E)))(m)=H ol(B(p,ω ), Fm)ψ(F−1(m))η(B(Fm,ω),E).
If we forget the factor Hol(B(p,ω), Fm)ψ(F−1(m)) we see that each semi-
classical state η(B(Fm,ω),E)is sitting in a point m∈M. Therefore the opera-
tor U(ω,F )introduces discontinuities in the state, since for two neighboring
points m1, m2the states η(B(Fm1,ω),E)and η(B(Fm2,ω ),E)are not necessarily
close, because the singularities added to Ain the two states are sitting over
different paths, namely over Fm1and Fm2.
We can see the state U(ω ,F )ξ(A,E)as a kind of integral over the η(B(Fm,ω),E),
and therefore also as an integral over the representations of the holonomy-
diffeomorphism algebra induced by the connections B(Fm,ω). However, al-
though the spectrum contains a lot of singular objects, and it does not
appear to contain any transition between different smooth connections (see
section 8), some of the expectation values in the GNS-construction bear re-
semblance to a transition between smooth connections. Let us for example
consider a closed flow F. The expectation value
U(ω,F )ξ(A,E)FU(ω ,F )ξ(A,E)
8Here we only concern ourselves with the effect on the connection. We are not sure
what happens to the field E.
30

will look like the expectation value of the connection A+ω′, where ω′depends
on Fand ω, i.e. it will look like the expectation value of a smooth connection
different from A.
Note that exponentiated vector-fields of the form (29) do in fact not
directly arise in the dQHD∗(M)algebra. However the difference between
the double commutators and the exponentiated vector-fields should be seen
as the difference between the usual Dirac-operator on a manifold and the
flows of vectorfields on the same manifold. The first does not directly moves
point on the manifold, but contains all the infinitesimal translations. We
therefore expect that the double commutators contain infinitesimal transla-
tions in the directions between Aand the B(p,ω)’s. More analysis is needed
to determine exactly how the algebra dQHD∗(M)affects the spectrum of
the semi-classical state.
3.8 A bimodule over volume-preserving diffeomorphisms
Let us end this section with a brief comment on the mathematical structure,
which we have obtained so far9. Before we do this we need however a
few definitions and some notation. Also, we shall in this subsection use
mathematical terms, which are explained in [5] and in the literature cited
there.
Consider again the representation of an element in HDnin Hnand H′
n
according to (7). We may, in a completely similar manner represent diffeo-
morphisms in Hnand H′
nas well as in H(ξn,dQHD∗
n), simply by replacing
the holonomy transforms in matrices like (7) by the two-by-two identity
matrix. We invite the reader to see [5] for details and shall here simply
introduce the notation Diffnand Diff(M)for the corresponding algebra of
diffeomorphisms restricted to Γnand the algebra of diffeomorphisms on M,
respectively. Furthermore, denote by Diffvol,n and Diffvol(M)the corre-
sponding algebras of volume-preserving diffeomorphisms. At the level of a
lattice Γnthe volume-preserving diffeomorphisms are the diffeomorphisms,
which are invertible.
Now, we first note that the state ξcdefines a Hilbert Diffvol(M)-module
with an action of both HD(M),HD∗(M)and dQHD∗(M). This arises
by leaving out the trace over the vn×vnmatrices in H(ξn,dQHD∗
n). In this
way the dQHD∗(M)algebra has a left-action and the Diffvol(M)algebra
a right action. The reason why we restrict ourselves to volume-preserving
9Here we shall assume that the issue with the invariance properties of Draised in
section 7.1 has been resolved. This means that the factor 2−nin Dshould be viewed as a
one-form.
31

diffeomorphisms – and not simply all diffeomorphisms – is given by our next
observation: the operator δcommutes with Diffvol(M). Thus, if δhad been
a Dirac-type operator we would have a Kasparov (HD(M),Diffvol (M))-
bimodule. Instead δis a commutator with a Dirac-type operator and thus
the construction is of a somewhat different nature.
4 Emerging elements of fermionic QFT
In this section we will analyze states in H(ξc,dQHD∗)from which a link to
fermionic QFT emerges in a semi-classical limit. We show that both a spatial
Dirac operator and the principal part of the Dirac Hamiltonian emerges from
the construction. This analysis built on the papers [9]-[12] and [17].
4.1 Infinitesimals
First we need to define infinitesimal elements in HD(M), where infinitesimal
is understood with respect to the manifold M. These infinitesimal elements
are again defined via the algebras HDnand are given by an infinite sequence
of lattice approximations, which we denote {dan}, where dan∈HDnconsist
of matrix entries, which are only non-zero if they correspond to adjacent
vertices: (dan)ij =gkor 0 if vi, vjare adjacent
0 else ,
where gkis an element of a copy of SU(2)assigned to the edge lkthat
connect adjacent vertices viand vjin Γn.
Next, we define corresponding elements in HD∗
nand HD∗(M), which
are both infinitesimal and non-trivial with respect to the Clifford algebra.
The matrix entries of these elements are of the form
(d˜an)ij =2−nea
kσagkor 0 if vi, vjare adjacent
0 else ,(30)
where the configuration of vertices is as above. Note that
d˜an=1
κδndan.
which means that d˜ais, in the language of non-commutative geometry, a
one-form with respect to the space Aof connections.
32

4.2 A two-point lattice state
Let us for simplicity first consider a lattice Γnwith only the two vertices
v1and v2and an edge lkconnecting them, and let danand d˜andenote the
corresponding element in HD∗
nthat connects them. Also, for the remainder
of this section we shall ignore the second term in (18) since this term does
not play a role in the semi-classical analysis, which we are concerned with
here. Although both terms in (18) are proportional to κ2the first term is
effectively of order κsince the right-invariant vector field absorbs one order
of κin the semi-classical limit (see for instance equation (24)).
Consider the state
ρn=(ψ(v1)+22nd˜aψ(v2)da∗)ξn
where ψ(vi)is a 2 ×2-matrix associated to vi. The double factor of 2n
corresponds to two one-dimensional delta-functions in the n→∞limit,
which means that this state does, strictly speaking, not correspond to a
continuum state in the Hilbert space H(ξc,dQHD∗). These delta-functions
correspond to the localization of the infinitesimal element da in HD(M)in
the direction of its flow. Ignoring this issue we are going to compute the
expectation value of Dnto lowest order in κon this state. Due to the trace
over the Clifford algebra we find
ρnDnρn=ξn(ψ(v1)+22nd˜aψ(v2)da∗)Dn(ψ(v1)+22nd˜aψ(v2)da∗)ξn
=22nξnψ(v1)Dnd˜aψ(v2)da∗ξn
+ξnd˜aψ(v2)da∗Dnψ(v1)ξn.
To proceed we need to denote the classical limit κ→0 of the basic operators
on the state ξ(n,k). We do this in a manner parallel to (24) and (25):
lim
κ→0ξ(n,k)κLea
kξ(n,k)=iEµ
a(x)(31)
lim
κ→0ξ(n,k)⊗vgkξ(n,k)⊗v=(v, Hol(lk, A)v)(32)
where v∈C2and where (,)denotes the inner product hereon. Also, the
index µ∈{1,2,3}corresponds to the spatial orientation of lkand ’x’ refers
to the endpoint of lk. We take the set (E, A)to be a pair of conjugate
Ashtekar variables – Eµ
a(x)is an inverse densitized triad field and Aa
µ(x)an
SU (2)connection – so that H ol(l , A)denotes the holonomy of a connection
33

Aalong l. With this in place we find
lim
κ→022nξnψ(v1)Dnd˜aψ(v2)da∗ξn
=lim
κ→02nξnψ∗(v1)κLea
kσagkψ(v2)g∗
kξn
=2nTrM2(ψ∗(v1)iEµ
aσaHol(lk, A)ψ(v2)Hol(lk, A)∗).
We are now going to let the two vertices approach each other. With a
deliberate misuse of notation, we call this the limit n→∞despite the fact
that we, for now, keep n=2 fixed. The point is that we shall, in a next
step, consider a full lattice, for which the n→∞limit implies that adjacent
lattice points converge onto each other with respect to the lattice metric,
i.e. limn→∞2−n=dx. Thus, keeping this in mind, we may write
lim
n→∞Hol(lk, A)=1+dxAµ,lim
n→∞ψ(v2)=ψ(v1)+dx∂µψ(v2)
which implies
lim
n→∞Hol(lk, A)ψ(v2)Hol(lk, A)∗=(1+dx∇µ)ψ(v1)
where ∇µ⋅=∂µ+[Aµ,⋅]is the covariant derivative. Adding all this up we
find
lim
κ→0lim
n→∞ρnDnρn
=TrM2(ψ∗(x)Eµ
aiσa∇µψ(x)−(∇µψ(x))∗Eµ
aiσaψ(x))
=TrM2(ψ∗(x)(Eµ
aiσa∇µ+∇µEµ
aiσa)ψ(x))
=TrM2ψ∗(x)
Dψ(x)(33)
where we wrote ψ(v1)as ψ(x)and where we in the second line permitted
ourselves a partial integration although we will not have this until we con-
sider a full lattice and its continuum limit. Note that with our conventions
iσaequals the standard Pauli matrices.
To get this far we had to start with a state ρnwith a factor of 2nin
front of each infinitesimal element dan, which means that this state does
not correspond to a state in H(ξc,dQHD∗)in the n→∞limit. The issue is
clear: when we act with δnwe get a factor of 2−nfrom Dn, which means
that δnan, with an∈HD∗
n, has exactly the right factor to give a well-
defined line-integral in the limit n→∞. This, however, becomes problematic
when we have an infinitesimal element in HD∗
n, since the corresponding line
integral just gives a dx, which vanishes. This explains the factor of 2n,
34

which effectively corresponds to a one-dimensional delta-function. Perhaps
one can compare this to the situation in quantum mechanics, where the
plane wave is not square integrable. It may be that the strict localization of
da in the direction of the flow is too idealized a point of view, albeit not of
principal concern. Note also that δnρndoes descent to a well-defined state
in H(ξc,dQHD∗).
4.3 An emergent one-particle state
Let us now consider a sequence of lattice approximations {Γn}and consider
the state at the n’th level of approximation:
ρn=(ψ+22n
3

m=1
d˜a(m)
nψda(m)∗
n)ξn(34)
where da(m)
nand d˜a(m)
nare now infinitesimal elements of HD∗
n, oriented in
the m’th direction, m∈{1,2,3}, and otherwise as defined previously. By ψ
we now denote a vn×vndiagonal matrix, where each diagonal entry (j, j)
is a M2(C)-valued field ψ(vj). In the continuum limit ψbecomes a spinor
ψ(x).
It is now straight forward to repeat the computation of the previous
section. We are now dealing with vn×vnmatrices and their limit as
ngrows infinitely large, where the trace over M∣vn∣(C)converges onto a
Riemann integral, but at the level of each matrix entry the situation is
identical to the situation encountered above with the exception that we are
now including all three spatial directions. Thus, we write
lim
κ→0lim
n→∞ρnDnρn=lim
κ→0ρDρ=Md3xTrM2ψ∗(x)
Dψ(x)(35)
where the domain of the integral is determined by the spatial domain of the
holonomy-diffeomorphism da.
Note that in the above derivation we did not pay particular attention
to the order of the two limits, κ→0 and n→∞, which is justified since
these limits commute. This is one major difference between the computation
given here and what we found in the papers [9]-[12] and [17], where the setup
entailed that these limits did not commute: the semi-classical limit had to
be taken before the continuum limit.
To obtain the Dirac Hamiltonian in a semi-classical limit we perform a
transformation in the M2(C)-factor in H(ξn,dQHD∗
n)in order to include the
lapse and shift fields. Thus, consider an vn×vnmatrix Mnwith diagonal
35

entries given by two-by-two self-adjoint matrices M(vi), together with a
sequence of such matrices assigned to lattice approximations Γn. Consider
the transformation ρnDnρn→ρnMnDnρn
which gives
lim
κ→0lim
n→∞ρnMnDnρn
=Md3xTrM2ψ∗(x)N(x)
Dψ(x)+ψ∗(x)Nµ(x)e∂µψ(x)
+zero order terms (36)
where we wrote M(vi)∶=M(x)=N(x)+Na(x)σawith N(x)and Na(x)
playing the role of the lapse and shift fields.
Thus, we find that the principal part of the Dirac Hamiltonian emerges
in a semi-classical limit of the gravitational degrees of freedom from a natu-
ral class of states in H(ξn,dQHD∗
n). We obtain the lapse and shift fields from
a local transformation in the spinor degrees of freedom represented by the
M2(C)-factor in H(ξn,dQHD∗
n). This establishes a connection to fermionic
quantum field theory and in particular it shows that the framework of quan-
tum holonomy theory has the potential of producing canonical matter de-
grees of freedom coupled to quantum gravitational degrees of freedom.
It is not clear what geometrical significance the structure of the one-
particle state (34) has. In terms of non-commutative geometry it involves a
one-form, and if we consider the quantity d˜ada∗alone, which we write (up
to a constant) as [D, da]da∗,
then we recognize the quantity, which arises from a fluctuation of the Dirac
type operator with by an infinitesimal inner automorphism:
D→daDda∗=D−[D, da]da∗.
It is, however, not clear to us if this is of significance. Alternatively one
could reproduce the above results as the inner product between states, that
involve the double commutator with Dn. This interpretation would be more
in line with the emphasis put on the algebra dQHD∗
nand would circumvent
the fact that the Dirac type operator is not a well defined operator in the
Hilbert space H(ξc,dQHD∗).
One might, however, speculate whether the algebra of differential forms
generated by the Dirac type operator is related to a type of Fock-space
36

structure, where one-forms corresponds to one-particle states and n-forms
to n-particle states. This idea was put forward in [12], but the results on
n-particle states obtained there cannot be incorporated in the setting of this
paper in a straight forward manner.
The introduction of the Dirac type operator and the Clifford algebra
does entail the interesting concept of orthonormal diffeomorphisms, which
might be of significance when one is to determine to what extend more
elements of fermionic quantum field theory can be uncovered from quantum
holonomy theory. Therefore, let us end this section by developing the notion
of orthogonality among diffeomorphisms. First, we define a class of elements
in HD∗
n, which are built from elements similar to the d˜an’s. A characteristic
of the operator d˜anis that it is not unitary. We can remedy this by changing
its definition in (30) to (see [12])
(d˜an)ij =2−(n+1)(ea
kσa+ie1
ke2
ke3
k)gk,(alternative definition)
and we check that d˜anwith this definition is unitary. In [12] we also found
that
TrC l((d˜an)abM(d˜an)∗
ba)=M0
where Mis a two-by-two matrix, which we write as M=Maσa+M01, and
where ’ab’ refers to a pair of adjacent vertices in Γn. Thus, conjugation with
d˜ancorresponds to taking the trace in the M2(C)-factor of H(ξn,dQHD∗
n).
This fact played a crucial role in the analysis carried out in [12].
We now define a class of elements in HD∗
nas products of these infinites-
imals (we omit the subscript ’n’)
˜a=d˜a1d˜a2...d˜ak,with a=a1a2...ak.
Notice that these elements in HD∗
nare mutually orthogonal
˜a˜a′=δa,a′
where δa,a′equals one if a=a′and zero else. This, too, will apply in the
continuum limit. Thus, this introduces a notion of orthogonality among
holonomy-diffeomorphisms.
5 On a dynamical principle
We are now going to consider the dynamics of general relativity, which in
the formulation in terms of Ashtekar variables is encoded in two constriants:
37

the Hamilton and the diffeomorphism constraints. The Hamilton constraint
has the form10
H(N)=Md3xNab
cFc
µν Eµ
aEν
b,(37)
where Nis the lapse field, Eµ
a(x)the inverse densitized dreibein and Fµν(x)
the field-strength tensor of the Ashtekar connection Am(x). The diffeomor-
phism constraint has the form
D(¯
N)=Md3xNνFc
µν Eµ
c,(38)
where Nνis the shift field. The aim of this section is twofold: first to
investigate how we can obtain these classical constraints in a semi-classical
limit from operators acting in H(ξc,dQHD∗), and second to check whether
these constraint operators satisfy an algebra comparable to the classical
constraint algebra. The latter issue is crucial since it determines to what
extend general covariance is maintained in the quantized theory11. The
classical constraint algebra modulo the Gauss constraint has the form:
H(N), H (N′)PB =D(N∂N′−N′∂N )(39)
H(N), D(¯
N)PB =H(N∂ ¯
N−¯
N∂N )(40)
D(¯
N), D(¯
N′)PB =D(¯
N∂ ¯
N′−¯
N′∂¯
N)(41)
5.1 The Hamilton constraint operator
In the following we first provide a geometrical motivation for choosing our
candidate for a Hamilton constraint operator. Then we compute the com-
mutator between two Hamilton constraint operators – which differ by choice
of lapse fields – and derive from that a candidate for a diffeomorphism con-
straint operator. With that we then compute the rest of the operator con-
straint algebra.
As already mentioned, a one-form in the vocabulary of noncommutative
geometry is given by an operator of the form a[D, b]where a, b are elements
of the particular algebra at hand and Dis a Dirac operator. Here we need
10Here we write the densitized Hamilton constraint [1] in which the integral is not
invariant but depends on scale due to an extra factor of e. Also, as pointed out elsewhere,
this Hamiltonian corresponds either to gravity with Euclidian signature - if the connection
takes values in su(2), or to gravity with Lorentzian signature - if the connection takes
values in the dual sector of sl(2,C), see section 9.
11see [23] for an interesting discussion on this issue in a framework comparable to the
one presented here.
38

one-forms which are infinitesimal with respect to the manifold Mand which
have a certain orientation. We denote by dbn,i,i∈{1,2,3,4,5,6}infinites-
imal elements of HDn, where dbn,1translates in the positive x-direction,
dbn,2in the negative x-direction, dbn,3in the positive y-direction and so
forth. We built two distinct one-forms by
Rn=right

i,j
dbn,i[Dn, dbn,j ]Ln=left

i,j
dbn,i[Dn, dbn,j ]
where the ’right’ sum picks out combinations (dbn,i , dbn,j )that form an angle
in the lattice with a positive orientation and where the ’left’ sum picks out
terms with negative orientation. Recall that the brackets are graded. Note
that the conjugation switches between the ’right’ and ’left’ sectors. We also
write down the corresponding curvature operators
FRn=[Dn, Rn]+1
2[Rn, Rn],
FLn=[Dn, Ln]+1
2[Ln, Ln],
which can be understood as two curvature operators over the approximate
space of connections Anand consequently in terms of curvature operators
over Ain the n→∞limit.
In the following we shall partly be concerned with the lowest orders in κ
in a semi-classical limit. Note that in a semi-classical limit one factor of κwill
be absorbed by a vector-field Lea
jdue to (31) whereas a commutator [Dn, an]
with an∈HDnwill produce a factor of κ. Furthermore, in line with the
previous section, we shall equip each infinitesimal element db with a factor
of 2n. Again, this implies that these elements do in fact not correspond
to operators in H(ξc,dQHD∗), but we shall permit ourselves this discrepancy
and refer to the comments of the previous section.
Let us now consider the matrix entries in Lnand Rn. They are of the
form
κ2nea
jgigjσa,
where giand gjcorrespond to adjacent edges in Γn, which do not have the
same spatial direction. Correspondingly, matrix entries in FRnand FLnare
of the form
2nκ(κLea
j)gigjσa+”three terms” ,(42)
where ”three terms” refer to three additional terms, which we shall for the
moment ignore since they do not affect our analysis. We will return to these
terms later in this section.
39

Let us consider the following operator:
hn=i
2FRn(FLn)∗−FLn(FRn)∗
=ImFLn(FRn)∗.(43)
We shall shortly explain why we choose this particular combination, but let
us first write down the form of the matrix entries that arise. Using (42) we
find
22nκ2(κLea
j)σagigjgkglσb(κLeb
k),(44)
where it is important to notice that the two derivatives in (44) will be asso-
ciated to adjacent edges due to the algebraic structure of (43). Due to the
particular choice of orientations we are certain that in those contributions,
which are diagonal in the M∣vn∣(C)matrix, the four infinitesimal elements
will form a closed loop, which gives the field strength tensor in the semi-
classical limit:
lim
κ→0lim
n→∞ξngigjgkglξn=TrM212+dx2Fµν +O(dx3),(45)
where the indices µand νcorresponds to the plane in which the loop sits.
The first term in (43) contribute with three loops in the three different
lattice planes, all with the same orientation, whereas the second term in
(43) contribute with the inverse of these same three loops. Thus, we have
the algebraic structure
loop −(loop)−1.
This is the reason why we choose the particular L-Rcombination in (43),
because with this algebraic structure, and keeping the first condition in (12)
in mind, we see that the identity terms of the loops as in (45) cancels out
whereas the term, which gives the field strength tensor Fµν in the semi-
classical limit, add up. Also, the reason why we split the one-forms into
”R” and ”L” sectors is to avoid backtracking. With this we can now take
the expectation value of (43) and find
lim
κ→0lim
n→∞κ−2ξnhnξn=Md3xEµ
aEν
bFc
µν ab
c.(46)
Comparing this to equation (37) we see that the densitized Hamiltonian of
general relativity emerges in the semi-classical limit.
The particular form of the operator (43) must be rooted in symmetry
considerations. Without going into details let us here just note that it seems
40

to be invariant under a symmetry that involves a change of orientation of
Mas well as a gauge transformation of the ”connections” Rnand Ln. It
seems natural that the operator (43) should have the canonical form
FAn(FAn)∗
where Anis a one-form that involves edges of both ”right” and ”left” ori-
entation. Such a construction must, however, involve some sort of grading
with respect to orientation, which we at the moment do not know how to
implement. Thus, we work with the operator (43).
Before we continue let us first deal with the ”three terms” in equation
(42). The first term is of the form
κ2ea
ieb
jgiσagjσb.
This term can, however, due to the Clifford algebra elements only give some-
thing non-zero via a backtracking, which will not arise due to the argumen-
tation given above. The second term is of the form
κ2ea
jeb
jgigjσaσb,
which can provide a non-trivial contribution. This, however, will be of higher
orders in κand shall, therefore, not concern us here. The third term comes
from the commutators [Ln, Ln]or [Rn, Rn], but again we find that this term
will vanish since it is non-trivial with respect to the Clifford algebra.
The operator hncan, as it stands, not be interpreted as a Hamilton
operator, since it does not involve a trace over M∣vn∣(C), which in the n→∞
limit gives an integral over M, nor does it involve a trace over M2(C).
Without this trace the operator has no chance of satisfying the right operator
constraint algebra. Therefore, we consider instead
H′
n=Trpartial (hn)∶=dx3
i
H′
vi(47)
where H′
videnotes the i’th diagonal matrix entry of hnand where the partial
trace is a normalized trace over the matrix factors M∣vn∣(C)and M2(C).
Note that it is not certain that this partial trace exist in the n→∞; we here
simply assume that it does.
One possibility is now to define the Hamilton operator Has the con-
tinuum limit of H′
n. Here, however, we run into the same problem that
41

we encountered with the operator Eωand with the Dirac type operator D,
namely that there is a conflict between 1) the existence of the expectation
value of the operator itself on the vacuum state and 2) its commutators
with elements of HD(M)being non-zero. With the definition (47) we will
get the correct expectation value on the vacuum state in the n→∞limit
– the classical Hamiltonian – but commutators between Hand elements of
HD(M)will all vanish.
The solution we choose to circumvent this discrepancy is to change def-
inition (47) to
H′
n∶=dx 
i
H′
vi,(new definition) (48)
which gives an operator that is only well defined in terms of commutators
with elements in dQHDn.
5.2 The operator constraint algebra
With definition (48) we are in a position where we can start computing
the operator constraint algebra and use the bracket [H′
n(N),H′
n(N′)] to
derive the diffeomorphism constraint operator. Before we do that we shall,
however, write down yet a new version of (48) where the vector-fields are
ordered symmetrically and where we also include a lapse field. This defi-
nition of the Hamilton operator is a simplification of (48) since it involves
three instead of twelve loops based at each vertex, which greatly simplifies
the computations. Thus we write
Hn(N)=dx 
i
Hvi(Nvi)(49)
with the vertex operators Hvi(Nvi)given by
Hvi(Nvi)=H(xy)
vi(Nvi)+H(yz)
vi(Nvi)+H(zx)
vi(Nvi)
and
H(xy)
vi(Nvi)=22n−2NviLvi
eb
y
,Lvi
ea
x,Tr σaÈ
Lvi
xy σb
H(yz)
vi(Nvi)=22n−2NviLvi
eb
z
,Lvi
ea
y,Tr σaÈ
Lvi
yz σb
H(zx)
vi(Nvi)=22n−2NviLvi
eb
x
,Lvi
ea
z,Tr σaÈ
Lvi
zx σb (50)
where Nis the element in M∣vn∣(C)⊗M2(C), which is diagonal in the first
factor and which gives rise to the lapse field. Nviis the two-by-two diagonal
42

matrix entry assigned to vi. We use the shorthand notation
È
Lvi
µν =Lvi
µν −Lvi
µν −1
where Lvi
xy is again the loop in the xy-plane based in vi, where we write
Lvi
xy =g1g2g−1
3g−1
4. Likewise for Lvi
yz and Lvi
zx. Note again that we have
here simplified the operator by only considering loops that start out in the
positive x,yand zdirections. The vector-fields are defined as
Lvi
ea
x=1
2Lvi
ea
2+Lvi
ea
4,Lvi
ea
y=1
2Lvi
ea
1+Lvi
ea
3,
where the right-invariant vector-field Lvi
ea
jis the vector-fields interacting with
the gjelement in the loop based in vi. This means that the vector-fields
in the operators (50) appear in all possible adjacent pairs. The vector-
fields appearing in the operators in the ’yz’ and ’zx’ planes are constructed
in the same way. This notation is not completely unambiguous for the
computational setup we shall encounter, but we trust that no confusion will
arise. In appendix A.1 we show that
Hn(N),Hn(N′)=Dn(¯
N(N, N ′))+Ξn(N, N ′)(51)
where
Dn(¯
N)=dx 
vi
1
2¯
Nx
vi,Dvi
x+¯
Ny
vi,Dvi
y+¯
Nz
vi,Dvi
z
and
Dvi
x=22nDvi(xy)
x−Dvi(zx)
x
Dvi
y=22nDvi(yz)
y−Dvi(xy)
y
Dvi
z=22nDvi(zx)
z−Dvi(yz)
z
where
Dvi(xy)
y=1
2Lvi
ea
x,Tr σaÈ
Lvi
xy ,Dvi(xy)
x=1
2Lvi
ea
y,Tr σaÈ
Lvi
xy ,
Dvi(yz)
z=1
2Lvi
ea
y,Tr σaÈ
Lvi
yz  ,Dvi(yz)
y=1
2Lvi
ea
z,Tr σaÈ
Lvi
yz  ,
Dvi(zx)
x=1
2Lvi
ea
z,Tr σaÈ
Lvi
zx ,Dvi(zx)
z=1
2Lvi
ea
x,Tr σaÈ
Lvi
zx ,(52)
43

and with
¯
Nµ(N, N ′)=
vi
ν=x,y,z N ∂νN′−N′∂νN1
4Lvi
ea
ν,Lvi
ea
µ,○
where we wrote (N∂νN′−N′∂νN)instead of its lattice approximation in-
volving factors like 2n(NviN′
vi+1−N′
viNvi+1)and thereby anticipating its con-
tinuum limit. We also wrote a ’○’ indicating that ¯
N(N, N ′)appears in (51)
in terms of anti-commutators with the Dvi’s.
Thus, we find that the commutator (51) successfully reproduces the
structure of the classical Poisson bracket (39) up to a potentially anomalous
term Ξn(N, N ′). The operator Ξn(N, N ′)consist of terms, which involve a
factor (Avi+1−Avi)where Aviis either a vector-field or a loop operator or a
combination of both. In order to rule out anomalies in this sector of the con-
straint algebra we need to show that all commutators with Ξn(N , N′)van-
ishes in a non-trivial sector of H(ξn,dQHD∗
n)in the continuum limit n→∞.
This will happen if the factor (Avi+1−Avi)produces a factor of dx in this
limit, which is the case at least on H(ξn,HDn)due to the requirement that
expectation values of all powers of the vector-fields be smooth with respect
to M.
The question is whether the factors (Avi+1−Avi)produces factors of
dx on all of H(ξn,dQHD∗
n)in the continuum limit. In fact, we expect that
this will not be the case since exponentiated vector-fields will produce fi-
nite translations on the configuration space of connections (see section 3.7),
which may cause the factor (Avi+1−Avi)to remain finite in the continuum
limit. Put differently, in section 7.3 we show that the action of the diffeomor-
phism group on holonomy-diffeomorphisms is not strongly continuous when
we also include the vector-fields – i.e. on dQHD∗(M)– and therefore there
will be no infinitesimal generators of diffeomorphisms. Thus, one cannot
expect the constraint algebra to close on H(ξc,dQHD∗)and in fact the mean-
ing of diffeomorphisms on the entirety of H(ξc,dQHD∗)may be questionable.
Since the holonomy-diffeomorphisms are a part of the conjugate variables, it
seems plausible that diffeomorphism invariance and covariance, which, after
all, must be regarded as classical concepts, cannot be maintained in the full
quantum theory.
This line of reasoning only holds, however, whenever we are dealing
with operators, which entail finite translations on the configuration space
of Ashtekar connections. If we restrict ourselves to finite orders in κ, which
means finite polynomials of vector-fields and thus infinitesimal translations
only, then we can be certain that a factor (Avi+1−Avi)will produce a factor
44

of dx in the continuum limit. This means that the bracket (51) will at least
close off-shell at all finite orders in κwithout anomalies.
Thus, we arrive at the conclusion that the ’Hamilton-Hamilton’ sector
of the quantum constraint algebra closes off-shell without anomalies at all
finite orders in perturbation theory.
Notice again that the operators on the right hand side of Eq. (51) are
only well defined in terms of commutators with operators on H(ξn,dQHD∗
n).
In fact, had we instead provides the Hamilton constraint operator in Eq.
(49) with a factor of dx3, so that its vacuum expectation value existed, then
the commutator (51) would vanish.
The result (51) provides us with a candidate for a diffeomorphism con-
straint operator, which is Dn(¯
N). If we multiply this operator with a dx2
we can check that for n→∞its semi-classical limit in fact coincides with
the classical diffeomorphism constraint.
The next step is to compute the commutator [Dn(¯
N),Dn(¯
N′)]. This is
done in appendix A.2. Here we find, however, that the operator Dn(¯
N)does
not entail commutators that matches the structure of the classical Poisson
brackets (40) and (41). Furthermore, the anomalous terms are of a nature,
which suggest that our choice of diffeomorphism constraint operator must
be flawed and thus that our initial ansatz for a Hamilton constraint operator
is in need of modification.
To be specific, we find that terms emerge, which are divergent because
they involve non-vanishing contributions from vertex operators associated
to the same vertex. We also find terms, which correspond to classical terms,
that involve a derivative of a triad field. Such terms also appear in the
computations of the classical Poisson bracket except that here they fail to
be covariant. Had they been covariant they could either be attributed to on-
shell closure or to an anomaly involving a torsion operator, but with ordinary
derivatives appearing and with divergent terms we believe this shows that
our initial ansatz simply cannot be correct.
The commutator [Hn(N),Dn(¯
N)] produces similar anomalous terms
and we do not write it down.
It is possible to choose a diffeomorphism constraint operator from (51),
which is quadratic in the vector-fields. Classically this amounts to a diffeo-
morphism constraint, which takes values in the Lie-algebra and thus couples
to a lapse field with an su(2)index. This choice of diffeomorphism constraint
operator, which may be attractive for other reasons that we comment on
shortly, does, however, not solve the problems with anomalies.
45

Let us consider what goes wrong (see appendix A.2 for a more detailed
discussion). When we computed the ’Hamilton-Hamilton’ sector of the con-
straint algebra we found that certain commutators have the general algebraic
structure
Av1Bv2+Av2Bv1(53)
with an overall factor given by the lapse fields. This structure, which secures
that no misplaced derivatives survives, is absent in the ’diffeomorphisms-
diffeomorphisms’ sector of the constraint algebra due to the richer index
structures given by the lapse fields. We suspect that with an initial ansatz
for a Hamilton constraint operator that is more symmetrical one might be
able to obtain the algebraic structure (53) and thereby avoid the problematic
derivative terms and also the divergent terms. Indeed, there exist a number
of different operators within our framework, which all have the classical
constraints as their semi-classical limit and the operator (49) is not the
most general candidate possible. Let us therefore provide a list of possible
modifications to (49), which we think could improve the situation:
1. First of all, the operator Hn(N)only involves three loops based in
each vertex, whereas twelve are possible. The curvature operator hn
in (43) in fact involves these twelve loops.
2. The symmetrization of the vector-fields leading to (49) does not respect
the algebraic structure given by the operator hn, that dictates that a
loop based in vihas vector-fields associated to adjacent edges, which
are not directly connected to vi.
3. The operator Hn(N)only involves right-invariant vector-fields; none
left-invariant.
A significant computational effort is required to explore the space of possible
Hamilton constraint operators. We would like to stress, however, that we
see no fundamental reason why such an endeavor could not succeed.
Let us end this section with three remarks: first, we suspect that the
number of consistent quantum theories of gravity is very limited, probably
equal to one. Thus, the operator constraint algebra will only close once
we have all components of the theory in place. In this light it is perhaps
no surprise that the constraint algebra, which we compute, is anomalous,
since there is still a number of issues, which we have not yet dealt with.
Rather, we see it as a great encouragement that the commutator between
two Hamilton operators in fact has the right off-shell structure.
46

Second, we believe that the closure of the constraint algebra must be
dictated by a powerful principle of symmetry. Our partial derivation of
the Hamilton constraint operator from a curvature operator involving the
Dirac type operator is an attempt to formulate such a principle. From a
more mathematical vantage point we think it would be interesting to use
the modular operator from Tomita-Takesaki theory as a candidate for a
Hamilton operator.
In fact, if we choose a diffeomorphism constraint operator, which is
quadratic in the vector-fields then the three commutators in the constraint
algebra will all have the same general algebraic structure and we would be
able to combine the Hamilton constraint and the diffeomorphism constraint
operators in a single operator as
¯
Na
viDa
vi+NviHvi∶=Tr (MviMvi)(54)
where Mvi=Nvi+¯
Na
viσais a self-adjoint two-by-two matrix and where Mvi
would be a vertex operator quadratic in the vector fields. The constraint
algebra would be of the form
[M(M),M(M′)]=M(M ∂M′−M′∂M )
where M(M)=∑viMvi(Mvi). It is an operator like (54) that we specu-
late might be related to a modular operator coming from Tomita-Takesaki
theory.
Thirdly, our computations on the constraint algebra in the ’diffeomorhism-
diffeomorhism’ sector indicated that an anomaly proportional to a torsion
operator could arise. Despite the fact that these computations showed that
our diffeomorphism constraint operator cannot be correct, we think that
the general computational structure leading to this potential anomaly could
carry over to a computation involving a more realistic diffeomorphism con-
straint operator.
It may not be a surprise if an anomaly involving torsion shows up in
a setting involving Ashtekar variables, since the Ashtekar connection only
becomes Levi-Civita via the equations of motion. We need more analysis,
however, before the existence of such an anomaly can be confirmed.
6 Emergence of an almost-commutative algebra
The algebra HD(M)can be formulated as the closure of the semi-direct
product
HD(M)=C∞
c(M)⋊FI . (55)
47

in the norms described in [6], where Fis the group generated by flow opera-
tors eX, where Iis an ideal given by certain reparametrizations of flows (see
[6] for details) and where C∞
c(M)is the algebra of smooth functions with
compact support. The semi-direct product comes with the multiplication
relation
f1F1f2F2=f1F1(f2)F1F2,
where F1, F2∈F.
In [5] we observed that HD(M)reduces in the semi-classical limit to
the algebra (C∞
c(M)⊗M2(C))⋊Diff(M),(56)
where Diff(M)is the group of diffeomorphisms on M. This is so because
the holonomies on a fixed classical geometry generate a two-by-two matrix
algebra12. Thus we find the almost commutative algebra C∞
c(M)⊗M2(C)
as a sub-algebra.
This is interesting because the non-commutative geometrical formula-
tion of the standard model of particle physics coupled to general relativity
is based on an almost-commutative geometry, where the matrix factor is
related to the gauge sector of the standard model. The algebra (56) sug-
gest that the almost-commutative algebra on which the standard model is
based might have its origin in a purely quantum gravitational setting. Of
course, there is some way to go between the algebra C∞
c(M)⊗M2(C)and
the almost-commutative algebra of standard model, but one has to bear in
mind that there are several issues in the present construction, which we have
not yet addressed. Let us therefore provide a list of issues, which we think
are relevant here:
1. so far we have dealt with a real Ashtekar connection, not the self-dual
SL(2,C)connection, that corresponds to a Lorentzian signature. A
construction based on the latter connection may give rise to a richer
structure than (56) in a semi-classical limit.
2. the formulation of the Standard Model in terms of non-commutative
geometry is in a Lagrangian setting whereas the analysis presented here
plays into a Hamiltonian setting. Thus, a straight forward comparison
is not possible.
3. in the non-commutative formulation of the Standard Model, the finite
algebra is represented on a corresponding finite-dimensional Hilbert
12We assume we are considering a semi-classical analysis around a irreducible connec-
tion.
48

space. In the present case, the finite sector of the algebra, which
emerges in the semi-classical limit, acts directly on the spinors.
4. several natural components from the toolbox of non-commutative ge-
ometry has not been introduced in our construction. For instance the
real structure.
In fact, the original motivation for studying non-commutative algebras
of holonomies was the hope that such algebras might reduce to an almost-
commutative algebra in a semi-classical limit, see [16].
7 Background independency and action of the dif-
feomorphism group
In this section we address the question of background and lattice indepen-
dency. In contrast to the algebra QHD(M), which is manifestly background
independent, the algebra dQHD∗(M)has been defined via lattice approxi-
mations and is therefore not necessarily background independent. The same
applies to the Dirac type operator Dand the state ξc. We therefore need to
assess to what extend these objects depend on the lattice approximations
and the coordinate system, which these represent. First we shall address this
question within the lattice formulation and next commence the construction
of a lattice-independent formalism.
7.1 Invariance properties of Dand dQHD∗(M)
First we want to look at the invariance properties of the Dirac type oper-
ator. In order to do this we will look at the classical value of the double
commutator of the Dirac type operator with a flow, or rather we will only
look at what happens for a path.
Thus, let pbe a path, and let pbe parametrized by t→p(t)by arc-length
in ⋅1. Furthermore let p<xbe the path [0, x]∋t→p(t)and let p>xbe the
path [x, L(p)]∋t→p(t). We put
A(x)=Hol(A, p<x)Dω(˙p(x))Hol(A, p>x),
where Dω is the M2valued object
(Dω)µ=Eµ
aσa,(57)
49

with Ebeing a classical field. If we look at the formula (28) we see that the
classical expectation value of {D, [D, ∇(p)]} is
L(p)
0(ψ(p(L(p))), A(x)ψ(p(0))dx. (58)
Now, it is important to note that with the present definition of the Dirac
type operator formula (58) will depend on the path being parametrized by
arc-length in the 1-metric. This indicates that the Dirac operator is only
invariant under coordinate changes which preserves the 1-metric, which in
3-dimensions consists of 48 coordinate changes.
This is of course not an acceptable state of affairs, since we want the
algebra dQHD∗(M)to be background independent. What is missing from
our construction is that the insertion (57) together with the factor 2−nshould
be a one-form, which would improve the invariance properties of formula
(58). One way to obtain this is to ignore the Dirac type operator and
simply promote by hand the factor 2−nto a one-form dxµ, which then turns
the insertion (57) into a one-form. Using conditions (12) we can then in
fact secure covariance for the dQHD∗(M)algebra at least within the GNS
construction of ξcand to all finite orders in perturbation theory.
If, however, we wish to keep the Dirac type operator we need to change
the construction so that the factor 2−nin the definition of the Dirac type
operator (14) corresponds to a one-form. This change would also have to
involve the Clifford algebra and in particular the relation (15), which comes
into play when one computes the double commutator (58). We see no ob-
stacles for such a modification but shall not work out the details here.
7.2 Unitary equivalence
We now turn to the question of indepence of the Hilbert space construction
of the chosen lattice. We will argue that at the level of the holonomy-
diffeomorphism algebra the state is independent of the chosen lattice in the
sense that the GNS-constructions for the holonomy-diffeomeorphism algebra
for different lattices are unitarily equivalent.
Let us first look at what the semi-classical state looks like on a path p.
Let pbe approximated by a family of paths {pn},pn∈Γn. From the formula
(13) we see that the expectation value of pn={l1,....lk}is approximated
by
(ψ(e(pn)), H ol(l1, A)e−2−nB(l1)⋯Hol(lk, A)e−2−nB(lk)ψ(s(pn))),
50

where e(pn)and s(pn)denotes the end- and start-point of pnand where
B(lk)denotes the value of B(n,i)at the endpoint of lk. Since B(n,i)is
stricktly positive we get factors c, C >0 satisfying the estimate
ce−L(pn)(ψ(e(pn)), H ol(pn, A)ψ(s(pn)))
≤(ψ(e(pn)), H ol(l1, A)e−2−nB(l1)⋯Hol(lk, A)e−2−nB(lk)ψ(s(pn)))
≤Ce−L(pn)(ψ(e(pn)), Hol(pn, A)ψ(s(pn))).
Therefore in the continuum limit we get the estimate
ce−L(p)(ψ(e(p)), H ol(p, A)ψ(s(p)))
≤ξ(A,E),∇(p)ξ(A,E)≤Ce−L(p)(ψ(e(p)), Hol(p, A)ψ(s(p))).
Similarly the expectation value on a flow is just integration over the paths
in the flow, and hence we get similar estimates in this case as well.
If we have two lattices they induce two different 1-metrics and define two
different coherent states ξ(A,E,1)and ξ(A,E,2). These coherent states give rise
to states ρ1, ρ2on the flow algebra via
ρi(F)=ξ(A,E,i)Fξ(A,E,i)=lim
n→∞ξ(n,A,E,i)Fnξ(n,A,E,i), i ∈{1,2},
where Fnis again a lattice approximation to F. We see from the form of the
expectation value, that their absolute value on a path pdiffer by a factor
exp(D1L1(p)−D2L2(p)),
where L1and L2denote the 1-lengths induced by the two lattices, and
D1, D2>0. It hence follows that as long as we only consider flows of vector-
fields with compact support then for each element Fin the HD(M)algebra
there exist c1, c2>0 with
ρ1(F)≤c2ρ2(F), ρ2(F)≤c1ρ1(F).
We let (H1,⋅,⋅1)and (H2,⋅,⋅2)be the Hilbert spaces of the GNS repre-
sentations induced by the HD(M)algebra and the states, i.e. Hiconsists
of the closure of the elements in the HD(M)algebra with the inner product
F1F2i=ρi(F∗
1F2).
The bilinear form H1×H1∋(F1, F2)→ρ2(F∗
1F2)
51

is densely defined and positive, and gives therefore rise to a positive densely
defined operator Von H1with F1F22=F1V F21. Furthermore Vcom-
mutes with the action of the HD(M)algebra on H1since
F1F3V F21=F∗
3F1V F21=F∗
3F1F22=F1F3F22=F1V F3F21.
The square root U=V1
2fulfills
F1F22=UF1UF21,
and hence Udefines an isometry U∶H2→H1via
H2∋F→UF ∈H1
which commutes with the action of the HD(M)algebra.
Like we have constructed Uwe can likewise construct an isometry U′∶
H1→H2which commute with the action of the HD(M)algebra and fulfills
UU ′=idH1,U′U=idH2.
We conclude that the two GNS-representations of the HD(M)algebra
are unitarily equivalent and that the equivalence is implemented by U.
The above argument requires a more detailed analysis, especially since
Vis unbounded. We will not give the analysis here. The argument is taken
from [25], where the case of two states, which are uniformly bounded by
each other is treated.
Also further analysis is needed to extend the above argument to the
dQHD∗(M)algebra. Especially one needs to deal with the transformation
properties of the Dirac operator, see section 7.1.
7.3 Action of the diffeomorphism group
Since we have assumed that the C2-bundle over Mis trivial, the group of dif-
feomorphisms acts naturally on the Holonomy-Diffeomorpism algebra. We
will now look at what continuity properties this action has, in particular we
will look at the continuity properties of the action on the GNS-construction
of a semi-classical state.
In order to discuss the continuity properties of the action we need a
topology on the group of diffeomorphisms. We will consider the weak topol-
ogy. This is defined in the following manner: Choose a Riemmannian metric
on M. We consider the metrics
dK,r(Ψ,Φ)=sup
m∈K
d(Ψ(m),Φ(m))+r

n=1DnΨ(m)−DnΦ(m),
52

where K⊂Mis compact. The weak topology is the topology induced by
the metrics {dK,r}r∈N,K⊂Mcompact.
The action of the diffeomorphism group on the Holonomy-Diffeomorphism
algebra is not continuous, if we consider the Holonomy-Diffeomorphism al-
gebra with the C∗-norm defined in section 2.1. This follows since the action
of Ron L2(R)is not norm continuous.
On the other hand, we can also consider HD(M)with the strong topol-
ogy induced by the GNS-construction of a semi-classical state. With this
topology the action of the diffeomorphism group will be continuous. This
follows because the expectation value of a the semi-classical state on a paths
depends smoothly on the start and endpoint on the path, see the continuum
limit of formula (7.2).
However if we consider the algebra dQHD∗(M)with the weak topology
induced by the GNS-construction, the action of the diffeomorpism group will
no longer be continuous. This stems from the following fact: Consider two
paths p1and p2that can be composed. The double commutator with Dcon-
sists of inserting vector-fields along the paths. If p1does not overlap with p2,
in the product {D, [D , p1]}{D, [D, p2]}there will be no interference between
the vector-fields in {D, [D, p1]} and the holonomy part of {D, [D, p2]}, and
vice versa. If however p1and p2do overlap, there will be an extra quantum
correction stemming from the interference of the vectorfields in p1and the
holonomy part of {D, [D, p2]}, and vice versa. Therefore if we consider a
family of diffeomorpisms Φtsuch the Φt(p1)can be composed with p2, and
such that Φt(p1)has no overlap with p2when t=0, but do have an overlap
when t=0, we see
lim
t→0ξcΦt({D, [D, p1]}){D , [D, p2]}ξc
=ξcΦ0({D, [D, p1]}){D , [D, p2]}ξc,
which shows that the action of the diffeomorphisms is not weakly continuous,
and therefore also not strongly continuous.
7.4 A lattice-independent formulation of Dand dQHD∗(M)
We will in this section try to formulate the dQHD∗(M)algebra and the
Dirac type operator Dwithout the use of lattice approximations. We will
mainly describe what happens on paths, since flows are just families of paths.
Let pbe a path, and let p∶[a, b]→Mbe a parametrization of the path.
Consider a coordinate system (x1, x2, x3)and the infinitesimal translation
53

operator ˆ
Eα
a(x)=fxEσadxα, see equation (5). We will consider the operator
Epof the following form: for t∈[a, b]we set
pt=∇(p<t)σa˙pα(t)ˆ
Eα
a(p(t))∇(p>t)
and define
Ep=b
aptdt.
One checks that
EpEq=Epq.
With Fbeing an element of HD(M)we then also obtain the operator EF,
which is the flow consisting of a family of operators Epand which then
satisfy
EF1EF2=EF1F2.
Next we define
p(2)
t=E(p<t)σa˙pα(t)ˆ
Eα
a(p(t))∇(p>t)+∇(p<t)σa˙pα(t)ˆ
Eα
a(p(t))E(p>t)
as well as
E(2)
p=b
ap(2)
tdt,
which then also gives us E(2)
Ffor F∈HD(M).
In a straightforward generalization we also define E(n)
F, which involves
ninsertions of vector-fields in the flow F.
In a next step we introduce a grading to HD(M)and let ˜
Fbe the
odd-graded element that corresponds to the even-graded flow F∈HD(M).
Thus, ˜
Fsatisfies the same commutator identities as F, just with a graded
bracket. Likewise we introduce graded elements ˜
E(n)
F, which are odd-graded
elements that corresponds to E(n)
F.
We define an abstract Dirac type operator Dvia its commutator relations
with elements in HD(M)and with elements E(n)
Fand ˜
E(n)
F. These are as
follows: D, E(n)
F=˜
E(n)
F,D, ˜
E(n)
F=E(n+1)
F
where we define E(0)
F=Fand ˜
E(0)
F=˜
F. The commutators
[E(n)
F1, E(m)
F2],[˜
E(n)
F1, E(m)
F2],[˜
E(n)
F1,˜
E(m)
F2]
are non-trivial and shall not be worked out here. In general we can say
that a non-trivial commutator with a vector-field E(n)
Fwill insert a Pauli
54

matrix in the corresponding location in a flow. This will result in two Pauli
matrices with contracted indices, located at the same point but inserted in
different flows.
The lattice-independent dQHD∗(M)algebra is then the ⋆-algebra gen-
erated by HD(M)and all commutators between Dand elements in HD(M).
Note that we have not provided a proof that the abstract dQHD∗(M)
algebra defined here is in fact identical to the dQHD∗(M)algebra defined
previously with the aid of lattice approximations.
8 The overlap function
In section 3.6 we argued that the states ξcon dQHD∗(M)will always be
associated to a phase-space point of Ashtekar variables and we explicitely
constructed states, which are peaked over such a phase-space point. The
question therefore arises what the overlap function between two states as-
sociated to different phase-space points amounts to. This is the subject of
this section.
To ease the analysis we shall restrict ourselves to the case where the state
ξcis constructed from coherent states on SU (2),ξc=ξκ
(A,E), as we did in
section (3.6), and we shall consider two phase-space points, which only differ
in the Ashtekar connection. Thus, the overlap function we are interested in
is this
Ωκ(A, A′, E)=ξκ
(A,E)ξκ
(A′,E).
We first define the overlap function for a single edge li
ωκ
i(A, A′, E)=φs
(A,dxE,i)φs
(A,dxE,i),
where we recall that s=κdx and that φs
(A,dxE,i)is Hall’s coherent state on
one copy of SU (2). To find Ω(A, A′, E)one must work out the continuum
limit
Ωκ(A, A′, E)=lim
n→∞Π∣ln∣
i=1ωκ
i(A, A′, E)
where the index iruns over all edges in Γn. We write
Ωκ(A, A′, E)=lim
n→∞exp 
−∣ln∣

i=1
ln ωκ
i(A, A′, E)−1
(59)
and since ωκ
i(A, A′, E)≤1 we know that ln ωκ
i(A, A′, E)−1is positive.
Next, since the sum ∑∣ln∣
i=1in (59) becomes an integral over the manifold M
55

in the large nlimit
∣ln∣

i=1
n
Ð→M
it is important to count the powers of dx =2−nin order to check whether the
measure that arises, matches the dimensions of the integral. The manifold
Mis 3-dimensional and therefore we need a factor 2−3nin order to obtain
a volume form. In this way we can estimate whether and to what the
expression in (59) converges by simple power-counting.
The infinitesimals come from two sources: from the parameter sin the
coherent state φs
i(A,E)and from the connection A. Further, if we assume
that we have coherent states based on a Laplace operator complexifier, then
the relevant parameter is the difference z=(A−A′). Since
lim
s→0Ωκ(A, A′, E)∝δ(A, A′),
we assume that at least one term in ln(ωs
i(A, A′, E)) comes with a factor
s−1. We do not have a general proof for the validity of this assumption, but
in the special case with a Laplace operator complexifier we know it holds
true (see [24]).
Furthermore, since ln ωs
i(A, A′, E)−1is positive its Taylor expansion in
(A−A′)can only involve even powers and since Ωs(A, A, E)=1 its lowest
possible power is quadratic.
ln ωs
i(A, A′, E)−1=a1z2+a2z4+. . . .
On the other hand, for the sum in (59) to converge we need four powers of
dx to arise from (A−A′)in ωs
i(A, A′, E)since smay provide one negative
power due to assumption made above. Thus, if there are terms in the Taylor
expansion of ωs
i(A, A′, E), which provides fewer powers than four, then the
sum will diverge and the overlap function will vanish. Also, if there are only
terms strictly higher than four, then the sum will vanish and the overlap
function will equal 1.
Thus, in order for the overlap function in (59) to obtain a nonzero value
less than one we must require that the lowest power of (A−A′)in the Taylor
expansion of ln(ωs
i(A, A′, E)−1)will be exactly four and that this term also
involves a 1sfactor. In particular, this means that the quadratic term is
required to vanish.
This seems questionable. One can check that this condition is not sat-
isfied for the special case where the complexifier is the Laplace operator
on SU (2)(use formula 4.55 in [24]), and we are inclined to think that it
56

will not be satisfied for any choice of complexifier coherent state, which
also satisfies standard requirements for semi-classical states (minimizing the
uncertainly conditions, for instance). We do, however, not have a general
proof. Whether it holds for any state on HD(M)is also unknown.
Thus, for the special class of semi-classical states with a Laplace-operator
complexifier, we conclude that the overlap function between two different
semi-classical approximations vanishes
Ωκ(A, A′, E)=0.
The overlap function is important since it is the expectation value of Uω
ξκ
(E,A)ξκ
(E,A′)=ξκ
(E,A)Uωξκ
(E,A), A′=A+ω .
and therefore tells us about the possibilities of shifting between classical ge-
ometries. If our reasoning presented here holds in general, then it means that
each classical geometry gives rise to a vacuum state and a GNS construc-
tion over it, which are all isolated from each other. This does, however, not
necessarily imply that there cannot be finite changes in geometries caused
by quantum effects, as our analysis in subsection 3.7 shows. But it implies
that there cannot be quantum interference between semi-classical approxi-
mations associated to different classical geometries. We find this to be an
interesting possibility.
9 The complex Ashtekar connection
So far we have based our construction on the group S U (2). We choose
SU (2)because it plays into the framework of canonical quantum gravity
formulated in terms of Ashtekar variables. The original Ashtekar connection,
however, is a complex connection, that takes values in the self-dual sector
of sl(2,C), the Lie-algebra of SL(2,C). The SU (2)connection corresponds
to a metric with Euclidian signature. Since SU (2)is a compact group it is
simpler to start with this type of connection. In order to encompass also
Lorentzian signatures it is, however, necessary to consider how a complex
connection can be built into or emerge from our construction.
The Lie group SL(2,C)is generated by the six generators {1
√2σi,i
√2σi},
i∈{1,2,3}, and the self-dual sector is characterized by invariance under the
exchange of generators: 1
√2σi←→ i
√2σi
57

which corresponds to interchanging rotations with Lorentz boosts. Thus, in
terms of degrees of freedom, the self-dual sector of SL(2,C)matches those
of SU (2). In terms of algebras generated by holonomies, however, the two
cases are distinctly different.
There are two natural strategies to encompass the complex Ashtekar
connection. Either we repeat the entire construction of the algebra generated
by holonomy-diffeomorphisms and semi-classical states thereon with SU(2)
replaced by SL(2,C), or, alternatively, we try to obtain a complexification
of SU (2)by doubling the Hilbert space.
Let us briefly discuss the first option. The key difference between holonomy-
diffeomorphisms of SU (2)and S L(2,C)connections is that the latter does
not correspond to bounded operators. The norm given in section 2.1 diverges
for sl(2,C)connections
sup
∇∈Asl eX
∇=∞,Asl =space of sl(2,C)connections.
which means that we do not have a C∗-algebra. It is therefore not possi-
ble to define HD(M, S, Asl ). There is, however, still a ∗-algebra and thus
HD(M , S, Asl)is available for a Hilbert space representation.
The conditions for a state to exist on HD(M , S, Asl)are identical to the
SU (2)case and we strongly expect states on HD(M, S, Asl )as well as on
dQHD(M , S, Asl), which is the ∗-algebra version of dQHD(M, S, Asl), to
exist. A GNS construction around such a state would then involve only
unbounded operators.
Once a Hilbert space representation of dQHD(M , S, Asl)is obtained we
can proceed to construct the Hamilton operator in the same manner as
we did for SU (2)and project into the self-dual sector. This will give the
complex Hamiltonian that corresponds to the Lorentzian signature.
It is beyond the scope of this paper to work out the details for a con-
struction based on SL(2,C), but we would like to stress that we do not see
any major obstacles for such a construction to exist.
10 Interpreting Din terms of the volume of M
The Dirac type operator Dhas an interesting interpretation in terms of a
quantization of the volume of the manifold M. To see this we first write the
volume of M
Vol(M)=Meµν ρdxµdxνdxρ
58

∆x1
vi
lj
∆S1
j
Figure 5: The surfaces ∆S1
j, which are orthogonal to the line interval ∆x1
corresponding to a subdivision of Min cubic lattices, are associated to flux
variables located at the edges lj.
where e=det(ea
µ), and rewrite it in terms of a triad field
Vol(M)=Mdxaeeµ
aµνρ dxνdxρ
=MdxadFa(60)
where dFa=eeµ
aµνρ dxνdxρis a sum over the three infinitesimal flux variable
in the xµ’s directions, which is conjugate to the holonomy of the Ashtekar
connection, see [7]. If we rewrite the Riemann integral in (60) as a limit of
lattices we then we find
Vol(M)=lim
n→∞
i,a (∆xa)iF∆Sa
i
where the sum runs over two indices: the index iover all edges13 in the
lattice approximation and the ”flat” index a∈{1,2,3}. Here F∆Siis a lattice
approximation of one of the three xµ-components in dF awhere the surface
∆Siis perpendicular to the direction of xµ. Thus, here the sum runs over
all surfaces Siin all three directions. In the quantization procedure, which
involves holonomies of Ashtekar connections, this flux variable corresponds
to a right-invariant vector-field Lea
ion a copy of SU (2)associated to the
edge li, see figure 5, which gives us:
Vol(M)quant.
Ð→ lim
n→∞
i,a (∆xa)iLea
i,(61)
13In fact, this sum could equally well run over vertices. The point is that the ’dxa’ does
not depend on the three directions in the lattice and thus only ’sees’ the vertices.
59

There is a striking similarity between (61) and the Dirac operator Dnin
(14), where the Clifford elements play the role of the infinitesimal element
dxaor its lattice approximation ∆xa. Note too that this is in sync with
the our conclusions in section 7.1, where we found that the Clifford algebra
must be related to the exterior algebra of M.
11 Summary and discussion
In this paper we have presented quantum holonomy theory, which is a non-
perturbative and background independent theory of quantum gravity cou-
pled to quantized degrees of matter.
Four central objectives are met in this paper. The first is the formulation
of a first principle – namely the QHD(M)algebra –, which serves as the
foundation for this approach to a theory of quantum gravity. It is the
conceptual simplicity of the QHD(M)algebra, which makes it attractive.
What could be more natural, more poetic, as a foundation for a theory of
quantum gravity than an algebra that simply encodes how tensorial degrees
of freedom – i.e. stuff – are moved in space? We find it surprising that
this algebra, which encodes the mathematical setup of canonical quantum
gravity, has not, to the best of our knowledge, been studied before.
The second central objective met in this paper is the finding that semi-
classical states exist on the algebra dQHD∗(M), which is the algebra ob-
tained from QHD(M)by forming a canonical Dirac type operator and
considering its commutators with the algebra of holonomy-diffeomorphisms.
A state gives us a kinematical Hilbert space and a stage for a semi-classical
analysis. It is remarkable that where other non-perturbative approaches to
quantum gravity are challenged by the necessity of producing semi-classical
states, we find that semi-classical states appear extremely natural in our ap-
proach. Also and perhaps no less surprising is the evidence we find that the
overlap function between two different semi-classical states might vanish.
The third central objective met in this paper is the formulation of a
geometrical principle, which provides us with a Hamilton constraint operator
from which the classical Hamilton constraint emerges in a semi-classical
limit. This means that we do obtain general relativity in a semi-classical
limit from our construction.
The key step to obtain the Hamilton constraint operator is again the
construction of the Dirac type operator. Since this operator provides us
with a canonical metric structure on the spectrum of the HD(M)algebra
it gives us access to geometrical notions such as curvature. Indeed, it is a
60

scalar curvature operator that provides us with a candidate for a Hamilton
constraint operator. The construction of this scalar curvature operator does,
however, involve a certain measure of ad-hoc reasoning, which is likely to
be the cause of the failure of the operator constraint algebra to close. Nev-
ertheless, the fact that our provisional candidate for a Hamilton constraint
operator entails off-shell closure in the ’Hamilton-Hamilton’ sector of the
operator constraint algebra is an encouraging result, which should motivate
further analysis in this direction.
The fourth central objective met in this paper is the identification of ele-
ments and mechanisms of unification. This point is essentially an adaptation
of results already published. First of all, the HD(M)algebra is inherently
non-commutative, which immediately places our construction well within the
domain of non-commutative geometry with its toolbox of unifying mecha-
nisms. The fact that the HD(M)algebra produces an almost-commutative
algebra in the semi-classical limit suggest a link to the basic mathemati-
cal setup behind the non-commutative formulation of the standard model.
Secondly, we find states in the kinematical Hilbert space from which the
expectation value of the Dirac type operator gives a spatial Dirac operator
in a semi-classical limit and from which we are also able to obtain the Dirac
Hamiltonian in the same limit. This provides a link to fermionic quantum
field theory.
The results presented in this paper raises a number of both conceptual
and technical questions. If we start in the visionary end of the spectrum,
then one may speculate whether there exist a link between the constraint
operators and Tomita-Takesaki theory. For the operator constraint algebra
to close off-shell in a non-trivial sector of the Hilbert space there must be
a powerful symmetry principle that dictates such closure. We suspect this
to be Tomita-Takesaki theory, which under certain conditions prescribe the
existence of a one-parameter group of automorphisms, that is unique up to
inner automorphisms. Indeed, with a state on the dQHD∗(M)algebra we
are in a position where we can attempt to derive the one-parameter group
and the modular operator. We speculate that the latter is related to the
constraint operators and that the eventual off-shell closure of the operator
constraint algebra will turn out to be a consequence of this theory.
Note again that because the action of the diffeomorphism group on the
dQHD∗(M)algebra is not strongly continuous and thus will not have
infinitesimal generators one cannot expect the constraint algebra to close
on the entire kinematical Hilbert space but merely in a sector defined by
finite perturbation theory in the quantization parameter. This observa-
61

tion appears to be generic and may reflect a deeper conceptual issue. In-
deed, since the diffeomorphisms themselves are quantized – the holonomy-
diffeomorphisms are part of the conjugate variables –, one may question to
what extent it makes sense to talk about diffeomorphisms and diffeomor-
phism invariance outside the perturbative regime.
Concerning the constraint operators it is clearly possible to push the
analysis commenced in this paper much further, i.e. by brute force calcula-
tions of the constraint algebra, to determine which candidates for a Hamil-
ton constraint operator are physically realistic. As a first step it would be
interesting (and rather demanding) to compute the operator constraint al-
gebra with the new candidate for a Hamilton operator, that we obtain in
the appendix and which involves twelve loops in each vertex operator. It
would also be favorable to tighten the geometrical argument, that we use to
derive our candidate for a Hamilton constraint operator from a curvature
operator on the configuration space of Ashtekar connections. To do this one
would have to consolidate the non-commutative geometrical notions, which
we have already built, i.e. move further towards a spectral geometry over a
space of spatial connections.
It is an open question what support the measure on the configuration
space of Ashtekar connections, which the semi-classical state provides, has.
Our analysis on the overlap function suggest that the measure could be
localized over a single classical point and that this point is the only mea-
surable one in its support. If this result holds in general it would render
a theory of quantum gravity, which has a measure markedly different from
the Ashtekar-Lewandowski measure used in loop quantum gravity. There
may, however, be finite geometrical changes caused by transitions generated
by the dQHD∗(M)algebra, as our analysis of how the algebra affects the
spectrum of the states suggest, but there will not be quantum interference
between vacuum states associated to different classical geometries. On the
other hand it is also possible that states exist, which render a non-zero over-
lap function. If this should be the case one would have to work out whether
both types of states are physically feasible.
The fact that we use lattice approximations for much of our analysis
raises the question wether or not the construction is lattice and background
independent. The algebra QHD(M)is manifestly background indepen-
dent, but the bulk of our analysis is based on a version of dQHD∗(M),
which is constructed using lattice approximations. We have a formulation
of dQHD∗(M)and a Dirac type operator, which is independent of the
lattices, but we have not provided a proof that these objects are in fact
identical to the ones on which our analysis is based. Also, the construction
62

of the state has so far only been carried out using lattice approximations.
It seems, however, very plausible that also the state – and in fact the entire
construction presented in this paper – can be constructed independently of
lattice approximations. To do this is a primary task.
So, can this approach to quantum gravity be said to be background
independent? The fact that the kinematical Hilbert space is provided by
a GNS construction over a semi-classical state of course implies that the
Hilbert space always depends on a classical ”background” metric. But this
is not the background dependency, that is usually meant by the term. The
background dependency encountered here comes naturally out of a mani-
festly background independent construction – the QHD(M)algebra – as a
consequence of representation theory.
Concerning the lattice approximations, it is in fact not clear whether they
approximate diffeomorphisms or merely analytic diffeomorphisms. Note that
if the latter should be the case this does not necessarily imply that this
framework only harbors analytic diffeomorphisms. It would only imply that
the lattices approximations do. Also, since exponentiated vector-fields have
a non-measurable effect on the spectrum of the holonomy-diffeomorphism
algebra, it is not clear with respect to what norm the C∗-algebra is built.
The dQHD∗(M)algebra does not directly involve exponentiated vector-
fields, but such non-measurable effects could still occur and could indicate
that the holonomy-diffeomorphism algebra should be build over the counting
measure instead of a measure coming from a Riemannian metric. On the
other hand the algebra should tie the various non-measurable effects together
in a measurable way, so that we in the end should end up with a Riemannian
metric.
Another question is to determine the nature of the yet tentative connec-
tions to fermionic quantum field theory and the non-commutative formula-
tion of the standard model. Overall, we believe that this construction begs
for a deeper application of non-commutative geometry. For instance, is the
emergence of elements of fermionic QFT related to the algebra of differential
forms generated by the Dirac type operator? And what is the spectral ac-
tion? One serious obstacle for a direct comparison to the non-commutative
formulation of the standard model is that we are operating with a Hamilto-
nian setup whereas the former is in the Lagrangian formalism.
Let us end by commenting on the issue of the complex Ashtekar connec-
tion. Within the framework presented in this paper it is entirely plausible
that a formulation set with SL(2,C)exist. The way the complex Ashtekar
connection takes values in the self-dual sector of sl(2,C)seems, however,
to suggest that it should be obtained in a more intrinsic manner, probably
63

v2
v3
x
v4
v1
y
z
Figure 6: The location of the vertices {v1, v2, v3, v4}in a lattice plaquette.
The circular arrow indicates the orientation of the ”xy” loop.
by doubling the Hilbert space and using methods of non-commutative ge-
ometry. It would then be natural that the real structure should be used to
define a reality condition.
Acknowledgements
We would like to express our thanks to Mario Paschke for helpful comments
and suggestions.
A The operator constraint algebra
A.1 The commutator [H(N),H(N′)]
To compute the commutator
Hn(N),Hn(N′)(62)
we first compute all possible commutators between vertex operators H●●
vi,
which will contribute with a factor N(x)∂xN′(x)or N′(x)∂xN(x)in the
continuum limit. Once we have this we can add up the three spatial direc-
tions.
64

Thus, we start by computing the commutator between the two ’(xy)’-
operators at two adjacent vertices v1and v2, see figure 6
H(xy)
v1(Nv1),H(xy)
v2(N′
v2)
=24n−4Nv1N′
v2Lv1
eb
y
,Lv1
ea
x,Tr σaÈ
Lv1
xy σb,Lv2
ed
y
,Lv2
ec
x,Tr σcÈ
Lv2
xy σd
=24n−4Nv1N′
v2Lv1
eb
y
,Lv1
ea
x,Tr σaÈ
Lv1
xy σb,Lv2
ed
y,Lv2
ec
x,Tr σcÈ
Lv2
xy σd
+24n−4Nv1N′
v2Lv2
ed
y
,Lv2
ec
x,Lv1
eb
y
,Tr σcÈ
Lv2
xy σd,Lv1
ea
x,Tr σaÈ
Lv1
xy σb .
In a next step we evaluate the commutator between loops and vector-fields
and use (12) to obtain
Lvi
eb
x
,Tr(σaÈ
Lvi+1
xy σc)=±1
2Tr(σaσbσc)+Odx2,(63)
where Lvi
eb
xmay be interchanged with Lvi
eb
yand where the sign depends on
which part of the loop the vector-field interacts with. If the vector-field
interacts with either g1or g2in the loop, then the sign is plus; else minus.
In obtaining relation (63) we are explicitly using that we are operating within
a GNS construction of the semi-classical state ξnand its limit. Since we are
only interested in the continuum limit we may ignore all terms except those
at lowest orders of dx.
We continue the computation and find:
H(xy)
v1(Nv1),H(xy)
v2(N′
v2)
=−24n−4Nv1N′
v2Lv2
eb
x
,Lv1
eb
y
,Lv1
ea
x,Tr σaÈ
Lv2
xy
−Lv1
ea
x,Lv2
ea
x,Lv1
eb
y
,Tr σbÈ
Lv2
xy
+Lv2
ea
y,Lv1
ea
x,Lv2
ec
x,Tr σcÈ
Lv1
xy
−Lv2
ea
x,Lv1
ea
x,Lv2
ed
y
,Tr σdÈ
Lv1
xy .(64)
Here the operator in the bracket has the general structure
Av1Bv2+Av2Bv1
which we rewrite as
Av1Bv1+Av2Bv2+(Av2−Av1)(Bv1−Bv2).
65

We use this, together with definition (52), to rewrite (64) as
H(xy)
v1(Nv1),H(xy)
v2(N′
v2)
=−24n−3Nv1N′
v2Lv2
eb
x
,Lv2
eb
y
,Dv2(xy)
y−Lv2
ea
x,Lv2
ea
x,Dv2(xy)
x
+Lv1
ea
y,Lv1
ea
x,Dv1(xy)
y−Lv1
ea
x,Lv1
ea
x,Dv1(xy)
x
+Ξ(xy,xy)
v1v2(N, N ′)(65)
where Dvi(µν)
µare defined in (52) and where Ξ(xy,xy)
v1v2(N, N ′)is an operator
that consist of terms of the form
24n−3Nv1N′
v2(Av2−Av1)(Bv1−Bv2).(66)
We see that the general structure of the classical Poisson bracket (39)
emerges in equation (65) with Ξ(xy,xy)
v1v2(N, N ′)as a possible candidate for
an anomaly. Note also that there is an additional factor 1
2in equation (65)
due to the fact that contributions at both vertices v1and v2show up. When
we eventually add all contributions to the commutator (62) this factor will
be needed.
We obtain the opposite commutator
H(xy)
v2(Nv2),H(xy)
v1(N′
v1)
simply by interchanging Nand N′in equation (65) and adding a sign.
Adding these two commutators will produce the factor Nv1N′
v2−N′
v1Nv2,
which in the continuum limit converges to dx(N(x)∂xN′(x)−N′(x)∂xN(x)).
Thus, in the following computations we shall not explicitly write down the
opposite commutators but simply write dx(N(x)∂xN′(x)−N′(x)∂xN(x))
in the end.
Next we work out the commutator between two vertex operators involv-
ing perpendicular loops. We first compute
H(xy)
v1(Nv1),H(yz)
v2(N′
v2)
=−24n−4Nv1N′
v2acb Lv2
ed
z
,Lv1
eb
y
,Lv1
ea
x,Tr σcÈ
Lv2
yz σd
+24n−4Nv1N′
v2cbd Lv2
ed
z
,Lv2
ec
y,Lv1
ea
x,Tr σaÈ
Lv1
xy σb (67)
66

as well as
H(zx)
v1(Nv1),H(yz)
v2(N′
v2)
=24n−4Nv1N′
v2adb Lv1
eb
x
,Lv1
ea
z,Lv2
ec
y,Tr σcÈ
Lv2
yz σd
−24n−4Nv1N′
v2cad Lv2
ed
z
,Lv2
ec
y,Lv1
eb
x
,Tr σaÈ
Lv1
zx σb (68)
The second term in (67) appears to be a problem since it has a spatial index
structure that cannot correspond to the diffeomorphism constraint. There
will, however, be a term with the same index structure coming from the
commutator
H(xy)
v3(Nv3),H(yz)
v2(N′
v2) (69)
where v3is the vertex neighboring v1in the z-direction, see figure 6. This
term will have the opposite sign since the intersection Lv2
yz ∩Lv3
xy has the
opposite orientation as that of Lv2
yz ∩Lv1
xy. Also, this term will be proportional
to Nv3N′
v2, which in the n→∞limit gives a N(x)(∂x+∂y)N′(x). The
N(x)∂xN′(x)-term will cancel the first term in (67) and the N(x)∂yN′(x)-
term will contribute to the computation in the y-direction. This term coming
from the commutator (69) will of course only be identical to the second term
in (67) up to another term, which involves a factor
Lv1
ea
x,Tr σaÈ
Lv1
xy σb−Lv3
ea
x,Tr σaÈ
Lv1
xy σb (70)
which we must keep in mind in the end of this computation.
In a completely parallel manner the second term in (68) also has a wrong
index structure and again we find that this term will cancel – up to a term
similar to (70) – with a term coming from a commutator
H(zx)
v4(Nv4),H(yz)
v2(N′
v2)
where v4is the vertex neighboring v1in the y-direction, see figure 6.
Notice that two identical terms come from the commutator involving v3
and v2and the commutator involving v4and v2.
Let us now continue our analysis of the first term in both (67) and (68).
67

We find
(67) and (68)first terms
=24n−3Nv1N′
v2−Lv2
ea
z,Lv1
ea
x,Lv1
eb
y
,Tr σbÈ
Lv2
yz 
+Lv2
eb
z
,Lv1
eb
y
,Lv1
ea
x,Tr σaÈ
Lv2
yz 
+Lv1
eb
x
,Lv2
eb
y
,Lv1
ea
z,Tr σaÈ
Lv2
yz 
−Lv1
eb
x
,Lv1
ea
z,Lv2
ea
y,Tr σbÈ
Lv2
yz  (71)
Here the Lvi
ea
zand Lvi
ea
yvector-fields display the structure
Av1Bv2+Av2Bv1
which again permits us to rewrite (71) as
(67) and (68)first terms =
=24n−3Nv1N′
v2−Lv2
ea
z,Lv2
ea
x,Dv2(yz)
z+Lv2
ea
x,Lv2
ea
y,Dv2(yz)
y
+Ξ(xy,yz)
v1v2(N, N ′)+Ξ(zx,yz)
v1v2(N, N ′)(72)
where Ξ(xy)(yz)(N, N ′)+Ξ(zx)(yz)(N, N ′)is an operator similar to (66), that
involves a factor of the form (Av2−Av1)(Bv1−Bv2).
We do not write down the ’zx-zx’ commutators since they are identical
to the ’xy-xy’ commutators just with ’zx’ replacing ’xy’.
Thus, we have now identified and analyzed all parts of the commutator
(62), which involves a derivative of the lapse fields Nand N′in the x-
direction. Adding all this up and also adding the y- and z-directions we
find
Hn(N),Hn(N′)=Dn(¯
N(N, N ′))+Ξn(N, N ′)
where Ξn(N, N ′)is a potentially anomalous term that reads
Ξn(N, N ′)=
vi,vj
Ξvivj(N, N ′)(73)
where the sum runs over adjacent vertices viand vjand where Ξvivj(N , N′)
is given by the sum of all additional contributions like (66), which consist of
68

terms that involve an operator of the form (Lvi
ea
µ−Lvj
ea
µ)and else are identical
to terms contributing to Dnin (51). In Ξn(N , N′)are also the additional
terms involving factor like (70), which arise when terms from commutators
involving the pairs of vertices (v1, v2)cancel with commutators involving
(v3, v2)and (v4, v2)- as described above.
A.2 The commutator [D(¯
N),D(¯
N′)]
We continue the computation of the operator constraint algebra. The task
now is to compute the commutator between two diffeomorphism constraint
operators [Dn(¯
N),Dn(¯
N′)].(74)
Since we now have three different vertex operators, one for each spatial
direction, we first need to check whether we must also include commuta-
tors between different vertex operators associated to the same vertex. The
relation Lvi
eb
µ
,Tr(σaÈ
Lvi
xy σc)=Odx2,
implies that we do not need to worry about vertex operators, that involve
the same loop, since such contributions will be of such orders in dx that
converges to zero in the continuum limit. There will, however, be a contri-
bution from vertex operators, which are associated to the same vertex and
which involve perpendicular loops. Such contributions will be problematic
since they do not have the right orders of dx, i.e. they will be divergent.
Thus, already here do we find that our computation of the commutator (74)
runs into severe difficulties.
Nevertheless, let us for now ignore this problem and continue the com-
putation in order to assess how close we come to the wished for result. Thus,
we compute the commutator between to vertex operators locates at adjacent
vertices and which both involve loops in the ’xy’-plane. We find
¯
Nx
v1Dv1(xy)
x−¯
Ny
v1Dv1(xy)
y,¯
N′x
v2Dv2(xy)
x−¯
N′y
v2Dv2(xy)
y
=−24n−2¯
Nx
v1¯
N′x
v2Lv1
ea
y,Tr σaÈ
Lv2
xy+Lv2
ea
y,Tr σaÈ
Lv1
xy
+24n−2¯
Nx
v1¯
N′y
v2Lv2
ea
x,Tr σaÈ
Lv1
xy+¯
Ny
v1¯
N′x
v2Lv1
ea
x,Tr σaÈ
Lv2
xy
(75)
where the loops are again based in v1and v2, see figure 6. To obtain (75)
69

we used Lvi
ea
y,Tr σbÈ
Lvi+1
xy =±1
2δab +Odx2
where the sign again depends on the interaction between the vector-field
and the loop, see equation (63).
Here again do we encounter difficulties. The commutator (75) is not – as
we found in the previous section when we computed the [Hv1(N),Hv2(N′)]
commutator – of the general form Av1Bv2+Av2Bv1multiplied with a factor
involving the shift fields. This means that what amounts in the semi-classical
limit to a derivative of an inverse, densitized triad field or a field strength
tensor will arise that must be taken into account. This also happens in the
computation of the classical Poisson bracket (40) but there the derivative is
covariant, which is not the case here.
Nevertheless, we continue the computation to see how close we get to
the wished for result and in doing so simply ignore any further problems
with non-covariant derivatives. Thus, we first write
¯
Nx
v1Dv1(xy)
x−¯
Ny
v1Dv1(xy)
y,¯
N′x
v2Dv2(xy)
x−¯
N′y
v2Dv2(xy)
y
=−24n−1¯
Nx
v1¯
N′x
v2Dv2(xy)
x+Dv1(xy)
x
+24n−1¯
Nx
v1¯
N′y
v2Dv1(xy)
y+¯
Ny
v1¯
N′x
v2Dv2(xy)
y+Υ(xy,xy)
v1v2(¯
N, ¯
N′),(76)
with
Υ(xy,xy)
v1v2(¯
N, ¯
N′)=−24n−2¯
Nx
v1¯
N′x
v2Lv1
ea
y−Lv2
ea
y,Tr σaÈ
Lv2
xy
+Lv2
ea
y−Lv1
ea
y,Tr σaÈ
Lv1
xy
+24n−2¯
Nx
v1¯
N′y
v2Lv2
ea
x−Lv1
ea
x,Tr σaÈ
Lv1
xy
+¯
Ny
v1¯
N′x
v2Lv1
ea
x−Lv2
ea
x,Tr σaÈ
Lv2
xy(77)
Next we consider again the commutator involving two loops perpendic-
70

ular to each other
¯
Nx
v1Dv1(xy)
x−¯
Ny
v1Dv1(xy)
y,¯
N′y
v2Dv2(yz)
y−¯
N′z
v2Dv2(yz)
z
=24n−2¯
Nx
v1¯
N′y
v2Lv2
ea
z,Tr σaÈ
Lv1
xy
+24n−2¯
Nx
v1¯
N′z
v2Lv1
ea
y,Tr σaÈ
Lv2
yz 
−24n−2¯
Nx
v1¯
N′z
v2Lv2
ea
y,Tr σaÈ
Lv1
xy
−24n−2¯
Ny
v1¯
N′z
v2Lv1
ea
x,Tr σaÈ
Lv2
yz (78)
Similar to what happened when we computed the [Hn(N),Hn(N′)] com-
mutator we here see that the first and third term in (78) match similar terms
coming from the commutator
¯
Nx
v3Dv3(xy)
x−¯
Ny
v3Dv3(xy)
y,¯
N′y
v2Dv2(yz)
y−¯
N′z
v2Dv2(yz)
z.
We continue with
¯
Nz
v1Dv1(zx)
z−¯
Nx
v1Dv1(zx)
x,¯
N′y
v2Dv2(yz)
y−¯
N′z
v2Dv2(yz)
z
=24n−2¯
Nz
v1¯
N′y
v2Lv1
ea
x,Tr σaÈ
Lv2
yz 
−24n−2¯
Nx
v1¯
N′y
v2Lv1
ea
z,Tr σaÈ
Lv2
yz 
+24n−2¯
Nx
v1¯
N′y
v2Lv2
ea
z,Tr σaÈ
Lv1
zx
−24n−2¯
Nx
v1¯
N′z
v2Lv2
ea
y,Tr σaÈ
Lv1
zx (79)
where we again note that the last two terms match similar terms coming
from the commutator
¯
Nz
v4Dv4(zx)
z−¯
Nx
v4Dv4(zx)
x,¯
N′y
v2Dv2(yz)
y−¯
N′z
v2Dv2(yz)
z.
Thus, we continue with
(78) and (79)relevant terms =24n−2¯
Nx
v1¯
N′z
v2Lv1
ea
y,Tr σaÈ
Lv2
yz 
−24n−2¯
Ny
v1¯
N′z
v2Lv1
ea
x,Tr σaÈ
Lv2
yz 
+24n−2¯
Nz
v1¯
N′y
v2Lv1
ea
x,Tr σaÈ
Lv2
yz 
−24n−2¯
Nx
v1¯
N′y
v2Lv1
ea
z,Tr σaÈ
Lv2
yz  (80)
71

Here we immediately notice the two middle terms, which do not appear to
have the index structure required for a diffeomorphism constraint operator.
If we look at the front factors to these terms and also include those of
the opposite commutators, where ¯
Nand ¯
N′are interchanged and a sign is
added, and if we consider the continuum limit of these factors, then we find
a total derivative:
−¯
Ny
v1¯
N′z
v2+¯
Nz
v1¯
N′y
v2+¯
N′y
v1¯
Nz
v2−¯
N′z
v1¯
Ny
v2
continuum
Ð→ ∂x¯
Ny¯
N′z−∂x¯
Nz¯
N′y−∂x¯
N′y¯
Nz+∂x¯
N′z¯
Ny
=∂x¯
Ny¯
N′z−¯
Nz¯
N′y.(81)
Here we ignored that there is also a contribution in (81) without a derivative,
which will be divergent. Further, if we consider also the classical correspon-
dent to the operator 1
2Lv1
ea
x,Tr σaÈ
Lv2
yz , which is Ex
aFa
yz =Tr (ExFyz),
then we find
Tr Dx ¯
Ny¯
N′z−¯
Nz¯
N′yExFyz 
=−¯
Ny¯
N′z−¯
Nz¯
N′yTr (ExDxFy z)(82)
where we used partial integration and interchanged ∂xwith the covariant
derivative Dx. Now, we use the Bianchi identity to proceed
−¯
Ny¯
N′z−¯
Nz¯
N′yTr (ExDxFy z)
=−¯
Ny¯
N′z−¯
Nz¯
N′yTr ((DyEx)Fz x)−∂y¯
Ny¯
N′z−¯
Nz¯
N′yTr (ExFz x)
−¯
Ny¯
N′z−¯
Nz¯
N′yTr ((DzEx)Fxy )−∂z¯
Ny¯
N′z−¯
Nz¯
N′yTr (ExFxy )
If we add the classical correspondents to the remaining two terms in equation
(80) and the classical correspondents to (76) as well as the ’zx −zx’ com-
mutators, which we again do not write down, then we obtain the classical
expression
−¯
Nα∂α¯
N′µ−¯
N′α∂α¯
NµEν
aFa
µν +”e.o.m. terms” (83)
where ”e.o.m. terms” are terms which involve a DµEν
a. Thus, here too do
we encounter derivatives of the triad fields, but now they are, as they should
be, covariant. In the classical computation such derivative terms cancel out
and leave only a Gauss constraint. Here they remain and form the torsion
tensor coupled to the field strength tensor and the lapse fields.
Of course, this final ’ad-hoc’ analysis of equation (80) is classical and
should be elevated to the level of operators. This means formulating partial
integration and the Bianchi identity for this setting.
72

We see, however, that we come close to reproducing the structure of
the classical constraint algebra. Apart from terms like the one we found
in (77), which fail to give at the operator level what corresponds to co-
variant derivatives, and apart from terms, which are simply divergent, we
discovered the possibility that an anomaly proportional to a torsion operator
might emerge. Despite the fact that the computational setup that leads to
this potential anomaly is flawed, we think that the overall structure of this
computation might reflect something more general, that could carry over
into a setting, that involves a more realistic candidate for a diffeomorphism
constraint operator.
Our analysis of the commutator (74) can also be used to shed light on
what a more realistic candidate for a Hamilton constraint operator should
look like. First of all, in order to avoid the initial problem that vertex
operators associated to the same vertex do not commute, we think one
needs to built a Hamilton constraint operator, which fully incorporates the
algebraic structure dictated by the operator hnin (43). This means that
vector-fields should only be associated to edges, which do not have the base-
point of the loop as a start or endpoint. Secondly, the problem with the
non-covariant derivatives of the triad fields might be avoided if one is able
to obtain the algebraic structure Av1Bv2+Av2Bv1multiplied with a factor
involving the lapse fields. To obtain this structure from the commutator
of two constraint operators one needs to include vertex operators, which
involve loops in all possible directions. In the present setup, the Hamilton
and diffeomorphism constraint operators only involve three loops based in
a given vertex. However, twelve loops are possible and it is these loops that
we think needs to be included.
What we have in mind is a Hamilton constraint operator built from
vertex operators of the form
Hvi(Nvi)=12

k=1
Hk
vi(Nvi)
where the sum runs over all 12 possible infinitesimal loops based in each
vertex vi, with
Hk
vi(¯
Nvi)=22n−2NviL(vi,k)
ea
i,L(vi,k)
eb
j
,Tr σaÈ
Lvi
kσb (84)
where L(vi,k)
ec
jis the vector-field associated to the edge ljin the k’th loop
based in viand where the edges liand ljare the two edges in the loop,
which are not connected to vi.
73

It requires, however, a significant computational effort to check the com-
plete operator constraint algebra with such an operator, and thus we leave
it to be done elsewhere.
The commutator [D(¯
N),H(N)]exhibits similar anomalies and we shall
not write it down.
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