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Gribov Problem and Stochastic Quantization
Adithya A Rao
Department of Physics
Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat
Supervised by
Prof. Laurent Baulieu, Sorbonne University, Paris
Dr. Vikash K. Ojha, SVNIT, Surat
Subimitted in partial fulfillment of the requirements for the degree of
Master of Science in Department of Physics, SVNIT, Surat
2023 - 2024
arXiv:2406.15059v1 [hep-th] 21 Jun 2024
ABSTRACT
The standard procedure for quantizing gauge fields is the Faddeev-Popov quantization,
which performs gauge fixing in the path integral formulation and introduces additional
ghost fields. This approach provides the foundation for calculations in quantum Yang-
Mills theory. However, in 1978, Vladimir Gribov showed that the gauge-fixing procedure
was incomplete, with residual gauge copies (called Gribov copies) still entering the path
integral even after gauge fixing. These copies impact the infrared behavior of the theory
and modify gauge-dependent quantities, such as gluon and ghost propagators, as they rep-
resent redundant integrations over gauge-equivalent configurations. Furthermore, their
existence breaks down the Faddeev-Popov prescription at a fundamental level. To par-
tially resolve this, Gribov proposed restricting the path integral to the Gribov region,
which alters the gluon propagator semiclassically in a way that points to gluon confine-
ment in the Yang-Mills theory.
In this thesis, we comprehensively study the Gribov problem analytically. After reviewing
Faddeev-Popov quantization, the BRST symmetry of the complete Lagrangian and the
Gribov problem in depth, we detail Gribov’s semi-classical resolution involving restric-
tion of the path integral to the Gribov region, outlining its effects on the theory. Further,
we elucidate stochastic quantization prescription for quantizing the gauge fields. This
alternate quantization prescription hints towards a formalism devoid of the Gribov prob-
lem, making it an interesting candidate for quantizing and studying the non-perturbative
regime of gauge theories.
Contents
I The Gribov Problem ............................ 1
1 Prelude - Gauge Field Theories ........................ 2
1.1 Introduction.................................. 2
1.2 Gauge Freedom And Yang-Mills Theories . . . . . . . . . . . . . . . . . . 3
2 Quantizing the Gauge Fields ......................... 7
2.1 The Failure of the Traditional Quantization . . . . . . . . . . . . . . . . 7
2.2 Faddeev-PopovGhosts ............................ 9
2.2.1 Feynman Rules For The Complete Yang-Mills . . . . . . . . . . . 10
2.3 Remarks On The Faddeev-Popov Method . . . . . . . . . . . . . . . . . . 12
2.4 TheBRSTSymmetry ............................ 12
2.5 Non-Trivial Structure Of The Gauge Group . . . . . . . . . . . . . . . . 16
3 The Haunting of the Gribov Copies ..................... 18
3.1 TheGribovProblem ............................. 18
3.2 Does The Gribov Problem Exist? . . . . . . . . . . . . . . . . . . . . . . 19
3.3 TheGribovRegion.............................. 22
3.4 Gribov’s No-Pole Condition . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Restricting the Path Integral Using No-Pole Condition . . . . . . . . . . 28
II Stochastic Quantization ......................... 31
4 Prelude - Probability Theory and Stochastic Processes ......... 32
4.1 StochasticProcesses ............................. 32
4.2 BrownianMotion ............................... 34
4.2.1 Focker-Planck equation from the Langevin equation . . . . . . . . 35
4.2.2 The thermal equilibrium . . . . . . . . . . . . . . . . . . . . . . . 36
5 Quantum Field Theory as a Brownian Motion .............. 39
5.1 Stochastic Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Quantizing the Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3 Stochastic Quantization of Gauge Fields . . . . . . . . . . . . . . . . . . 43
6 Gauge Fixing without Fixing the Gauge .................. 46
6.1 Gauge Fixing as a Non-Conservative drift force . . . . . . . . . . . . . . 46
6.2 Properties of the Gauge Fixing Force . . . . . . . . . . . . . . . . . . . . 49
III Concluding Remarks .......................... 52
List of Figures .................................... 55
Bibliography ..................................... 56
I
PART
THE GRIBOV PROBLEM
1
CHAPTER 1
PRELUDE - GAUGE FIELD THEORIES
1.1 Introduction
The quantization of non-Abelian gauge theories, such as quantum chromodynamics (QCD),
is a fundamental challenge in theoretical physics. QCD is the theory that describes the
strong interaction, one of the four fundamental forces of nature, and governs the behavior
of quarks and gluons, the fundamental constituents of nucleons.
While QCD has been successful in explaining many phenomena in the perturbative
regime, where the coupling strength is small, its non-perturbative aspects, which domi-
nate at low energies and large distances, remain a significant challenge. This is because
the coupling strength in QCD decreases with an increase in energy, and therefore at
low energies the perturbative techniques which rely on writing an asymptotic series in
the coupling constant break down. One of the key non-perturbative features of QCD is
confinement, which refers to the fact that at low energies, quarks and gluons are never
observed as free particles but are always confined within nucleons. This phenomenon is
believed to be related to the non-Abelian nature of QCD and the self-interactions of the
gluon field. Although there is no experimental evidence for the confinement of gluons,
one expects them to be confined owing to them being colorful, and the gluon confinement
is expected to be purely a quantum effect.
One of the problems in the quantization of non-Abelian gauge theories like QCD is the
Gribov problem, which arises due to the redundant degrees of freedom present even after
performing the gauge fixing. This redundancy leads to an overcounting of equivalent
field configurations, resulting in an ambiguity in the path integral measure and the prop-
agators of the theory. The Gribov problem was first identified by Vladimir Gribov in
the quantization of Yang-Mills theories in the Coulomb gauge. He discovered that the
Coulomb gauge condition does not completely fix the gauge freedom, leading to the ex-
istence of multiple gauge copies, known as Gribov copies, which correspond to the same
2
CHAPTER 1. PRELUDE - GAUGE FIELD THEORIES
physical configuration but have different gauge field representations, all of them counted
in the path integral.
This ambiguity in the gauge fixing procedure has far-reaching consequences for the non-
perturbative aspects of gauge theories like QCD. The Gribov problem is believed to play a
crucial role in the confinement mechanism and the dynamical generation of a gluon mass
scale. Over the years, various approaches have been proposed to address the Gribov
problem, including the Gribov-Zwanziger framework. However, these formulations often
rely on certain approximations and may not fully capture the non-perturbative dynamics
of the theory.
An alternative approach to addressing the Gribov problem is through stochastic quanti-
zation, a formalism introduced by G. Parisi and Yong-Shi Wu. Stochastic quantization
reformulates the path integral of a quantum field theory as an equilibrium limit of a
stochastic process, effectively replacing the time evolution of the system with a fictitious
time evolution governed by a Langevin equation.
In this thesis, we will look at the origin and effects of the Gribov problem and also present
the stochastic quantization formulation as an alternative quantization prescription that
has the potential to circumvent some of the issues associated with the Gribov problem
and provide insights into the non-perturbative aspects of gauge theories like QCD
1.2 Gauge Freedom And Yang-Mills Theories
The concept of gauge invariance was introduced by Hermann Weyl to describe electrody-
namics in analogy to Einstein’s theory of gravity, with local gauge freedom corresponding
to Einstein’s coordinate invariance, and the vector potential Aµplaying the same role as
the Christoffel’s symbols Γν
λµ.
To see exactly how the gauge fields arise, consider a theory with Nspinor fields, ψ(x)≡
ψa
α(x)where αdenotes the Dirac indices, and a= 1, . . . , N . This theory associates each
spacetime point xwith an N-dimensional vector space Ψxin which the spinors are de-
fined. Gauge refers to the basis for each of the vector spaces Ψx. The physical content
of the theory is required to be independent of the choice of basis of these vector spaces,
and this freedom in choice of basis is called the gauge freedom.
The naive Dirac Lagrangian for the spinor fields in this theory can be written as
L=¯
ψ(x)(i/
∂−m)ψ(x)(1.1)
3
CHAPTER 1. PRELUDE - GAUGE FIELD THEORIES
This Lagrangian is naive because it does not account for the freedom in the choice of
basis for the spaces Ψx. It contains the regular spacetime derivative ∂µwhich depends
on comparing spinors at infinitesimally separated points, and since these live in different
spaces spanned by different basis, there is no way to compare the spinors at two points,
and consequently, the definition of derivative breaks down.
There is a simple way to circumvent this problem - choose one set of basis and enforce
it on all the vector spaces at all points. But the derivative defined in this way is not
gauge-invariant and is not something we are looking for. What we are looking for is an
object that transforms a vector in one space into a vector in the space at an infinitesimally
separated point. To see how we can define such an object, first, let us write the spinor
fields in terms of the basis at each point.
ψα(x) = ψa
α(x)ua(x)(1.2)
where uaforms a basis for the space Ψx.
Now when going from xto x+dx let the actual value of the spinor change by an amount
dψ =ψ(x+dx)−ψ(x)(1.3)
The ∂µψcontains the contributions from both the actual change in the value of the spinor
and the change in the orientation of the basis. Therefore
dψ = (∂µψa)dxµua+ψadua(1.4)
Suppose in going from xto x+dx, the basis of the internal spaces rotates by dθk=
(∂µθk)dxµwhich is a set of Ngangles. Since the rotation group on Ndimensional complex
vectors is SU (N),Ngis the number of generators for SU(N). The rotation operator,
which would be an element of SU(N)is therefore given by
U(dx) = exp −iq dθkTk
ab(1.5)
where Tk
ab are the generators of the SU (N)group.
This rotates the basis as
U(dx)ua= exp −iq dθkTk
abub(1.6)
Expanding this upto first order in dx, we get
ua+dua=δab −iq (∂µθk)dxµTk
abub(1.7)
4
CHAPTER 1. PRELUDE - GAUGE FIELD THEORIES
which gives
dua=−iq (∂µθk)Tk
abubdxµ(1.8)
If we call (∂µθk(x))Tk
ab = (Aµ)ab the connection, then the total change in ψis given as
dψ = ((∂µψa)δab −iq(Aµ)abψa)ubdxµ(1.9)
From this, we can write the gauge-covariant derivative operator defined as called as the
covariant derivative as
Dµψa= (∂µδab −iq(Aµ)ab)ψb(1.10)
The comparator has values as a function of spacetime points and therefore defines a
field on its own which we call the gauge fields, and since we require that the covariant
derivative transform covariantly under gauge transformations, we see that the gauge fields
must transform as
Aµ(x)→exp(iθ(x)kTk)Aµ(x) exp(−iθ(x)kTk)−i
q(∂µexp(iθ(x)kT)k) exp(−iθ(x)kTk)
(1.11)
Infinitesimally, this transformation looks like
Ak
µ→Ak
µ−1
q∂µαk+fklmAl
µαm=Ak
µ−1
qD(adj) kl
µαl(1.12)
where fklm are the structure constants for the SU (N)group, and D(adj)denotes the ac-
tion of covariant derivative in the adjoint representation.
With this, we can write a gauge invariant Dirac Lagrangian density as
LDirac =ψ(x)( /
D−m)ψ(x)(1.13)
To make the vector field Adynamic, we add Lorentz covariant and gauge invariant
terms, quadratic in derivatives and fields. Noting that the covariant derivative transforms
covariantly and that the commutator [Dµ, Dν]has terms consisting of only Aand its
derivatives, we can construct a Lagrangian density as
LA=1
2
1
ig T r[Dµ, Dν]2(1.14)
Calling 1
ig [Dµ, Dν]ab =Fµν ab =Fk
µν Tk
ab and using the normalization for the trace of
generators of the group, we write
LA=1
4Fk
µν Fµν
k(1.15)
5
CHAPTER 1. PRELUDE - GAUGE FIELD THEORIES
The field theory with a non-abelian gauge group SU(N)and a Lagrangian density of the
gauge fields specified by the equation (1.15) is called a Yang-Mills theory.
Fk
µν , which is also called the field strength tensor, can be written in terms of Aas
Fk
µν =∂µAk
ν−∂νAk
µ−gf klmAl
µAm
ν(1.16)
The Lagrangian density constructed as prescribed contains the terms cubic and quartic
in Aand its derivatives. These higher-order terms arise purely due to the non-abelian
nature of the gauge group, and the existence of these terms implies that the field Aitself
is a charged field, and thus there is a multitude of self-interaction vertices in even the
simplest Feynman diagrams which is the reason for the beautiful complexity and richness
of the theory.
6
CHAPTER 2
QUANTIZING THE GAUGE FIELDS
In this chapter, we provide a comprehensive introduction to the Faddeev-Popov quantiza-
tion procedure which forms the basis for all quantum field theory calculations. Developed
by Ludvig Faddeev and Victor Popov in the 1960s, this method addresses the issue of
redundant degrees of freedom present in gauge field theories. In the context of QCD, the
gauge symmetry leads to an overcounting of physically equivalent configurations, which
can result in inconsistencies and ambiguities in the quantization process. The Faddeev-
Popov procedure introduces auxiliary fields, known as ghost fields, to cancel out the
unphysical degrees of freedom associated with the gauge redundancy. The ghost fields,
while unphysical themselves, play a crucial role in ensuring the gauge invariance of the
quantized theory and maintaining the correct counting of physical degrees of freedom.
2.1 The Failure of the Traditional Quantization
Given a field theory in field Aand a classic action functional S[A]defined on it, the
traditional quantization method is to take the classic action functional and write a path
integral for it defined as
Z[J] = ZCDAe(−S[A]−Rd4xJ·A)(2.1)
where Jis a function of spacetime and not fields, and is introduced as a mathematical
tool for generating Green’s functions. The integral over the configuration space Cwith
the (ill-defined) measure DArefers to integration over all possible field configurations,
and to make the measure well-defined, we further assume suitable regularisations.
The Yang-Mills Lagrangian inherently contains terms cubic and quartic in fields, and
therefore it does not describe a free theory but rather describes an interacting theory. To
quantize Yang-Mills perturbatively, we consider only the quadratic part of the Lagrangian,
and take the self-interaction terms as perturbations. In such a setting, the free path
7
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
integral for the Yang-Mills is written as
Zquad[J] = ZCDAexp −Zd4x1
4(∂µAk
ν−∂νAk
µ)2+Zd4x Jk
µAk
µ
=ZCDAexp −Zd4x1
2Ak
ν(∂µ∂ν−δµν ∂2)Ak
µ+Zd4x Jk
µAk
µ(2.2)
One can solve the Gaussian integral to obtain
Zquad[J] = 1
det(Gµν )exp 1
2Zd4x Jk
µG−1
µν Jk
ν(2.3)
where Gµν =∂µ∂ν−δµν ∂2.
In a usual quantum field theory, G−1
µν would be called the propagator, and by making use
of Wick’s theorem one writes all the correlation functions in terms of the above-defined
propagator. One could do the same in the case of Yang-Mills too, if the inverse of Gµν
existed. But there exist zero modes of the form ∂µλfor which (∂µ∂ν−δµν ∂2)∂µλ= 0,
making the operator non-invertible and therefore making Zquad[J]ill-defined. This is a
consequence of the gauge freedom, which makes the operator ∂µ∂ν−δµν ∂2a many-to-one
map, breaking its invertibility.
Another shortcoming of this method can be seen as follows. The path integral is formally
an integration over all possible field configurations, and gauge freedom means that all the
gauge equivalent configurations are counted multiple times too, meaning that the path
integral gives rise to an infinity, which is the size of the gauge transformation group. This
makes the path integral infinite and therefore lose its meaning.
To consistently define the theory, we need to eliminate the gauge degree of freedom. One
way to do this would be to introduce a constraint on the space of configurations that is
satisfied by only one out of the infinite gauge copies. A simple constraint would be of the
form ∂µAµ=√α b.
In the Abelian gauge theory (QED), we proceed by promoting this constraint to an
equation of motion, and we write a modified Lagrangian of the form
LQED =1
4(∂µAν−∂νAµ)2+α
2b2+b∂µAµ(2.4)
The equation of motion from this Lagrangian implies that ∂A follows the free field equa-
tion, □∂A = 0. This means that we can freely assign to it the field band this does not
modify the physical content of the theory. Introducing this term breaks gauge invariance
and hence makes the quadratic operator invertible.
8
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
In the case of Yang-Mills, one cannot follow such a gauge fixing procedure due to the
presence of the self-interaction terms. The equation of motion for ∂A does not follow
the free field equation, which means that assigning to it the field bnaively truly modifies
the physical content of the theory. Therefore the gauge fixing procedure that made QED
consistent fails in the case of Yang-Mills.
2.2 Faddeev-Popov Ghosts
Faddeev and Popov, in their 1967 work, suggested a way to quantize gauge fields without
performing gauge fixing in the Lagrangian. They did the quantization by introducing a
delta function in gauge constraint directly into the path integral. This would mean that
the path integral counts only those configurations that satisfy the gauge condition and
discards those that do not.
Consider the identity in the form
ZDF(GA)δ(F(GA)−λ) = 1 (2.5)
where F(GA)−λ= 0 is the gauge condition, with GAreferring to the gauge transformed
field.
We can change the integration measure from DF(GA)to DG
ZDG
det δF(GA)
δG
δ(F(GA)−λ)=1 (2.6)
with the determinant arising as the Jacobian of the variable change.
Since varying over λsimply gives a class of gauge fixing conditions that give rise to the
same physical theory, we can integrate over λwith a Gaussian measure to obtain
ZDλexp −Zd4xλ2
2αZDG δ(F(GA)−λ)
det δF(GA)
δG
=C(α)(2.7)
This identity can be inserted into the path integral eq (2.1)
Z[J] = NZDGZCDA
det δF(GA)
δg
exp −S[A]−Zd4xF(GA)2
2α+J.A
(2.8)
Since S[A]and DAare gauge invariant, we can perform two consecutive changes of
9
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
variables A→GAand then GA→Awithout altering the integration, giving
Z[J] = NZDGZCDA
det δF(GA)
δG G=1
×exp −S[A]−Zd4xF(A)2
2α−J.A (2.9)
Now, since the integrand is independent of G, the integral ZDGsimply gives the volume
of the gauge group factored out of the path integral and therefore can be absorbed into
the normalization without altering the values of observables.
In this thesis, we will mainly deal with the gauge fixing condition of the form ∂µAµ= 0.
This means the operator inside the determinant is of the form (since infinitesimal gauge
transformation is of the form Ak
µ→Ak
µ−Dk
µα)
det δ∂µ(Ak
µ(x)−D(adj) kl
µαl(x))
δαl(y)!α=0
= det(−∂µD(adj)kl
µδ(x−y)) (2.10)
Using the property of Grassman integrals, we write this determinant as a path integral
det(−∂µD(adj)kl
µδ(x−y)) =
ZDΩD¯
Ω exp −Zd4xd4y¯
Ωk(x)(∂µD(adj)kl
µδ(x−y))Ωl(y)(2.11)
where Ωand ¯
Ωare Grassman-valued scalars that transform in the same representation
as A.
With this introduction, we can write the complete Lagrangian of a consistent Yang-Mills
theory as
L=1
4(Fk
µν )2+1
2α(∂µAk
µ)2+¯
Ωk∂µD(adj)kl
µΩl(2.12)
The fields Ω & ¯
Ωsatisfy the Klein-Gordon equation but are also anticommuting fields.
Since this is a violation of the spin-statistics theorem, the excitations of these fields can not
be physical. These fields can only enter the equations as internal lines and not asymptotic
states. Thus due to their non-physical nature, they are called as Faddeev-Popov ghosts.
2.2.1 Feynman Rules For The Complete Yang-Mills
From the Lagrangian (2.12), we can write down the free propagators and the interaction
vertices for the gauge and ghost fields as follows
•Free Gauge Propagator:
10
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
The quadratic part of the Lagrangian in gauge fields is 1
4(∂µAk
ν−∂νAk
µ)2+1
2α(∂µAk
µ)2.
Thus the free gauge propagator would be the inverse of the operator ∂µ∂ν−δµν ∂2−
1
α∂µ∂ν. In momentum space, the free propagator is
⟨Ak
µ(p)Al
ν(−p)⟩=Ak
µAl
ν
pµ
=−iδkl
p2+iϵ δµν −(1 −α)pµpν
p2(2.13)
•Free Ghost Propagator:
The free part of the ghost Lagrangian is simply that of the complex scalar field
theory, and so the free ghost propagator in momentum space would be
⟨¯
Ωk(p)Ωl(−p)⟩=¯
ΩkΩl
pµ
=−iδkl
p2+iϵ (2.14)
•Gauge-Gauge Vertices:
There exist two gauge-gauge vertices, one for the interaction term gAk
νfklmAl
µ∂µAm
ν
with the vertex factor gpµfklm with pµbeing the momentum of Am
ν, and another
with interaction term g2fklmAl
µAm
νfknoAn
µAo
νwith the vertex factor g2fklmfkno
Am
ν
Al
µAk
ν
gf klmpµ
pµ
qµ
p′
µ
Al
µAm
ν
An
µAo
ν
g2fklmfkno
pµ
qµ
p′
µ
q′
µ
•Gauge-Ghost Vertex:
There also exists a gauge-ghost vertex corresponding to the interaction term ¯
ΩkfklmAl
µ∂µΩm
with the vertex factor being gf klmpµ, where pµis the momentum of Ωm
Al
µ
¯
ΩkΩm
gf klmpµ
qµ
p′
µ
pµ
•Negative Sign of Ghost Loops:
Since the ghost fields are anticommuting, the Feynman diagrams with ghost loops
get multiplied by −1.
11
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
These Feynman rules consistently define Yang-Mills theory encapsulating both gauge and
ghost fields.
2.3 Remarks On The Faddeev-Popov Method
In the previous discussion, we overlooked a few crucial considerations, which seem trivial
but are very important for the construction of the theory.
In the identity, eq (2.5), the change of integration variable should be done as
ZDG δ(F(GA)−λ) = YX 1
δF(GA)
δG
(2.15)
where the summation stands over all the zeros of the gauge fixing functional. We made
an assumption here that the gauge fixing condition is satisfied by a single configuration
out of all gauge equivalent configurations. This reduces the ugly expression in the right-
hand side of the above, to a simple product of the absolute value of the eigenvalues of
the Faddeev-Popov operator δF(GA)
δG . This can be then written as the determinant of
the said operator.
If the gauge fixing condition is satisfied by multiple equivalent configurations, this pro-
cedure will break down.
Another assumption we made was of the positive definiteness of the Faddeev-Popov op-
erator, which allowed us to ignore the absolute value requirement of the determinant.
With only this assumption, it was possible to introduce the ghost fields and the Grass-
man integral. If the Faddeev-Popov operator has negative modes, then we cannot drop
the absolute value operation, and therefore we cannot introduce the ghost fields. Further,
if the FP operator has zero modes, then for that specific configuration, the determinant
is zero, and hence the contribution of that configuration to the path integral is nullified.
It is these very considerations that later come back to bite us which will be discussed
later in our later discussion of the Gribov problem.
2.4 The BRST Symmetry
Let us take a detour and discuss the implications that the Faddeev-Popov quantization
has for the theory. Since the introduction of the gauge fixing term and ghost fields breaks
12
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
the gauge symmetry, we look for a new symmetry that encompasses all these fields, also
encapsulating the residual gauge symmetry of the Lagrangian. This symmetry, called the
BRST symmetry, was discovered by Carlo Maria Becchi, Alain Rouet, Raymond Stora,
and Igor Viktorovich Tyutin, as a complete symmetry of the Faddeev-Popov effective
action.
The ghost Lagrangian is invariant under the scaling of (anti)ghost fields as
Ω→eγΩ
¯
Ω→e−γ¯
Ω(2.16)
where γis a real, spacetime-independent parameter.
Noether’s theorem implies a conserved current corresponding to this symmetry, which we
will call the ghost number. We can compute the ghost number for the fields, Ωcarries
a ghost number +1 while ¯
Ωcarries a ghost number −1. Since Aµhas no (anti)ghost
components, it has ghost number 0.
We adopt here a matrix notation in adjoint representation, where we saturate spacetime
indices of fields (if present) with dxµ. For example, A=Ak
µTkdxµ. This is a matrix in
adjoint representation with elements being one-forms in spacetime. The elements of this
matrix are anticommuting under exterior product. Similarly Ω=ΩkTk, the elements of
which are Grassmann numbers, and hence are anticommuting.
Along with this, we introduce a universal graded bracket, [X, Y ] = X±Ywhere the +
sign occurs only in the case Xand Yare both anticommuting.
It is immediately obvious that the matrix elements anticommute if its form degree is odd
and the ghost number is even or the other way round. The matrix elements are always
commuting when both the form degree and ghost number are odd or even. Thus we can
assign a generalized grading, which defines if the matrix elements are commuting or anti-
commuting. This grading is defined as the sum of the form degree and the ghost number.
If the grading is even, then the matrix elements commute. Otherwise, they anticommute.
Here we can introduce a structure, what we will call a generalized n-form, (Ωn)a
b, a+b=n
whose ghost number is aand form degree is b. The (anti)commutative nature of such a
structure depends on n (mod 2), similar to their n-form counterparts.
In the space of these structures, there must exist an operator ˆ
d, which behaves like the
exterior derivative that acts on the space of n-forms. The operator ˆ
dmust increase the
grading by 1, which can be achieved in two ways. The first way is to increase the form
degree by 1, while the second way is to increase the ghost number by 1. That is, we
13
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
require the action of ˆ
don (Ωn)a
bas ˆ
d(Ωn)a
b= (Ωn+1)a
b+1 + (Ωn+1)a+1
b.
There already exists the regular exterior derivative dthat increases the form degree by
1, and hence we demand a new operator s, which we will call the BRST operator, that
increases the ghost number by 1. With this we would have ˆ
d=d+s, with both dand s
being operators of grading 1.
The operator ˆ
dconstructs an anti-commuting structure from a commuting one and vice
versa, and hence it must hold that ˆ
d2=d2+ds +sd +s2= 0. Since sand dact on
orthogonal spaces, [d, s] = ds +sd = 0 should hold. Thus, we arrive at the condition
s2= 0, i.e., the BRST operator should be nilpotent of order 2. This condition, along
with sd +ds = 0 defines how the operator sacts on the fields.
To see where such a geometrical structure arises from, we see that if we consider spacetime
to be spanned by {x}and the gauge group to be spanned by {y}, then the spinor field
at any point in the extended space (x, y)can be written as
˜
ψa(x,y) = exp iykTk
abψb(x)(2.17)
The local connections on the extended space would be the generalized one form
˜
Ak=Ak
µdxµ+ Ωk
mdym(2.18)
where the first term is the regular gauge field, and the second term, which is the local
connection in the gauge space, can be recognized with the ghost fields.
In this extended space, ghosts are not some artifacts that need to be introduced to the
theory to make it consistent but rather are the fundamental objects of the gauge theory.
The BRST operator arises in this model simply as the exterior derivative in the yspace.
s=dym∂m(2.19)
The deduction that s2= 0 from the previous discussion is now straightforward since s
is the exterior derivative on yspace, which means it should be nilpotent. This operator
can be, in a sense, interpreted as a generator of translations in the gauge space. Gauge
invariance implies invariance under translations in the gauge space, which means that we
expect the Lagrangian to be invariant under such translations. To check this, we proceed
as follows.
From the conditions s2= 0 and [d, s] = 0, it is pretty straightforward to construct the
14
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
action of son the fields
sA =1
gdΩ+[A, Ω] = D(adj)Ω
sΩ = 1
2g[Ω,Ω]
s¯
Ω = −1
gα (∂µAµ)
(2.20)
These equations can also be constructed from the “horizontality” condition on the gener-
alized curvature ˜
F, which states that for a physical theory, the only non-zero components
of the curvature tensor should be the dxµ∧dxνcomponents. The above equations are
precisely the equations that state that the other components of ˜
Fare zero.
As expected, we can notice from the above equations that the action of the operator ηs
on the gauge fields is exactly that of an infinitesimal gauge transformation, if we recog-
nize ηΩwith the field of gauge parameters, where ηis an infinitesimal g-valued constant
introduced to make the gauge transformations boson valued.
It can be seen that the Lagrangian eq (2.12) is indeed invariant under these transforma-
tions of the fields, which means that the BRST operator generates a symmetry group of
the Lagrangian.
A very important consideration that we overlooked here is that for this Lagrangian,
s2¯
Ω∝∂D(adj)Ωis 0only when the EOM for the ghost field, ∂D(adj) Ω=0is imposed
by hand. This means that the current Lagrangian is in an on-shell representation of the
BRST algebra.
To get a representation that closes off-shell too, we re-introduce the non-dynamical auxil-
iary bosonic field bof mass dimensions 2, whose equations of motion give the gauge fixing
condition. Such a Lagrangian would be
L=1
2F2+α
2b2+b∂A +¯
Ω∂D(adj)Ω(2.21)
Because the mass dimension of bis 2, it can never enter the Lagrangian dynamically, and
hence it behaves as a background field. But at the same time, its introduction makes the
BRST operator closed off-shell hence completing its definition.
15
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
With the introduction of the auxiliary field, the action of the BRST operator becomes
sA =1
gdA + [A, Ω]
sΩ = 1
2g[Ω,Ω]
s¯
Ω = b
sb = 0
(2.22)
We immediately see that these transformations close off-shell too. Thus, the BRST sym-
metry forms a symmetry of the complete Lagrangian.
Many authors have suggested that rather than arriving at the BRST symmetry as the
symmetry group of the Lagrangian, the right thing to do would be to consider a symme-
try group with nilpotent generators and construct a Lagrangian that is invariant under
the transformations generated by this group, stating the BRST invariance to be the first
principle from which a gauge theory should be constructed.
Nevertheless, BRST symmetry is not simply some mathematical curiosity, rather it plays
a vital role in the renormalisability of the theory. Imposing the condition that the physical
states of the theory should be annihilated by sremoves all the unphysical states of the
theory from the physical subspace, and also ensures the unitarity of the theory [1]. Some
authors have also suggested that the violation of BRST symmetry due to the Gribov
problem might also lead to confinement effects.
2.5 Non-Trivial Structure Of The Gauge Group
So far, we have not considered how the gauge group is structured, since it was irrelevant
to the discussion. But to address the Gribov problem, it is necessary to understand the
notion of small and large gauge transformations and a discussion of the structure of the
gauge group becomes necessary here.
The gauge group, in its simple sense, defined as G∞is trivial, considering that all the
gauge transformations can be smoothly connected to the identity. But this is not true
actually, since we impose some restrictions on the allowed gauge transformations, thus
making the structure non-trivial.
The conserved Noether charge associated with the gauge symmetry is called the color
16
CHAPTER 2. QUANTIZING THE GAUGE FIELDS
charge. One can construct the color charge operator from Noether’s theorem as [2]
Q|physical⟩=1
gZd3x ∂iEi(x)|physical⟩=1
glim
R→∞ ZS2
R
ds·E(x)|physical⟩(2.23)
where E(x)is the gauge field equivalent of the QED electric field.
Since we require that the color charge operator remain invariant under gauge transfor-
mations, the following expression must hold,
QU=1
glim
R→∞ ZS2
R
ds·U−1(x)E(x)U(x) = Q(2.24)
Thus at R→ ∞, we require that the gauge group element must behave as lim
x→∞ U(x) = U∞
where U∞should be a constant that commutes with all other members of the group.
Adding this restriction, it turns out that the allowed gauge transformations form disjoint
subsets of the original gauge group, each characterized by a winding number and with
elements smoothly connected to some paradigm. With the introduction of this non-trivial
structure, we can talk about small and large gauge transformations. A small gauge trans-
formation would be the one that would be smoothly connected to identity, while large
ones would be the one that would belong to other homotopy classes, disconnected from
identity. This disconnection is because any function that smoothly connects gauge trans-
formations that belong to two different homotopy classes should pass through regions
where the boundary condition is not satisfied. In such case, the gauge transformation
does not remain a pure gauge transformation, and hence any smooth function between
two homotopy classes must leave the domain of the allowed gauge transformations some-
where in between.
This property is reflected in the vacuum structure of the Yang-Mills theory. There are
multiple vacuums of the Yang-Mills, each with a different topological charge, and these
vacuums are not connected via small gauge transformations. The wavefunction can pick
a global phase factor parameterized by a vacuum angle θwhen the system transitions
from one vacuum to another, and this phase factor is physical observable.
The CP violation of the theory implies that the phase factor should have some value, but
so far the experiments have constrained the value of the phase angle to be <10−10. The
relative absence of such a phase factor is termed the strong CP problem, which is beyond
the scope of our discussion.
17
CHAPTER 3
THE HAUNTING OF THE GRIBOV COPIES
In this chapter, we will discuss the Gribov problem in detail with explicit examples. The
Gribov problem, first identified by Vladimir Gribov in the late 1970s, arises from the am-
biguity in the gauge-fixing procedure, where multiple gauge field configurations, known as
Gribov copies, can satisfy the same gauge condition while representing the same physical
state. We also outline Gribov’s original proposal for resolving this problem and describe
the implications of implementing the Gribov restriction in a semiclassical approxima-
tion. Specifically, we will show how the restriction modifies the standard Faddeev-Popov
approach and leads to the generation of a mass parameter for gluons.
3.1 The Gribov Problem
The ideal gauge fixing condition would be the one that will be satisfied by only one of the
gauge copies for a given configuration. Mathematically, the existence of the ideal gauge
fixing condition translates to a statement regarding the existence of a global section of
the principle bundle of the theory. We can write a base manifold as the set of all con-
figurations of the gauge fields {A}. For each A∈ {A}, we assign a set of all equivalent
configurations {GA}called the gauge orbit. The orbits form a fiber bundle over the base
manifold, the entire structure being called the principle bundle. The gauge fixing func-
tion, which picks representatives from each gauge orbit thus defines a continuous section
map from the base manifold to the fiber bundle.
The ideal gauge condition that we demanded, which requires a section map to intersect
the gauge orbit exactly once per orbit, translates to it being a global section. According
to a theorem in topology, a principle bundle will admit a global section if and only if it
is a trivial bundle. If the principle bundle is non-trivial, then no matter how ingenious
the gauge fixing condition is devised to be, there will be multiple intersections with at
least a few of the orbits, and the gauge fixing method will fail. Such a problem, if exists,
would be termed the Gribov Problem.
18
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
At this point, one might ask what are the consequences of the failure of gauge fixing.
First of all, the elevation of the right-hand side of eq (2.15) to the inverse of the determi-
nant would not be possible since the primary assumption in doing so was that the gauge
condition is satisfied only once per orbit. Secondly, the existence of multiple gauge copies
satisfying the gauge fixing condition implies the existence of αsuch that ∂µDµ(α)=0.
This implies that the FP operator has zero modes, meaning that the determinant in the
path integral becomes zero for some configurations, effectively causing their contribution
to the integral to be nullified.
The existence of zero modes also implies that perturbations around these zero modes can
create negative eigenvalues of the FP operator. We were able to introduce ghost fields by
dropping the absolute value operation owing to the assumption that the Faddeev-Popov
operator is always positive. The existence of these negative modes implies that the in-
troduction of the Grassmann integral and the ghost fields become meaningless.
Thus, the non-uniqueness of gauge fixing conditions would lead to profound problems in
the very foundations of the quantum Yang-Mills theory.
3.2 Does The Gribov Problem Exist?
Consider an explicit example of SU (2) Yang-Mills in the Coulomb gauge, A0= 0, ∂iAi=
0.
To simplify our discussion we consider one specific field configuration, the vacuum con-
figuration in which the field curvature vanishes implying that the fields are in pure gauge
Ai=U−1∂iU.
Following Gribov, we take spherically symmetric gauge transformations of the form
U(r) = exp i1
2α(r)σini= cos 1
2α(r)+niσisin 1
2α(r)(3.1)
where niis the unit vector xi/r, and σiare the generators of SU (2)
With such a gauge transformation, the gauge copies of the vacuum would be
19
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
Ai=cos 1
2α−niσisin 1
2α
×−1
2sin 1
2αdα
dr ni−1
r(δij −ninj)σjsin 1
2α+njσj
1
2cos 1
2αdα
dr ni
=1
2α′ni(niσj) + 1
2r(δij −ninj)σjsin (α) + sin21
2α1
rϵijk njσk
(3.2)
The Coulomb gauge condition in this case translates to the differential equation
r2α′′ + 2rα′−2 sin α= 0 (3.3)
With a change of variable r= exp(t), the equation becomes
α′′ +α′−2 sin α= 0 (3.4)
This equation is an equation of motion for a damped pendulum, with αbeing the angle
from the unstable equilibrium of the pendulum, with a periodic external driving force,
usually called the Gribov pendulum, with the pendulum initially being at α= 0, the
position of unstable equilibrium.
To avoid singularities at r= 0, we require the condition α(r=0)=0to be imposed.
This condition translates to lim
t→−∞ α(t) = δexp(t). Following this, we can construct three
different classes of solutions, δ= 0, δ > 0,&δ < 0.
The case δ= 0 gives the trivial vacuum α(r) = 0. The other two cases give topologi-
cally charged vacuum with charges ±1
2, with boundary conditions lim
t→∞ α±1
2=±π
2, with
α(r) = α1
2(r)and α(r) = −α1
2(r)respectively.
Therefore, the Coulomb gauge condition is satisfied by not one, but multiple vacuum
configurations each belonging to one of the three homotopically distinct classes, i.e. sep-
arated by large gauge transformations
A= 0
Ai= exp −iα1
2(r)σini∂iexp iα1
2(r)σini
Ai= exp iα1
2(r)σini∂iexp −iα1
2(r)σini
(3.5)
The three solutions that exist to the pendulum equation with the specified conditions are:
the pendulum never falls (which is the trivial vacuum), the pendulum falls clockwise, and
the pendulum falls counterclockwise.
20
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
Gribov, in his seminal paper, considered a more general case of a general spherically
symmetric gauge field configuration and showed that the gauge fixing condition fails in
all cases of the gauge fields. Further, Singer proved that for any gauge theory and any
covariant gauge condition, the gauge fixing procedure will always break down. [3]
A1A3
A2
UA1UA3
UA2
M(UA) = f
(a) Ideal case of gauge fixing where M(UA) =
fis the gauge condition. Our assumption had
it that each of the gauge orbits intersected the
gauge fixing hyperplane only once.
A
UA
M(UA) = f
(b) The actual situation with the gauge fixing
condition, where each orbit intersects multiple
times, the number of intersections also being
orbit dependent
Figure 3.1: The ideal and real gauge fixing situation
The presence of Gribov copies is closely tied to the existence of zero modes of the Faddeev-
Popov operator. Mathematically, it means the existence of α(x)(we are working back in
Landau gauge) such that
∂µ(∂µ+fklmAl
µ)α(x) = 0 (3.6)
This equation does have trivial solutions of the form α(x) = cIwhere Iis the identity
element in the gauge group and cis a constant independent of spacetime. These solutions
correspond to global gauge transformations and hence do not contribute to the problem
of Gribov copies. The only relevant solutions are the ones that are non-trivial in the
sense stated above.
From this, we can also see that in the case of QED, the above condition reduces to the
Laplacian in α,∂2α= 0, since f= 0 for U(1) group. The solutions to this equation are
the plane wave solutions, but these do not respect the boundary condition α(±∞) = c
and hence these αdo not belong to the subset of allowed gauge transformations. Thus
there exist no zero modes of the FP operator and hence QED is free of Gribov copies.
The same analysis also applies to the high momentum regime of the theory and also to
21
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
the perturbative calculations, where the gauge fields are small perturbations around zero.
In this case, again the FP operator approximately reduces to the Laplacian, and Gribov
copies become irrelevant to the analysis. But in the low momentum regime, where large
Aeffects become prominent, the theory is haunted by Gribov copies.
3.3 The Gribov Region
If we consider the configuration space of the gauge fields, then in small perturbations
around the vacuum, the FP operator is essentially the Laplacian, and hence it is positive
definite. As we move away from the vacuum in any direction, at some distance there
appears a zero mode.
Let us call this surface at which the Faddeev-Popov operator obtains its first zero mode
the Gribov horizon, and the region inside this horizon - the Gribov region. Beyond this
region, the FP operator is not ensured to be positive-definite and will have both negative
modes and zero modes.
Inside the region, the FP operator has neither zero modes nor negative modes. If one
restricts the path integral to this region alone, then both the problems encountered above
would be resolved. Such a restriction would allow us to perform the gauge fixing and
introduce the ghosts consistently.
One might ask what would be the consequences of such a restriction. Specifically one
might ask if any non-trivial configurations are left out of the path integral by this re-
striction. This question was answered partially by Gribov himself, who showed that for
every field infinitesimally close to the Gribov horizon, there exists a Gribov copy outside
the horizon. The general statement of whether all orbits intersect the Gribov region or
not is constructed using the Hilbert square norm along the orbit, and it is discussed below.
Thus the Gribov region can be defined as
C0={Ak
µ|∂µAk
µ= 0, ∂µDµ>0}(3.7)
and it has the following properties:
1. Every orbit intersects the Gribov region at least once:
Consider a Hilber square norm on a given gauge orbit of the form
||A||2=Zdx Tr(AµAµ) = 1
2Zdx Ak
µAk
µ(3.8)
22
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
As we vary the configuration along the gauge orbit, we obtain different values for
the Hilbert square norms. This norm attains its extremum along a given orbit when
δ||A||2= 2 Zdx Tr(δ(Aµ)Aµ)=0
= 2 Zdx Tr((Dµα)Aµ) = 0
= 2 Zdx Tr(∂µα Aµ+AµαAµ−αAµAµ)=0
= 2 Zdx Tr(∂µα Aµ+αAµAµ−αAµAµ)=0
=Zdx (∂µαk)Ak
µ= 0
=−Zdx αk(∂µAk
µ) = 0
(3.9)
This implies that the configuration is an extremum only when ∂µAk
µ= 0, i.e. only
when the gauge fixing condition is satisfied.
Further, the norm obtains a minima when
δ2||A||2=δZdx (−αk(∂µAk
µ)) >0
=Zdx αk(−∂µDkl
µ)αl>0
(3.10)
This implies that the norm obtains a minimum when the FP operator is positive.
Thus the Gribov region can be defined as the set of all minima of the Hilbert space
norm.
In [4], the authors have rigorously proved that the Hilbert space norm for each orbit
will certainly attain a global minimum, hence proving that all gauge orbits intersect
the Gribov region at least once.
2. The trivial vacuum belongs to the Gribov region:
Since around A= 0, the FP operator is essentially the Laplacian which is always
positive definite, the vacuum and perturbations of the vacuum belong to the Gribov
region. This means that all the perturbation theory calculations are free of the
Gribov problem.
3. The Gribov region is convex:
What this means is that given two configurations A1&A2belonging to the Gribov
region, then the affine combination A=τA1+ (1 −τ)A2, where τ∈[0,1], also
belongs to the Gribov region.
23
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
This can be easily seen as
∂µAµ=τ∂µA1µ+ (1 −τ)∂µA2µ= 0 (3.11)
and
∂µDµ[A] = ∂µ(∂µ+f(τA1µ+ (1 −τ)A2µ))
=∂µ(τ∂µ+τ f A1µ+ (1 −τ)∂µ+ (1 −τ)fA2µ)
=τ∂µDµ[A1µ] + (1 −τ)∂µDµ[A2µ]
(3.12)
Since both τand 1−τare positive, −∂µDµ[τ A1µ+ (1 −τ)A2µ]is positive. The
above two conditions imply that if A1and A2belong to the Gribov region, then
their affine combination also belongs to the Gribov region.
4. The Gribov region is bounded:
Since both Aand αbelong to the adjoint representation, [A, α]is traceless, and the
sum of all eigenvalues of this for a certain Ais zero. This implies that there exists at
least one eigenvector ωwith eigenvalue κ < 0,such that Rdxdy ω(x)[A(y), ω(y)] = κ
Now consider a field Aresiding inside the Gribov region. For a another field A′=
λA, with positive λ,
Zdxdy ω(x)∂µDµ[A]ω(y) = Zdxdy ω(x)∂2ω(y) + Zdxdy ω(x)[λA(y), ω(y)]
=Zdxdy ω(x)∂2ω(y) + λκ
(3.13)
since[λA, α] = λ[A, α], they both have the same eigenvectors but with eigenvalues
scaled by λ
For a sufficiently large λ, the second term in the above expression becomes larger
than the first term, and hence the overall sign of the inner product becomes negative,
which means that the FP operator is not positive-definite.
What this means is that for an Ain the Gribov region and large enough λ,A′=λA
live outside the Gribov region, which implies that the Gribov region is bounded on
all sides.
With these properties, one might be tempted to conclude that restricting the path in-
tegral to the Gribov region not only resolves the problems with the introduction of the
ghost fields but also removes Gribov copies from the integral. But there is one short-
coming to this argument. Since every minima of the Hilbert square norm on the gauge
orbit lies in the Gribov region, there still is a possibility that some orbits have multiple
local minima which will all enter the Gribov region, thus introducing copies in the region.
24
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
This problem can be resolved if one considers another region, contained inside the Gribov
region - the fundamental modular region, containing only those configurations that are
global minima of the Hilbert square norm. Since each gauge orbit has a unique global
minima [5], restriction to the fundamental modular region truly resolves the problem of
Gribov copies. But so far nobody has managed to construct a path integral with the
configuration space restricted to the fundamental modular region.
Nevertheless, the restriction of the configuration space to the Gribov region solves the
non-positive-definiteness problem of the Faddeev-Popov operator. Thus such a restriction
is not only possible but also necessary to define the path integral of the quantum Yang-
Mills theory.
3.4 Gribov’s No-Pole Condition
The inverse of the FP operator is nothing but the full propagator of the ghost fields. This
means that the positive definiteness condition on the Faddeev-Popov operator translates
to the condition that the ghost propagator should be positive and not have any non-trivial
poles.
To look at this condition more closely, we consider the ghost propagator in Fourier space
(without restriction to the Gribov region)
G(p2) = ZDA¯
Ω(−p)
1
−∂µDµ
Ω(p)det(∂µDµ) exp(−S[A]) (3.14)
Naive perturbation theory calculation of the above gives the result
G(p2) = 1
p2
1
1−11g2N
48π2ln Λ2
p2(3/22)(3/2−α/2) (3.15)
Where Λis the ultraviolet cutoff and αis the gauge parameter.
For large p2the above expression is both non-zero and positive real. But as we go to
lower p, problems start arising.
From the expression 3.14, poles in the propagator imply that we are approaching the
surface where the FP operator has zero modes. The trivial pole p2= 0 can be inter-
preted as approaching the Gribov horizon and is hence harmless. But the non-trivial pole
p2= Λ2exp −48π2
11g2Nimplies that we are approaching other surfaces with zero modes
of Faddeev-Popov operator which means we are outside the Gribov region. Another
25
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
noteworthy observation is that for values of p2<Λ2exp −48π2
11g2N, the propagator is
non-real which also indicates that one has left the Gribov region.
Thus the restriction of the path integral to the Gribov region should imply that the non-
trivial pole of the ghost propagator should be removed, and the propagator should be
real and positive. We can do so by looking at the explicit expression of G(p2), which we
calculate by considering the gauge fields as an external coupling. We do this by summing
the Feynman diagrams, i.e., we consider the following series
G(p2, A)kl =¯
ΩkΩl
p
+
Am
¯
ΩkΩl
q−p
p q
+
Am
¯
Ωk
An
Ωl
qq′
pp−qp−q−q′
and do term-by-term evaluation of it (here we consider terms only up to second order in g)
Since the external coupling should be integrated out, the gauge lines should be connected.
Thus the second term can not exist and in the third term the two gauge ends are connected
and q′=−q.
Thus the evaluation of the sum reduces to the evaluation of the following diagrams
1. ¯
ΩkΩl
p
This would be the free propagator for the ghost fields, which is simply
G(p2, A) (0th order) = δkl
p2(3.16)
2. Am
µ
Ωn
¯
Ωk
Ao
µ
¯
ΩpΩl
q
pp−qp
26
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
This diagram can be evaluated as
G(p2, A) (2nd order)
= (−i)2g21
VZdq
2π41
p2
δnp
(p−q)2
1
p2fkmn(p−q)µAm
µ(−q)fpolpνAo
µ(q)
=−g21
Vfkmnfnol 1
p4Zdq
2π41
(p−q)2(p−q)µpνAm
µ(−q)Ao
ν(q)
(3.17)
where the volume Vhas been introduced to maintain the dimensionality of the
Feynman graph, since Ahas been introduced as external coupling.
To get the correct normalization, we trace over the color indices k&land divide by
N2−1.
From this, we obtain the propagator (up to the second order in g) as
G(p2, A) = 1
p2−g21
V−Nδmo
N2−1
1
p4Zdq
2π41
(p−q)2(p−q)µpνAm
µ(−q)Ao
ν(q)
=1
p2+g21
V
N
N2−1
1
p4Zdq
2π41
(p−q)2(p−q)µpνAm
µ(−q)Am
ν(q)
=1
p2 1 + g21
V
1
p2
N
N2−1Zdq
2π41
(p−q)2(p−q)µpνAm
µ(−q)Am
ν(q)!
(3.18)
Calling σ(p2, A) = g21
V
1
p2
N
N2−1Zdq
2π41
(p−q)2(p−q)µpνAm
µ(−q)Am
ν(q), the propa-
gator, up to second order in g, becomes
G(p2, A) = 1
p2(1 + σ(p2, A)) (3.19)
Because we are summing over the entire series of Feynman diagrams, the complete prop-
agator can be appropriately approximated as
G(p2, A)≈1
p2
1
1−σ(p2, A)(3.20)
From this, the no-pole condition simply reads as σ(p2, A)<1.
This condition can be simplified further by considering that since we are working in the
Landau gauge, the transversality condition should hold, i.e. qµAµ(q)=0.
With this gauge, we can write
Aµ(−q)Aν(q) = C(A)δµν −qµqν
q2(3.21)
27
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
To find the form of C(A), we multiply the above equation by δµν and see that Aµ(−q)Aµ(q) =
C(A)(4 −1). Therefore
Aµ(−q)Aν(q) = 1
3Aρ(−q)Aρ(q)δµν −qµqν
q2(3.22)
With this, and using the transversality condition,
σ(p2, A) = g21
V
N
N2−1
1
3
pµpν
p2Zdq
2π41
(p−q)2Aρ(−q)Aρ(q)δµν −qµqν
q2(3.23)
Using Zd4q f(q2)qµqν
q2=1
4δµν Zd4qf (q2),σcan be written as
σ(p2, A) = g21
V
N
N2−1
1
4
pµpν
p2δµν Zdq
2π41
(p−q)2Aρ(−q)Aρ(q)(3.24)
This is a continuously decreasing function of p2. Therefore the maximum value of σis at
p2= 0. Thus, the no-pole condition can be restated as σ(0, A)<1, i.e.
σ(0, A) = lim
p2→0g21
V
N
N2−1
1
4Zdq
2π41
(p−q)2Aρ(−q)Aρ(q)
=g21
V
N
N2−1
1
4Zdq
2π41
q2Aρ(−q)Aρ(q)<1
(3.25)
Since this expression is independent of any momenta, the no-pole condition can be in-
serted into the path integral to restrict it to the Gribov region, in the form of Heaviside
step function θ(1 −σ(A)).
3.5 Restricting the Path Integral Using No-Pole Con-
dition
It is pretty straightforward to implement the no-pole step function into the path integral,
by using the integral representation of the theta function in the form
θ(1 −σ(A)) = Z∞+iϵ
−∞+iϵ
dβ 1
2πiβ exp(β(1 −σ(A))) (3.26)
Therefore the modified partition function would be
Z=ZDAZdβ eβ
2πiβ exp (−S[A]−βσ[A]−J·A) det(∂µDµ)(3.27)
Since this additional term is quadratic in the gauge fields, we expect it to modify the
28
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
form of the gauge propagator. To calculate the free gauge propagator, we consider only
the gauge part of the path integral, quadratic in fields.
It is convenient to work in Fourier space where the action is
Squad[A] = Zd4q
(2π)4Aν(−q)( δµν q2+qµqν1
α−1)Aµ(q)
and thus the partition function becomes
ZA[J] = Zdβ eβ
2πiβ ZDAexp(−Zdq Aν(−q)( δµν q2+βg21
V
N
N2−1
1
4
1
q2
+1
α−1qµqν)Aµ(q)−Jµ(−q)Aµ(q))
(3.28)
Calling Mkl
µν =δkl δµν q2+βg21
V
N
N2−1
1
4
1
q2+1
α−1qµqν, the gluon propaga-
tor can be easily calculated as
⟨Ak
µ(−p)Al
ν(p)⟩=Zdβ
2πiβ eβ1
pdet(M)M−1(3.29)
Using det M= exp Tr ln M, we can write
⟨Ak
µ(−p)Al
ν(p)⟩=Zdβ exp β−ln β−1
2Tr ln MM−1(3.30)
where all the constants have been absorbed into the normalization.
To perform the integration over β, we use the method of steepest descent. Since the
exponential is a monotonically increasing function, the most dominant term in the integral
would be the one for which the exponential would be the maximum (assuming that M−1)
doesn’t oscillate much. The maximum value of βcan be found as the solution to the gap
equation
d
dβ β−ln β−1
2Tr ln M= 0 (3.31)
The integration over βcan now be approximated as a simple substitution β→β0in the
integrand. Thus, the modified propagator becomes
⟨Ak
µ(−p)Al
ν(p)⟩=M−1β→β0
(3.32)
Where the exponential is dropped since it gets canceled by the normalization.
Calling β0g21
V
N
N2−1
1
4≡κ4, and taking the limit α→0, the propagator in Landau
29
CHAPTER 3. THE HAUNTING OF THE GRIBOV COPIES
gauge can be explicitly written as
⟨Ak
µ(−p)Al
ν(p)⟩=δkl p2
p4+κ4δµν −pµpν
p2 (3.33)
This is a propagator of a massive field, with a mass gap defined by κThus the restriction
to the Gribov region introduces a dynamic mass gap in the Yang-Mills theory.
This propagator has complex poles, p2=±iκ2, which leads to the violation of the reflec-
tion positivity axiom of the Osterwalder-Schrader axioms for quantum field theory. The
violation of the positivity of the Schwinger function implies that the gluon states can-
not enter the scattering matrix as asymptotic states. Such a theory is said to be confined.
The physical intuition behind the occurrence of the mass gap is pretty simple. Just like in
an infinite space, the momentum spectrum is continuous, but once one introduces periodic
boundary conditions on the said space, there arises a discretization of the momentum
spectrum; the confinement of the functional integral to the Gribov region introduces a
mass gap into the Yang-Mills theory.
Thus, by introducing the no-pole condition, we have not only introduced a mass gap into
the Yang-Mills theory but also have made a way for confinement.
30
II
PART
STOCHASTIC QUANTIZATION
31
CHAPTER 4
PRELUDE - PROBABILITY THEORY AND STOCHASTIC
PROCESSES
In this chapter, we discuss the fundamentals of probability theory and stochastic pro-
cesses. Stochastic calculus deals with the study of stochastic processes involving random-
ness. It provides a rigorous mathematical framework for working with random variables
and stochastic differential equations. We further discuss in-depth one of the most fun-
damental stochastic processes - the Brownian motion which models the erratic, unpre-
dictable movement of a particle subjected to random perturbations.
4.1 Stochastic Processes
Consider a set Ωwhose elements are to be interpreted as the possible outcomes of a
probabilistic experiment. On this set, we define a σ-algebra S, which is a set whose
elements are some subsets of Ω, such that the following properties are satisfied
•Ω∈ S
•∀E∈ S,∃EC≡Ω\E∈ S. That is, for every element Eof S,ECwhich is defined
as the complement of Ein Ωis also in the set S
•∀En∈ S, n = 1,2,..., ∪nEn∈ S. That is, the finite union of elements in Salso
belongs to S
With such an algebra defined, we call the tuple (Ω,S)a measurable space.
A measure is a function defined on a measurable space. It is a function µ:S → [0,∞),
that satisfies
•If E1⊂E2, then µ(E1)< µ(E2).
•If En∈ S, n = 1,2, . . . and Ei∩Ej=∅(i=j), then µ(∪nEn) = Pnµ(En)
32
CHAPTER 4. PRELUDE - PROBABILITY THEORY AND STOCHASTIC
PROCESSES
If we further impose a condition µ(Ω) = 1, then the measure so defined is called the
probability measure and is usually denoted by P. The tuple (Ω,S, P )is then called the
probability space.
Suppose (Ω,S)and (Ω′,S′)are two measurable spaces. The function X: Ω →Ω′is
called a random variable, if for every A∈ S′,X−1(A)∈ S. If the first measurable space
is equipped with a probability measure P, then the random variable induces a probabil-
ity measure on the second space given by P ≡ P◦X−1, called the distribution (law) of X.
In general, the second measurable space is taken to be (R,B), where Bis the Borel
σ-algebra on R. For this case, the distribution of Xis completely determined by the
distribution function defined as
FX(x) = P(X≤x)(4.1)
where X≤xis defined as the set {X≤x}={ω∈Ω|X(ω)≤x}.
The distribution function is continuously non-decreasing and gives the measure associated
with (x, x +dx)∈Bas d(FX(x))
A special case is when the distribution function can be written in the form
F(x) = Zx
−∞
p(x)dx (4.2)
In this case, the function p(x)is called the probability density function of X, and the
measure associated with dx is d(FX(x) = p(x)dx.
With the knowledge of the distribution function, one can define the expectation value of
any function h(X)as
Eh(X) = Zh(x)d(FX(x)) (4.3)
With these definitions, we now define a stochastic process. A stochastic process is an
indexed set of random variables Xt, t ∈T, where Tis called the index set and is usually
taken (in the continuous case) to be T= [0,∞). In a physical setting, the index is time
and a stochastic process can be interpreted as, for each t∈T,Xtpicks an event E∈ S
with probability P(E), and returns Xt(E).
To proceed, let us consider first a simple physical process that can be modeled by a
stochastic process, and extend the model to quantize a given system too.
33
CHAPTER 4. PRELUDE - PROBABILITY THEORY AND STOCHASTIC
PROCESSES
4.2 Brownian Motion
The Brownian motion is one of the oldest and simplest stochastic processes known to
physicists. Physically it is a diffusion process of a particle in a medium under a static
drift force K(x)while also accounting for the random collisions of the particle with the
medium.
The forces on this diffusing particle are the external drift force, K(x), a friction force
opposing the motion due to the mismatch in the number of collisions in the direction of
motion and the opposite direction, −αdx
dt , and a gaussian noise force due to the random
collisions the particle will have with the surrounding media, αη(t).
The η(t)s are random variables with Gaussian distributions, satisfying
⟨η(t)⟩η= 0
⟨η(t′)η(t′′)⟩η= 2Dδ(t′−t′′)(4.4)
where ⟨·⟩ denores the expectation value with p(η) = 1
Nexp −1
4DZdt η(t)2.
The Newton’s equation for this particle would be
md2x
dt2=−αdx
dt +K(x) + αη(t)(4.5)
We consider the situation where the mass of the particle is negligible compared to the
drift and friction forces. In this case, we can neglect the inertial term to give
dx
dt =1
αK(x) + η(t)(4.6)
This is the Langevin equation.
Calling 1
αas γ, for a given initial condition x(0) = x0, the above differential equation has
the formal solution
x[η](t) = x0+Zt
0
(γK (x) + η(t′)) dt′(4.7)
This is a functional of η, and since ηis a stochastic process, x[η]is also a stochastic process
with x[η](t)s being random variables. We can define the probability distribution for the
position xat time tas
p(x, t) = ⟨δ(x−x[η](t)) ⟩η(4.8)
This probability distribution follows the evolution equation
∂p(x, t)
∂t =d
dx Dd
dx −γK (x)p(x, t)(4.9)
34
CHAPTER 4. PRELUDE - PROBABILITY THEORY AND STOCHASTIC
PROCESSES
called the Focker-Planck equation.
4.2.1 Focker-Planck equation from the Langevin equation
To see how the Focker-Planck equation follows from the Langevin equation, start by
discretizing the Langevin equation
x(t+dt)−x(t) = γK (x)dt +1
√dt ˜η(t)dt (4.10)
where ⟨˜η⟩= 0 and ⟨˜ηt′˜ηt′′ ⟩= 2Dδt′t′′
(The 1
√dt is needed since δ(t′−t′′)has dimension t−1, but δt′t′′ is dimensionless making
1
√dt ˜ηthe right gaussian random variable for the discrete version).
Now consider a generic functional of x(t),f(x(t)). To compute df
dt , we obtain (upto first
order in dt)
f(x(t+dt)) −f(x(t)) = f(x(t) + γK (x(t))dt +√dt˜η)−f(x(t))
=γ∂f
∂x Kdt +∂f
∂x ˜η√dt +1
2
∂2f
∂x2˜η2dt
(4.11)
Dividing by dt and taking the average over ˜η, we find that the second term becomes zero
due to the zero mean of ˜ηand the equation becomes
df
dt =∂f
∂x γK +∂2f
∂x2D(4.12)
Using ⟨f(x)⟩=Zf(x)p(x, t)dx, where p(x, t)is the probability distribution function in
x, we get the left hand side as
df
dt =∂
∂t Zf(x)p(x, t)dx =Zf(x)∂p(x, t)
∂t dx (4.13)
and the right-hand side as
∂f
∂x γK +∂2f
∂x2D=γK(x)p(x, t)f(x)
∞
−∞ −Zγf (x)∂K(x)p(x, t)
∂x dx
+DP (x, t)∂f
∂x
∞
−∞ −ZD∂f
∂x
∂p(x, t)
∂x dx
(4.14)
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CHAPTER 4. PRELUDE - PROBABILITY THEORY AND STOCHASTIC
PROCESSES
Using the fact that p(x, t)goes to zero at both infinities, we get
∂f
∂x γK +∂2f
∂x2D=−Zγf (x)∂K(x)p(x, t)
∂x dx +ZDf(x)∂2p(x, t)
∂x2(4.15)
Since the above relation should hold for any function f(x), we see that it must hold that
Zf(x)∂p(x, t)
∂t +γ∂K(x)p(x, t)
∂x −D∂2p(x, t)
∂x2dx = 0
=⇒∂p(x, t)
∂t =−γ∂K(x)p(x, t)
∂x +D∂2p(x, t)
∂x2
=⇒∂p(x, t)
∂t =∂
∂x D∂
∂x −γK(x)p(x, t)
(4.16)
which is the Focker-Planck equation.
4.2.2 The thermal equilibrium
One important case of Brownian motion is when the drift force is conservative. That is,
when K(x)can be written in terms of the gradient of a potential
K(x) = −∂V (x)
∂x (4.17)
Let us, for the moment, consider γ= 1 and D= 1.
The Focker-Planck equation then becomes
∂p
∂t =∂
∂x ∂
∂x +∂V
∂x p(4.18)
Consider the distribution ψ(x, t) = p(x, t) exp V(x)
2
=⇒p(x, t) = ψ(x, t) exp −V(x)
2.
The distribution ψtherefore follows the equation
exp −V
2∂ψ
∂t =∂
∂x exp −V
2∂ψ
∂x + exp −V
21
2ψ∂V
∂x
=−exp −V
21
2
∂V
∂x ∂ψ
∂x +1
2ψ∂V
∂x
+ exp −V
2∂2ψ
∂x2+1
2
∂ψ
∂x
∂V
∂x +1
2ψ∂2V
∂x2
=⇒∂ψ
∂t = ∂2
∂x2−1
4∂V
∂x 2
+1
2
∂2V
∂x2!ψ
(4.19)
36
CHAPTER 4. PRELUDE - PROBABILITY THEORY AND STOCHASTIC
PROCESSES
Calling
H=−1
2 ∂2
∂x2−1
4∂V
∂x 2
+1
2
∂2V
∂x2!(4.20)
we get ∂ψ
∂t =−2H ψ (4.21)
which is a Schrodinger-type equation.
Since we can write
H=1
2−∂
∂x +1
2
∂V
∂x ∂
∂x +1
2
∂V
∂x (4.22)
Since ∂
∂x
T
=−∂
∂x , and ∂V
∂x is proportional to the identity operator, we see that the
operator His self-adjoint, and is in the form H=MTM.
Therefore the operator Hhas a non-negative real spectrum, with the eigenvectors forming
a complete orthonormal basis. Therefore, we can write the solution to the equation 4.21
as
ψ(x, t) =
∞
X
n=0
anψn(x) exp (−Ent) = ψ0(x) +
∞
X
n=1
anψn(x) exp(−Ent)(4.23)
where ψn(x)form an orthonormal basis with Hψn(x) = Enψn(x)and ψ0is the eigenfunc-
tion with E0= 0.
We immediately see that ψ0follows (Since ATAx = 0 =⇒Ax = 0),
Hψ0(x) = 0 =⇒∂
∂x +1
2
∂V
∂x ψ0= 0 =⇒ψ0∝exp −V
2(4.24)
Therefore the original probability distrubution phas the solution
p(x, t) = ψexp −V
2=a0exp (−V) +
∞
X
n=1
anψn(x) exp −V
2exp(−Ent)(4.25)
At thermal equilibrium, i.e. in the limit t→ ∞, all terms with n > 0become zero, while
only the n= 0 term survives, giving the equilibrium probability distribution as
peq(x) = lim
t→∞ p(x, t) = a0exp(−V(x)) (4.26)
where the constant a0is fixed by normalisation, i.e. a−1
0=Z∞
−∞
exp(−V(x))dx
The similarity of the structure of this probability distribution to the path integral of
quantum field theory is what allows one to define a stochastic quantization prescription
37
CHAPTER 4. PRELUDE - PROBABILITY THEORY AND STOCHASTIC
PROCESSES
that allows one to view quantum mechanics as a Brownian motion but in configuration
space.
38
CHAPTER 5
QUANTUM FIELD THEORY AS A BROWNIAN MOTION
In this chapter, we show how a quantum field theory can be written as an equilibrium
limit of a Brownian motion in the configuration space. Developed in the early 1980s by
Gert Parisi and Yong-Shi Wu, unlike traditional quantization methods, which rely on the
path integral formulation and the introduction of ghost fields, stochastic quantization
reformulates the quantum field theory as a stochastic process described by a Langevin
equation. We explicitly quantize the scalar field theory with stochastic quantization
prescription, and also discuss the stochastic quantization for gauge fields and the problems
associated with this formalism.
5.1 Stochastic Quantization
There is a striking similarity in the equilibrium distribution of the position of a particle
undergoing Brownian motion and the Euclidean Quantum Field Theory measure, the
correspondence being
Z−1
0exp −S(ϕ)
ℏ⇐⇒ a0exp −V(x)
Dwith D⇐⇒ ℏ(5.1)
This motivates a generalization of the Brownian motion to an infinite dimensional con-
figuration space, expecting the equilibrium limit of the probability distribution to be the
same as the Euclidean Quantum Field Theory measure.
Note that the probability distribution obtained in the case of Brownian motion was at
time t→ ∞, but in Euclidean Quantum Field Theory the distribution is expected for all
tand not only at t→ ∞. Therefore we see that there is a lack of a physical parameter
that can be sent to infinity to obtain the distribution. To overcome this, we introduce a
fictitious time parameter, in which a Brownian motion is supposed, and the equilibrium
limit is taken to obtain the Euclidean Quantum Field Theory measure.
Therefore, formally, the stochastic quantization prescription is to take a classical field
39
CHAPTER 5. QUANTUM FIELD THEORY AS A BROWNIAN MOTION
theory with fields ϕ(x)(the xhere stands now for both space and physical time dimen-
sions) and action S[ϕ]and extend the definition of the fields as ϕ(x)→ϕη(x, t), where
tis now the fictitious time parameter. With the action as a drift potential, construct a
Langevin equation for the fields as
∂ϕη(x, t)
∂t =−δS[ϕη]
δϕη(x, t)+η(x, t)(5.2)
with initial condition ϕη(x, 0) = C(x)where C(x)is a function independent of t.
The correlation functions of Euclidean Quantum Field Theory become averages on η,
⟨ϕη(x1, t1). . . ϕη(xn, tn)⟩η=ZDηexp −1
4Zη2(x, t)dxdtϕη(x1, t1). . . ϕη(xn, tn)
(5.3)
which, in the limit t→ ∞, gives the desired Euclidean Quantum Field Theory correlation
functions as
lim
t→∞ ⟨ϕη(x1, t1). . . ϕη(xn, tn)⟩η=⟨ϕη(x1). . . ϕη(xn)⟩(5.4)
For this, we can also write the corresponding Focker-Planck equation as
∂P (ϕ, t)
∂t =Zdx δ
δϕ(x, t)δ
δϕ(x, t)+δS[ϕ]
δϕ(x, t)P(ϕ, t)(5.5)
with initial condition P(ϕ, 0) = Qyδ(ϕ(y)).
The equilibrium limit t→ ∞ of P(ϕ, t)is expected to give the required Euclidean Quan-
tum Field Theory distribution,
w.lim
t→∞ P(ϕ, t) = e−S
RDϕe−S(5.6)
where the w.lim denotes a weak limit which means that the limit is taken only when the
probability density is applied to a string of fields.
In the following section, we show explicitly how the scalar field theory can be quantized
via stochastic quantization prescription.
5.2 Quantizing the Scalar Fields
For a scalar field theory governed by the action S=Zd4x1
2(∂ϕ(x))2−1
2(mϕ(x))2
the Langevin equation is ∂ϕη
∂t = (∂2−m2)ϕη+η(x, t)(5.7)
40
CHAPTER 5. QUANTUM FIELD THEORY AS A BROWNIAN MOTION
Consider the Fourier Transformed Langevin equation in kand tgiven as
∂
∂t ϕη(k, t)+(k2+m2)ϕη(k, t) = η(k, t)(5.8)
One can obtain ϕηby using Duhamel’s formula,
ϕη(k, t) = Zt
0
G(k, t′)η(k, t′)dt′(5.9)
where G(k, t −t′)is the solution to the Fourier transformed heat equation
∂
∂t G(k, t)+(k2+m2)G(k, t) = 0 (5.10)
which is given by
G(k, t) = e−(k2+m2)(t−t′)(5.11)
Therefore, using this in eq 5.9 and taking inverse Fourier transform, we get
ϕη(x, t) = 1
(2π)dZt
0Zeikx e−(k2+m2)(t−t′)η(k, t′)dk dt′(5.12)
which from the convolution property of the Fourier transform becomes
ϕη(x, t) = Z Z t
01
(2π)dZeik(x−y)−(k2+m2)(t−t′)dk η(y, t)dt′dy (5.13)
where the function
G(x−y, t −t′) = 1
(2π)dZeik(x−y)−(k2+m2)(t−t′)dk (5.14)
can be seen as the Green function for the free Langevin equation.
The propagator can be calculated as a two-point correlator as
D(x, y;t, t′) = ⟨ϕη(x, t)ϕη(y, t′)⟩η(5.15)
which can be evaluated as
D=Zdz Zdz′Zt
0
dτ Zt′
0
dτ′G(x−z, t −τ)G(y−z′, t′−τ′)⟨η(z, τ )η(z′, τ′)⟩η
= 2 Zdz Zdz′Zt
0
dτ Zt′
0
dτ′G(x−z, t −τ)G(y−z′, t′−τ′)δ(z−z′)δ(τ−τ′)
(5.16)
The integration over the time parameter should be done first on the one whose integration
41
CHAPTER 5. QUANTUM FIELD THEORY AS A BROWNIAN MOTION
range is larger since doing it the other way will leave parts of the range where the delta
function’s argument is not zero. Therefore, the above integration becomes
D= 2 Zdz Zmin(t,t′)
0
dτ G(x−z, t −τ)G(y−z, t′−τ)(5.17)
Let us, for a moment, consider the probability density obtained from this theory. Writing
P0(ϕ, t)as
P0[ϕ, t] = ZDηZDkexp −1
4Zη2(x′, t′)dx′dt′exp iZdx k(x) (ϕ(x)−ϕη(x, t))
(5.18)
which we can evaluate now, using the form of ϕηas
P0[ϕ, t] = ZDηZDkexp −1
4Zη2(x, t)dxdt
+iZdx k(x)ϕ(x)−Z Z t
0
G(x−y, t −t′)η(y, t)dt′dy
(5.19)
Performing the Gaussian integral over η, we obtain
P0[ϕ, t] =N−1ZDk
exp −Zdx1dx2k(x1)Zdy1dy2Zt
0
dτ1dτ2G(x1−y1, t −τ1)
G(x2−y2, t −τ2)δ(y1−y2)δ(τ1−τ2)k(x2)
·exp −iZdxk(x)ϕ(x)
(5.20)
which can be simplified to obtain
P0[ϕ, t] = N−1ZDkexp −Zdx1dx2k(x1)1
2D(x1, x2;t, t)k(x2)−iZdx k(x)ϕ(x)
(5.21)
Now performing Gaussian integration over k, we obtain the probability density to be the
distribution
P0[ϕ, t] = N−1exp −1
2Zdx1dx2ϕ(x1)D−1(x1, x2;t, t)ϕ(x2)(5.22)
42
CHAPTER 5. QUANTUM FIELD THEORY AS A BROWNIAN MOTION
The propagator Dcan be explicitly calculated as
D(x, y;t, t)=2Zdz Zt
0
dτ Zdk1dk2eik1(x−z)−(k2
1+m2)(t−τ)eik2(y−z)−(k2
2+m2)(t−τ)
= 2 Zdk1dk2Zt
0
dτ δ(k1+k2) exp ik1x+ik2y−(k2
1+k2
2+ 2m2)(t−τ)
= 2 Zdk Zτ
0
exp ik(x−y) + 2(k2+m2)(τ−t)
=Zdk eik(x−y)1−e−2t(k2+m2)
k2+m2
(5.23)
In the thermal equilbrium limit, the propagator reduces to
D(x, y) = Zdk eik(x−y)1
k2+m2=−1
∂2−m2(5.24)
which is the same as the propagator obtained from other quantization methods. From
this, we see that the equation 5.22 in the thermal equilibrium limit gives the same mea-
sure, P∝exp(−S), as in the case of other quantization methods.
5.3 Stochastic Quantization of Gauge Fields
Let us consider the simplest case of Abelian gauge fields, for which the Langevin equation
(which we will call Parisi - Wu equations henceforth) reads
∂Aµ(x, t)
∂t = (δµν ∂2−∂µ∂ν)Aν(x, t) + ηµ(x, t)(5.25)
The presence of the ∂
∂t term for the fields Aexplicitly breaks the gauge invariance of the
theory, meaning that one doesn’t need to worry about fixing the gauge at all. In princi-
ple, one expects this explicit breaking of the gauge freedom to let us quantize the system
without having to worry about fixing the gauge or the problems like Gribov Ambiguity
that come with it, but this is not the case as we can see below.
Consider the Langevin equation for the gauge fields in the Fourier space,
∂Aµ(k, t)
∂t = (δµν k2−kµkν)Aν(k, t) + ηµ(k, t)(5.26)
43
CHAPTER 5. QUANTUM FIELD THEORY AS A BROWNIAN MOTION
We can split the above equation into transverse and longitudinal parts, with
AT
µ=δµν −kµkν
k2Aν=OT
µν Aν
AL
µ=kµkν
k2Aν=OL
µν Aν
ηT
µ=δµν −kµkν
k2ην=OT
µν ην
ηL
µ=kµkν
k2ην=OL
µν ην
(5.27)
This splits the Parisi-Wu equation as
∂AT
∂t =−k2AT+ηT
∂AL
∂t =ηL
(5.28)
This equation has solutions
AT= exp(−k2t)AT
0+ exp(−k2t)Zt
0
exp(k2τ)η(τ)dτ
AL=AL
0+Zt
0
exp(k2(t−τ))η(τ)dτ
(5.29)
From this, we observe that the initial distribution in the transverse modes get dissipated
at infinite time due to the presence of the damping term, while in the case of longitu-
dinal mode, there is no damping term and hence the initial distribution persists even in
equilibrium, i.e. there exists no stationary distribution in the transverse modes. This is
directly the consequence of the gauge invariance.
To see how this affects calculations, let us look at the propagator for the gauge fields.
Dµν (k, t|k′, t′) = ⟨Aµ(k, t), Aν(k′, t′)⟩
=δ4(k+k′)1
k2OT
µν (exp(k2(t−t′)) −exp(−k2(t+t′)))
+ 2t<δ4(k+k′)kµkν
k2+Aµ(k, 0)LAν(k′,0)L
(5.30)
The translational invariance of the propagator requires that the propagator be propor-
tional to δ4(k+k′), which is broken by the last term in the above-derived propagator.
One way to circumvent the problem would be to select the initial configuration of the
longitudinal mode to be Aµ(k, 0) = 0.
But such a distribution is exceptional and therefore we consider a more general distribu-
44
CHAPTER 5. QUANTUM FIELD THEORY AS A BROWNIAN MOTION
tion that is symmetric around k= 0 as
AL
µ=kµ
k2ϕ(k)(5.31)
and for the propagator, we take another average over the distributions ϕ(k). Knowing
⟨ϕ(k), ϕ(k′)⟩ϕ=−αδ4(k+k′)(5.32)
where αis the width of ϕ(k), we get the limit of the propagator as
lim
t=t′→∞ Dµν (k, t|k′, t′) = δ4(k+k′)1
k2δµν −(1 −α)kµkν
k2+ 2tkµkν
k2(5.33)
which is nothing but the usual gauge fixed propagator, with a diverging term proportional
to t.
Therefore one can see that the stochastic quantization prescription allows one to quantize
gauge fields and get the propagator of the theory without having to resort to gauge fixing
explicitly, but as seen from the above calculations, the gauge freedom is reflected as a
choice in the initial distribution for the longitudinal mode of the gauge fields, the width
of the distribution αplaying the role of the gauge fixing parameter.
Another subtlety worth mentioning is that for gauge-invariant quantities, the term pro-
portional to tin the equilibrium limit goes to zero. But the gauge-variant quantities
diverge in the limit t→ ∞. This is mainly due to the absence of drift force in the
transverse part of Langevin’s equation, meaning that the stationary distribution for the
transverse part doesn’t exist, and a particle undergoing Brownian motion in the config-
uration space will drift forever in this direction. Speaking in terms of forces, the action
provides a force that constrains a particle to a gauge orbit in the configuration space, but
the particle is free to move along the gauge orbit indefinitely. It is this free motion along
the gauge orbits that causes the gauge-variant quantities to diverge in the equilibrium
limit.
Therefore, for gauge-invariant quantities, the stochastic quantization prescription gives
the same results as the regular quantization without having to resort to explicit gauge
fixing procedures, but in the case of gauge-variant observables, the theory again leads to
divergences.
45
CHAPTER 6
GAUGE FIXING WITHOUT FIXING THE GAUGE
In this chapter, we discuss how the addition of a non-conservative drift force along the
gauge orbit can provide a way to circumvent the problems associated with the stochastic
quantization of gauge fields. We also discuss the properties of such a force and also
discuss how such a formalism can be used to obtain the same effects as the restriction
of the path integral to the Gribov region. By avoiding the need for explicit gauge fixing
and the introduction of ghost fields, stochastic quantization therefore offers a promising
alternative for studying the non-perturbative aspects of gauge theories.
6.1 Gauge Fixing as a Non-Conservative drift force
As we saw from the previous sections, the divergences of gauge variant quantities at in-
finite time arise due to the free drift along the gauge orbits, where there are no damping
forces to provide a stationary limit. Therefore, a way to circumvent this problem would
be to introduce a drift force along the gauge orbits. In the proceeding section, we will
see how such a force can be introduced and how the introduction of such a force leads
naturally to the random walker being attracted to the Gribov region.
The presence of zeros of the Faddeev-Popov operator makes the friction force singular at
the Gribov horizon. It is well known that if a particle undergoing Brownian diffusion in
the presence of friction force encounters a boundary where the friction force is singular
and repulsive enough, it will get reflected. Therefore, in this case, the introduced force
being singular at the Gribov horizon implies that the Brownian motion in configuration
space never leaves the Gribov region at all, and therefore, one can expect the stochastic
quantization to inherently describe the same system as obtained by the regular quanti-
zation done by strictly restricting the path integral to the Gribov region.
Consider a one-dimensional Brownian motion, with the probability density following the
46
CHAPTER 6. GAUGE FIXING WITHOUT FIXING THE GAUGE
Gribov Horizon
Gribov Region
Figure 6.1: The path of a particle undergoing Brownian motion in the configuration space
in the presence of the singular friction force that is repulsive enough.
equation
˙
P(x, t) = ∂(∂−f)P(x, t) = −L∗P(x, t)(6.1)
where fis the static drift force (all the constants have been taken to be 1).
The above equation has the solution
P(x, t) = e−L∗tP(x, 0) ≡e−L∗tP0(x)(6.2)
Any functional that is dependent on x,Φ(x)has an expectation value given by
⟨Φ⟩t=Zdx Φ(x)P(x, t) = Zdx Φ(x)[e−L∗tP0(x)]
=Zdx Φ(x)(1 −L∗)P0(x)(upto first order)
=Zdx Φ(x)(P0+∂2P0−∂(fP0))
Performing integration by parts, and using P0(±∞)=0, we get
⟨Φ⟩t=Zdx (1 −∂2−f∂)Φ(x)P0(x)≡Zdx [e−LtΦ(x)] P0(x)
(6.3)
Where L=−(∂+f)∂.
Since as t→ ∞,⟨Φ⟩trelaxes to the ⟨Φ⟩independent of P0(x), so we can choose a trivial
P0(x) = δ(x−x′), and hence we arrive at
⟨Φ⟩= lim
t→∞ e−LtΦ(x)(6.4)
Thus given an initial data Φ(x, 0), we can calculate Φ(x, t)and the limit lim
t→∞ Φ(x, t) = ⟨Φ⟩.
47
CHAPTER 6. GAUGE FIXING WITHOUT FIXING THE GAUGE
In the case of gauge fields, consider Φ[A]which is a gauge invariant observable. Then the
equation of motion becomes
˙
Φ[A, t] = −LΦ[A, t](6.5)
with
L=−ZdDxδ
δA(x)−δS[A]
δA(x)δ
δA(x)(6.6)
with S[A]being the classical action functional.
An infinitesimal gauge transformation is given as δA =Dω =∂ω +fAω (spacetime and
color indices are understood).
Such a transformation changes the observable by
δΦ[A] = ZddxDω δ
δA(x)Φ[A](6.7)
If Φis a gauge invarient observable, then δΦ[A]should be zero.
Thus we can add a term ZddxDω δ
δA(x)to eq (6.6) without changing the value ⟨Φ⟩t
provided that Φis gauge invarient observable.
This is equivalent to adding a term Dω to the drift force f. It is convenient to choose
ωto be a Lorentz scalar, and also a local function linear in A. One such possibility is
ω∝∂·A. The proportionality constant is the dimensionless gauge parameter α−1, with
α= 1 being the Feynman Gauge and α→0being the Landau gauge.
Thus, we obtain the operator
Lα=−ZdDxδ
δA(x)−δS[A]
δA(x)−α−1D∂Aδ
δA(x)(6.8)
The term D∂A has two parts, ∂2Aand C= [A, ∂A]. The first term can be written as a
conservative force, derived from the potential
SG=1
2Zddx(∂A)2(6.9)
while the second part cannot be written in this form, and hence is a non-conservative
force. The SGprovides the necessary restoring force along the orbit, while the term C
is a pure circulation ∂.C = 0, ∂ ×C= 0 which in equilibrium produces a stationary
circulating current.
48
CHAPTER 6. GAUGE FIXING WITHOUT FIXING THE GAUGE
The equation followed by the probability density becomes
˙
Pα=−L∗
αPα(6.10)
and we see that the equilibrium distribution should follow L∗
αPα E = 0.
This condition is not followed by the traditional Faddeev-Popov distribution
Pα,FP[A] = Nexp −S−α−1SGDet ∂µDµ/∂2(6.11)
Therefore the the Pα E is not the same as the FP distribution.
6.2 Properties of the Gauge Fixing Force
Let us look at the properties the introduced friction force has.
1. The force is zero (equilibrium point) when ∂A = 0:
This is pretty clear from the form of FG=D∂A which vanishes when ∂A = 0.
This implies that the gauge fixing hyperplane ∂A = 0 is made up of equilibrium
configurations at which the friction force vanishes.
2. It is a restoring force elsewhere:
Consider the Hilbert square norm ||A||2= (A, A)where (., .)is the appropriate
inner product. The time evolution of this in the presence of the gauge fixing force
is given by (using dA
dt =FG=α−1D∂A)
d||A||2
dt = (A, dA
dt )+(dA
dt , A) = 2(A, α−1D∂A)(6.12)
This is equal to (doing integration by parts)
d||A||2
dt =−2α−1(DA, ∂A) = −2α−1(∂A, ∂A) = −2α−1||∂A||2≤0(6.13)
Therefore, the drift force reduces the Hilbert square norm and hence is a restoring
force. (provided α > 0). That is, a particle at any point on the configuration
space that is not in the hyperplane ∂A = 0 is attracted towards the gauge fixing
hyperplane and therefore drifts towards it along its corresponding gauge orbit.
3. The equilibrium inside the Gribov horizon is stable and outside is un-
stable:
Consider d||∂A||2
dt =−2α−1(∂A, −∂DA∂A)(6.14)
49
CHAPTER 6. GAUGE FIXING WITHOUT FIXING THE GAUGE
At a point close to the equilibrium locus, A=A0+ϵA1, where A0∋∂A0= 0, since
∂A0= 0the value of (upto lowest order in ϵ)
d||∂A||2
dt =−2α−1(∂A1,−∂DA0∂A1)ϵ2(6.15)
Thus, if A0is inside the Gribov Horizon, i.e. −∂DA0>0, then the ||∂A||2is
decreasing, and hence the point A0is stable.
If it is outside the Gribov horizon, ||∂A||2increases even for points close to A0and
hence the point A0is unstable.
Therefore the Gribov region Ωis an attractor of the gauge fixing force.
From the properties of the friction force, one can see what happens to a random walker
in the configuration space.
∂A
∂A = 0
Gribov horizon
Lines of friction force along the gauge orbits
Figure 6.2: The gauge fixing frictional force confining the particle motion to the Gribov
region
The motion of the random walker in the configuration space in the presence of the fric-
tional force along the gauge orbit can be classified into three categories based on its
location.
1. Outside the hyperplane ∂A=0:
The frictional force along the orbit the configuration belongs to will push the random
walker toward the Gribov region.
2. In the hyperplane and outside the Gribov region:
These points are equilibrium points of the force, and therefore they are unaffected
by the friction force. However, the thermal fluctuations due to the noise force will
push the walker away from the hyperplane, at which point it is pushed away from
the hyperplane along the orbit towards the Gribov region.
50
CHAPTER 6. GAUGE FIXING WITHOUT FIXING THE GAUGE
3. In the hyperplane and inside the Gribov region:
At these points, the thermal fluctuations cannot push the walker in the transverse
direction but can push the walker along the hyperplane in the direction that takes it
outside the Gribov region. In the case the force at the horizon is repulsive enough,
the walker is reflected and is therefore confined to the Gribov region. Even in the
case the force is not repulsive enough, the walker goes out of the Gribov region
to the region where the force stops being attractive. It again gets some transverse
component due to thermal fluctuations, and again gets pushed along gauge orbit
towards the Gribov region.
Therefore we see that a random walker starting at any position in the configuration
space drifts towards the Gribov region, and therefore there is no random drifting to
infinity like in the case with no friction force. This procedure eliminates the problems
with stochastic quantization prescription, while also allowing us to quantize the gauge
fields without having to explicitly perform the gauge fixing, while also circumventing the
Gribov problem.
51
III
PART
CONCLUDING REMARKS
52
The Gribov problem in quantum Yang-Mills theories arises due to the presence of mul-
tiple solutions to the gauge fixing functional. This leads to ill-defined propagators, and
incorrect calculations of observables due to overcounting of the gauge copies, along with
a complete breaking of the introduction of the Faddeev-Popov ghosts. Therefore, the
Gribov ambiguity presents a formidable problem to the non-perturbative studies of the
quantum Yang-Mills theories. One could argue that the gaps in our understanding of
the non-perturbative aspects of these theories are due to the non-trivial structure of the
configuration space which makes the analysis very difficult, and there is a need to study
analytically the configuration space of the theory to obtain deeper understandings into
the non-perturbative regimes.
Restricting the path integral to the fundamental modular region is a proposed resolution,
but so far no attempts have succeeded in imposing such a restriction. The semi-classical
restriction to the Gribov region does indeed circumvent the problem to an extent. but it
is an approximation that does not capture all the information about the theory. Some
authors have suggested that the restriction is expected to lead to the generation of the
mass gap in the theory and possibly confinement as seen in a rudimentary version of cal-
culations, but the exact mechanism behind this is largely unknown. Further, Zwanziger’s
attempts [6, 7] at resolving this problem have led to a non-local action formalism for the
quantum Yang-Mills, the discussion of which is beyond the scope of this thesis.
The stochastic quantization formalism, which quantizes the gauge fields as a Brownian
motion in the configuration space, provides an alternative quantization that avoids the
Gribov problem. This formalism quantizes the gauge theory by the use of Langevin’s
equation which has an operator ∂/∂t that breaks the gauge invariance of the theory and
allows us to quantize it. Further, one can circumvent the problem of the absence of
a stationary distribution in the transverse modes for the gauge fields by introducing a
restoring force along the Gribov orbit. The restoring force confines a random walker’s
probability density to the Gribov region, therefore reproducing the same effect as that
of restriction of the path integral to the Gribov region. Further, if the restoring force is
repulsive enough at the Gribov horizon, the random walker will never leave the Gribov
region at all, exactly reproducing the effects of the restricted path integral. However, the
exact behavior of the restoring force is unknown, and therefore the behavior of the walker
at the boundaries cannot be determined. Further Baulieu and Zwanziger have proved the
equivalence of the stochastic quantization prescription to the Faddeev-Popov prescription
[8] implying the equivalence of the calculations done via the stochastic methods. Research
is being conducted by researchers like Ajay Chandra [9, 10], Massimiliano Gubinelli [11],
Arthur Jaffe [12], and others towards a mathematically rigorous quantization of fields via
stochastic quantization.
53
In summary, the Gribov problem presents a significant obstacle to the non-perturbative
study of quantum Yang-Mills theories, and while various approaches have been proposed
to address this issue, a complete understanding remains elusive. The stochastic quanti-
zation formalism offers a promising alternative by circumventing the gauge-fixing proce-
dure and introducing a restoring force along the Gribov orbit, but further investigation
is needed to fully understand the behavior and the equilibrium limits of the system.
54
List of Figures
3.1 The ideal and real gauge fixing situation . . . . . . . . . . . . . . . . . . 21
6.1 The path of a particle undergoing Brownian motion in the configuration
space in the presence of the singular friction force that is repulsive enough. 47
6.2 The gauge fixing frictional force confining the particle motion to the Gribov
region ..................................... 50
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