Wilson lines in quantum field theory 2nd edition

Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (14.61 MB, 289 trang )

Igor O. Cherednikov, Tom Mertens, Frederik Van der Veken
Wilson Lines in Quantum Field Theory

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

De Gruyter Studies in
Mathematical Physics

|

Edited by
Michael Efroimsky, Bethesda, Maryland, USA
Leonard Gamberg, Reading, Pennsylvania, USA
Dmitry Gitman, São Paulo, Brazil
Alexander Lazarian, Madison, Wisconsin, USA
Boris Smirnov, Moscow, Russia

Volume 24

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

Igor O. Cherednikov, Tom Mertens,

Frederik Van der Veken

Wilson Lines in
Quantum Field
Theory
|

2nd edition

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

Physics and Astronomy Classification 2010
11.15-q, 11.15.Tk, 12.38.Aw, 02.10.Hh, 02.20Qs, 02.40.Hw, 03.65.Vf
Authors
Dr. Igor O. Cherednikov
Universiteit Antwerpen
Departement Fysica
Groenenborgerlaan 171
2020 Antwerpen
Belgium

Dr. Frederik Van der Veken
CERN
Beams Department
Esplanade des Particules 1

1211 Geneva
Switzerland

Dr. Tom Mertens
Abram-Joffe-Str. 6
12489 Berlin
Germany

ISBN 978-3-11-065092-1
e-ISBN (PDF) 978-3-11-065169-0
e-ISBN (EPUB) 978-3-11-065103-4
ISSN 2194-3532
Library of Congress Control Number: 2019951784
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available on the Internet at .
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Cover image: Science Photo Library / Parker, David
Typesetting: VTeX UAB, Lithuania
Printing and binding: CPI books GmbH, Leck
www.degruyter.com

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

Preface
Aristotle held that human intellectual activity or philosophical (in a broad sense)
knowledge can be seen as a threefold research program. This program contains metaphysics, the most fundamental branch, which tries to find the right way to deal with
“Being” as such; mathematics, an exact science studying calculable – at least, in
principle – abstract objects and formal relations between them; and, finally, physics,
the science working with changeable things and the causes of the changes. Therefore,
physics is the science of evolution – in the first place the evolution in time. To put it
in more ‘contemporary’ terms, at any energy scale there are things which a physicist
has to accept as being ‘given from above’ and then try to formulate a theory of how
do these things, whatever they are, change. Of course, by increasing the energy and,
therefore, by improving the resolution of experimental facility, one discovers that
those things emerge, in fact, as a result of evolution of other things, which should
now be considered as ‘given from above’.1
The very possibility that the evolution of material things, whatever they are, can
be studied quantitatively is highly non-trivial. First of all, to introduce changes of
something, one has to secure the existence of something that does not change. Indeed,
changes can be observed only with respect to something permanent. Kant proposed
that what is permanent in all changes of phenomena is substance. Although phenomena occur in time and time is the substratum, wherein co-existence or succession of
phenomena can take place, time as such cannot be perceived. Relations of time are
only possible on the background of the permanent. Given that changes ‘really’ take
place, one derives the necessity of the existence of a representation of time as the substratum and defines it as substance. Substance is, therefore, the permanent thing only
with respect to which all time relations of phenomena can be identified.
Kant gave then a proof that all changes occur according to the law of the connection between cause and effect, that is, the law of causality. Given that the requirement
of causality is fulfilled, at least locally, we are able to use the language of differential
equations to describe quantitatively the physical evolution of things. There is, however, a hierarchy of levels of causality. For example, Newton’s theory of gravitation
is causal only if we do not ask how the gravitational force gets transported from one
massive body to another. The concept of a field as an omnipresent mediator of all interactions allows us to step up to a higher level of causality. The field approach to
the description of the natural forces culminated in the creation in the 20th century of
the quantum field theoretic approach as an (almost) universal framework to study the
physical phenomena at the level of the most elementary constituents of matter.

1 It is worth noticing that this scheme is one of the most consistent ways to introduce the concept of
the renormalization group, which is crucial in a quantum field theoretical approach to describe the
three fundamental interactions.
/>
www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

VI | Preface
To be more precise, the quantitative picture of the three fundamental interactions is provided by the Standard Model, the quantum field theory of the strong, weak
and electromagnetic forces. The aesthetic attractivity and unprecedented predictive
power of this theory is due to the most successful, and nowadays commonly accepted,
way to introduce the interactions by adopting the principle of local (gauge) symmetry.
This principle allows us to make use of the local field functions, which depend on
the choice of the specific gauge and, as such, do not represent any observables, to
construct a mathematically consistent and phenomenologically useful theory. In any
gauge field theory we need, therefore, gauge-invariant objects, which are supposed
to be the fundamental ingredients of the Lagrangian of the theory, and which can be
consistently related, at least, in principle, to physical observables.
The most straightforward implementation of the idea of a scalar gauge invariant
object is provided by the traced product of field strength tensors
Tr [Fμν (x)F μν (x)] ,

(1)

Fμν (x) = 𝜕μ Aν (x) − 𝜕ν Aμ (x) ± ig[Aμ (x), Aν (x)],

(2)

where

Aμ (x) being the local gauge potentials belonging to the adjoint representation of the
N-parametric group of local transformations U(x), and g a coupling constant. Field
strength tensors are also local quantities, which change covariantly under the gauge
transformations:
Fμν (x) → U(x)Fμν (x)U † (x).

(3)

Interesting non-local realizations of gauge-invariant objects emerge from Wilson
lines defined as path-ordered (𝒫 ) exponentials2 of contour (path, loop, line) integrals
of the local gauge fields Aμ (z)
y

Uγ [y, x] = 𝒫 exp [±ig ∫dz μ Aμ (z) ] .
x
[
]γ

(4)

The integration goes along an arbitrary path γ:
z∈γ
from the initial point x to the end point y. The notion of a path will be one of the crucial
issues throughout the book.
2 The terminology and the choice of the signature ± will be explained below.

www.pdfgrip.com

Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

Preface | VII

The Wilson line (4) is gauge covariant, but, in contrast to the field strength, the
transformation law reads
Uγ [y, x] → U(y)Uγ [y, x]U † (x),

(5)

so that the transformation operators U, U † are defined in different space-time points.
For closed paths x = y, so that we speak about the Wilson loop:
Uγ ≡ Uγ [x, x] = 𝒫 exp[±ig ∮ dz μ Aμ (z)] ,
γ

(6)

which transforms similarly to the field strength
Uγ = Uγ [x, x] → U(x)Uγ U † (x).

(7)

The simplest scalar gauge invariant objects made from Wilson loops are, therefore,
the traced Wilson loops
𝒲γ = Tr Uγ .

From the mathematical point of view, one can construct a loop space whose elements are the Wilson loops defined on an infinite set of contours. The recast of a quantum gauge field theory in loop space is supposed to enable one to utilize the scalar

gauge-invariant field functionals as the fundamental degrees of freedom, instead of
the traditional gauge-dependent boson and fermion fields. Physical observables are
then supposed to be expressed in terms of the vacuum averages of the products of
Wilson loops
(n)

𝒲{γ} = ⟨0| Tr Uγ1 Tr Uγ2 ⋅ ⋅ ⋅ Tr Uγn |0⟩.

(8)

The concept of Wilson lines finds an enormously wide range of applications in
a variety of branches of modern quantum field theory, from condensed matter and
lattice simulations, to quantum chromodynamics, high-energy effective theories, and
gravity. However, there exist surprisingly few reviews or textbooks which contain a
more or less comprehensive pedagogical introduction into the subject. Even the basics
of the Wilson lines theory may put students and non-experts in significant trouble. In
contrast to generic quantum field theory, which can be taught with the help of plenty of
excellent textbooks and lecture courses, the theory of Wilson lines and loops still lacks
such a support. The objective of the present book is, therefore, to collect, overview and
present in the appropriate form the most important results available in literature, with
the aim to familiarize the reader with the theoretical and mathematical foundations
of the concept of Wilson lines and loops. We also intend to give an introductory idea
of how to implement elementary calculations utilizing Wilson lines within the context
of modern quantum field theory, in particular, in Quantum Chromodynamics.

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

VIII | Preface
The target audience of our book consists of graduate and postgraduate students
working in various areas of quantum field theory, as well as curious researchers from
other fields. Our lettore modello is assumed to have already followed standard university courses in advanced quantum mechanics, theoretical mechanics, classical
fields and the basics of quantum field theory, elements of differential geometry, etc.
However, we give all necessary information about those subjects to keep with the
logical structure of the exposition. Chapters 2, 3, and 4 were written by T. Mertens,
Chapter 5 by F. F. Van der Veken. Preface, Introduction and general editing are due to
I. O. Cherednikov. In our exposition we used extensively the results, theorems, proofs
and definitions, given in many excellent books and original research papers. For the
sake of uniformity, we usually refrain from citing the original works in the main text.
We hope that the dedicated literature guide in Appendix D will do this job better.
Besides this, we have benefited from presentations made by our colleagues at
conferences and workshops and informal discussions with a number of experts.
Unfortunately, it is not possible to mention everybody without the risk of missing
many others who deserve mentioning as well. However, we are happy to thank our
current and former collaborators, from whom we have learned a lot: I. V. Anikin,
E. N. Antonov, U. D’Alesio, A. E. Dorokhov, E. Iancu, A. I. Karanikas, N. I. Kochelev,
E. A. Kuraev, J. Lauwers, L. N. Lipatov, O. V. Teryaev, F. Murgia, N. G. Stefanis, and
P. Taels. Our special thanks go to I. V. Anikin, M. Khalo, and P. Taels for reading parts
of the manuscript and making valuable critical remarks on its content. We greatly
appreciate the inspiring atmosphere created by our colleagues from the Elementary
Particle Physics group in University of Antwerp, where this book was written. We are
grateful to M. Efroimsky and L. Gamberg for their invitation to write this book, and to
the staff of De Gruyter for their professional assistance in the course of the preparation
of the manuscript.

Antwerp, May 2014

I. O. Cherednikov
T. Mertens
F. F. Van der Veken

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

Contents
Preface | V
1

Introduction: What are Wilson lines? | 1

2
2.1
2.1.1
2.1.2
2.1.3
2.2
2.2.1
2.2.2
2.2.3
2.3
2.3.1
2.3.2
2.3.3
2.3.4

2.4
2.4.1
2.4.2
2.5
2.5.1
2.5.2

Prolegomena to the mathematical theory of Wilson lines | 6
Shuffle algebra and the idea of algebraic paths | 7
Shuffle algebra: Definition and properties | 7
Chen’s algebraic paths | 22
Chen iterated integrals | 41
Gauge fields as connections on a principal bundle | 48
Principal fiber bundle, sections and associated vector bundle | 48
Gauge field as a connection | 53
Horizontal lift and parallel transport | 59
Solving matrix differential equations: Chen iterated integrals | 61
Derivatives of a matrix function | 61
Product integral of a matrix function | 63
Continuity of matrix functions | 66
Iterated integrals and path ordering | 67
Wilson lines, parallel transport and covariant derivative | 70
Parallel transport and Wilson lines | 70
Holonomy, curvature and the Ambrose–Singer theorem | 71
Generalization of manifolds and derivatives | 76
Manifold: Fréchet derivative and Banach manifold | 77
Fréchet manifold | 82

3
3.1

Operator definition for PDFs | 163
Gauge invariant operator definition | 165
Collinear factorization and evolution of PDFs | 169
Semi-inclusive deep inelastic scattering | 176
Conventions and kinematics | 176
Structure functions | 178
Transverse momentum dependent PDFs | 180
Gauge-invariant definition for TMDs | 183

A
A.1
A.2
A.3
A.4
A.5
A.6
A.7
A.8
A.9
A.10
A.11
A.12
A.13
A.14
A.15
A.16
A.17
A.18
A.19
A.20

Advanced topics | 259
Calculating products of fundamental generators | 259
Calculating traces in the adjoint representation | 262

D

Brief literature guide | 265

Bibliography | 266
Index | 269

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

1 Introduction: What are Wilson lines?
The idea of gauge symmetry suggests that any field theory must be invariant under the
local (i. e., depending on space-time points) transformations of field functions
ψ(x) → U(x)ψ(x), ψ(x) → ψ(x)U † (x),

(1.1)

where the matrices U(x) belong to the fundamental representation of a given Lie
group. In other words, the Lagrangian has to exhibit local symmetry. We shall mostly
deal with special unitary groups, SU(Nc ), which are used in Yang–Mills theories. Although a number of important results can be obtained by using only the general form
of this gauge transformation, it will be sometimes helpful to use the parameterization1
U(x) = e±igα(x) ,

(1.2)

where
α(x) = t a αa (x), t a =

λa
,
2

and λa are the generators of the Lie algebra of the group U.
Consider for simplicity the free Lagrangian for a single massless fermion field ψ(x)
ℒfermion = ψ(x) i𝜕/ ψ(x), 𝜕/ = γμ

𝜕
.
𝜕xμ

(1.3)

We easily observe that the local transformations (1.1) do not leave this Lagrangian intact. The reason is that the derivative of the field transforms as
𝜕μ ψ(x) → U(x)[𝜕μ ψ(x)] + [𝜕μ U(x)]ψ(x).

(1.4)

The minimal extension of the Lagrangian (1.3), which exhibits the property of
gauge invariance, consists in the introduction of the set of gauge fields
Aμ (x) = t a Aaμ (x)

(1.5)

belonging to the adjoint representation of the same gauge group, which is required to
transform as
Aμ (x) → U(x)Aμ U † (x) ±

i
U(x)𝜕μ U † (x).
g

(1.6)

1 The coupling constant g can be chosen to have a positive or a negative sign. As this is merely a matter
of convention, we leave the choice open and will write ±g.
/>
www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

2 | 1 Introduction: What are Wilson lines?
Thus, the usual derivative has to be replaced by the covariant derivative:
Dμ = 𝜕μ ∓ igAμ ,

(1.7)

Dμ → U(x)Dμ U † (x).

(1.8)

Dμ ψ(x) → U(x)[Dμ ψ(x)].

(1.9)

which transforms as

Then

This procedure obviously makes the minimally extended Lagrangian gauge invariant
/ μ ψ(x).
ℒgauge inv. = ψ(x) iD

(1.10)

Consider now a little bit more complicated object, namely, the bi-local product of
two matter fields
Δ(y, x) = ψ(y)ψ(x).

(1.11)

Such products arise in various correlation functions in quantum field theory,2 in particular, they determine the most fundamental quantities, Green’s functions, via
G(y, x) = ⟨0|𝒯 ψ(y)ψ(x)|0⟩,

(1.12)

where the symbol 𝒯 stands for the time-ordering operation. It is evident that in such
a naive form the bi-local field products and Green’s functions are not gauge invariant:
Δ(y, x) → ψ(y)U † (y)U(x)ψ(x).

(1.13)

Therefore, the problem arises of how to find an operator T[y,x] , which transports the
field ψ(x) to the point y, so that
T[y,x] ψ(x) → U(y)[T[y,x] ψ(x)].

(1.14)

ψ(y)T[y,x] ψ(x) → ψ(y)U † (y)U(y)[T[y,x] ψ(x)] = ψ(y)T[y,x] ψ(x),

(1.15)

In this case, we have

so that the product (1.13) becomes gauge invariant.
Consider first the Abelian gauge group U(1). In this case
U(x) = e±igα(x) ,

(1.16)

2 See, in particular, references in the section ‘Gauge invariance in particle physics’, Appendix D.

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

1 Introduction: What are Wilson lines?

| 3

where α(x) is a scalar function. Then3
Aμ (x) → Aμ (x) + 𝜕μ α(x),

(1.17)

so it is straightforward to see that the ‘transporter’ T[y,x] is given by4
y

T[y,x]

= exp [sign ig ∫ dzμ Aμ (z)] .
x

[

(1.18)

]

Indeed, the product (1.15) transforms as
ψ(y)Uγ [y, x]ψ(x) →
∓igα(y)

ψ(y)e

y

exp [sign ig ∫ dzμ [Aμ (z) + 𝜕μ α(z)]] e±igα(x) ψ(x).
[

x

(1.19)

]

It is instructive to see that the choice of the sign in equation (4) depends on the parameterization of the symmetry transformation U(x). Taking the line integral for the
integrand 𝜕μ α explicitly, one concludes that in order to save the gauge invariance, the
sign should be chosen as
sign = ±.
In what follows we shall always specify the signature conventions we adopt.
In the non-Abelian case the situation is more involved. The fields at different
space-time points Aμ (z) and Aμ (z 󸀠 ), equation (1.5), do not commute, so the exponential of non-commuting functions is ill-defined. An infinitesimal version of equation
(1.15) suggests the following equation to the transporter T[y,x] :
d
T
= 𝒜γ (t)T[y,x] ,
dt [y,x]

(1.20)

where we introduce an arbitrary path γ along which T[y,x] ‘transports’ a field ψ(x) from
the point x to the point y. The need for this path stems from the fact that we do not
know (unless, for some reason, stated otherwise) along which trajectory we have to

transfer the fields from one point to another. The requirement of the gauge invariance
is not affected by the choice of path, but, as we will see, the transporter becomes a
functional of the path. The path γ is assumed to be parameterized by the coordinate
z∈γ
3 Note that the sign in front of 𝜕μ α(x) is independent on the sign choice of g.
4 Any ordering of the field functions is not needed in the classical Abelian case.

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

4 | 1 Introduction: What are Wilson lines?
depending on the parameter t in such a way that
dzμ = zμ̇ (t)dt, z(0) = x, z(t) = y.
The operator 𝒜γ (t) in the r. h. s. of equation (1.20) is given by
𝒜γ (t) = ±i gAμ [z(t)]zμ̇ (t).

(1.21)

It is easy to see that (1.18) solves equation (1.20) in the classical Abelian case.
Integrating equation (1.20) from 0 to t yields an integral equation instead of a
differential one:
t

T[y,x] − T[x,x] = T(t) − T(0) = ∫ 𝒜γ (t1 )T(t1 )dt1 .

(1.22)

0

Imagine that the coupling constant g can be considered as small and let us solve this
equation perturbatively. Namely, we assume that a solution can be presented as an
infinite series
T[y,x] (t) = T (0) + T (1) + T (2) + ⋅ ⋅ ⋅ + T (n) + ⋅ ⋅ ⋅

(1.23)

Suppose we have an initial condition
T(0) = T[x,x] = T (0) .

(1.24)

Then, for the first non-trivial term in the expansion (1.23) we obtain
t

T (1) (t) = [∫ 𝒜γ (t1 )dt1 ] T(0).
]
[0

(1.25)

T(0) is t1 -independent by construction and thus can be separated out from the integration. The next order gives
t

T (2) (t) = ∫ 𝒜γ (t1 )T(t1 )dt1
0

t1

t

= [∫ 𝒜γ (t1 ) ∫ 𝒜γ (t2 )dt1 dt2 ] T(0).
[0

0

(1.26)

]

We can rewrite equation (1.26) as
t t

1
T (t) = [𝒫 ∫ ∫ 𝒜γ (t1 )𝒜γ (t2 )dt1 dt2 ] T(0),
2
[ 0 0
]
(2)

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

(1.27)

1 Introduction: What are Wilson lines?

| 5

where the path-ordering operator reads
𝒫𝒜γ (t1 )𝒜γ (t2 ) = θ(t1 − t2 )𝒜γ (t1 )𝒜γ (t2 ) + θ(t2 − t1 )𝒜γ (t2 )𝒜γ (t1 ).

(1.28)

It is straightforward to see that generic n-order term is given by
T (n) (t) =

t

t

1
𝒫 ∫ ⋅ ⋅ ⋅ ∫ [𝒜γ (t1 ) . . . 𝒜γ (tn )dt1 dt2 ⋅ ⋅ ⋅ dtn ] T(0),
n!
0

(1.29)

0

with obvious generalization of the path-ordering to n functions 𝒜. Therefore, the final
solution can be presented in the form
t

t

0

0

1
𝒫 ∫ ⋅ ⋅ ⋅ ∫ [𝒜γ (t1 ) ⋅ ⋅ ⋅ 𝒜γ (tn )dt1 dt2 dtn ] T(0)
n!
n=0

T(t) = ∑

t

≡ 𝒫 exp [∫ 𝒜γ (t 󸀠 )dt 󸀠 ] T(0).
]
[0

(1.30)

Remembering the definition of 𝒜, equation (1.21) and taking the natural initial condition
T0 = T[x,x] = 1,
we have finally
y

T[y,x] = 𝒫 exp [±i g ∫ Aμ [z]dzμ ] ,
x
[
]γ

(1.31)

that is the Wilson line (4):
T[y,x] = Uγ [y, x],

(1.32)

with arbitrary path γ. In other words, the function (1.15) is gauge invariant, but path
dependent. The rest of the book will be devoted to mathematical motivation of the
above manipulations and to some applications of the Wilson line approach in Quantum Field Theory.

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

2 Prolegomena to the mathematical theory of Wilson
lines
In this part of the book we give the necessary conceptual thesaurus and overview the
main steps towards the construction of the mathematical theory of Wilson lines and
loops. To be more precise, a goal of this exposition is to demonstrate that gauge theories can be consistently formulated in the principal fiber bundle setting, where the
gauge fields (or potentials) are identified with pullbacks of sections of a connection
one-form in the gauge bundle. The gauge potentials give rise to a parallel transport
equation in the gauge bundle that can be solved by using product integrals. As a result,
we shall show that the solution of the parallel transport equation can be presented as
a Wilson line. We shall also discuss its relation to the standard covariant derivative in
gauge theories.
Then, an alternative way to construct a gauge theory will be discussed, which is
based on the use of the holonomies in the gauge bundle instead of the gauge potentials.

This possibility is based on the Ambrose–Singer theorem, which claims that the entire
gauge invariant content of a gauge theory is included in the holonomies. However, the
issues of overcompleteness, reparameterization invariance, and additional algebraic
constraints, coming from the matrix representation of the Lie algebra associated with
the gauge group, impede the straightforward application of the standard loop space
approach to gauge field theories. An interesting solution to these problems arises if
one extends this setting to the so-called generalized loops, first proposed by Chen and
further studied by Tavares (for references, see section ‘Algebraic paths’ in Literature
Guide D) within the framework of the generalized loop space (GLS) approach. Our exposition is based mostly on the original works by these authors.
Aiming towards the appropriate formulation of the generalized loop space framework and having in mind the demonstration of its relation to Wilson lines and loops,
we start with an introductory discussion of the most relevant algebraic concepts. Then
we make use of these concepts to construct Chen’s algebraic d-paths, and, consequently, the generalized loop space. We end the chapter with a discussion on the differential operators which can be defined in generalized loop space. Assuming that
gauge field theories can be recast within the GLS framework, and given that, to this
end, a suitable action could be found, one can use relevant differential operators to
generate the variations of the generalized degrees of freedom in the GLS, and hence,
to construct the appropriate equations of motion in the GLS. Let us mention that the
ambitious program of reformulation of gauge theories in the GLS setting has never
been fully accomplished and thus remains a challenge.
Note that we give complete definitions of the notions, formulations of theorems
and their proofs only when we find it necessary for the consistency of the exposition.
For an extended list of definitions and some helpful theorems and statements we refer
to Appendix A.
/>
www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

2.1 Shuffle algebra and the idea of algebraic paths | 7

2.1 Shuffle algebra and the idea of algebraic paths
2.1.1 Shuffle algebra: Definition and properties
2.1.1.1 Algebraic preliminaries
For our purposes it is sufficient to describe an n-dimensional manifold as a topological
space, wherein a neighborhood to each point is equivalent (strictly speaking, homeomorphic) to the n-dimensional Euclidean space. The fundamental geometrical object
in a manifold we will be concerned about is a path. One has a natural intuitive idea of
what a path or a loop in a manifold is. Mathematically one usually defines a path γ in
a manifold M as the map
γ : [0, 1] → M,

t 󳨃→ γ(t).

For closed paths, which are called loops, one just adds the extra condition that the
initial and final points of the path coincide
γ(0) = γ(1) ∈ M.
The straightforward idea of paths and loops can be generalized to the so-called
algebraic d-paths, where the d-paths are constructed as algebraic objects possessing
certain (desirable) properties. The resulting algebraic structure can then be supplied
with a topology, turning it into a topological algebra. The topology is used to complete
the algebraic properties with analytic ones, allowing the introduction of the necessary
differential operators.1
Several algebraic structures must be introduced before we begin the main discussion. Without going too deep into details, we define a ring as a set wherein two binary
operations of multiplication and addition are defined. Putting it another way, a ring is
an Abelian group (with addition being the group operation) supplied with an extra operation (multiplication). If the second operation is commutative, the ring is also called
commutative. The set of integers provides one of the simplest examples of a commutative ring. Otherwise we speak of noncommutative rings. The set of square matrices
is an example of a noncommutative ring.
Having introduced the notion of a ring, we are able to introduce another algebraic
structure, namely a field, which is defined as a commutative ring where division by
a nonzero element is allowed. It is evident that nonzero elements of a field make up

an Abelian group under multiplication. For example, the set of real numbers forms a
field.
1 Most of the material in this section is based on the original works by Chen (see Literature Guide),
where the proofs to a number of the stated theorems can also be found. For the sake of brevity we skip
those proofs which do not bring more insight than needed into the subject of the book.

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

8 | 2 Prolegomena to the mathematical theory of Wilson lines
One can then construct a vector space over a field. In this case, the elements of
the vector space are called vectors, while the elements of the field are scalars, and two
binary operations (addition of two vectors and multiplication of a vector by a scalar)
acting within the vector space should be defined. One easily captures the idea of a
vector space by thinking of the usual Euclidean vectors of velocities or forces.
The notion of a module over a ring generalizes the concept of a vector space over
a field: now scalars only have to form a ring, not necessary a field. For example, any
Abelian group is a module over the ring of integers. In what follows ‘K-module’ stands
for a module over a ring K.
2.1.1.2 Shuffle algebra
The generalization of the concept of paths in a manifold calls for the introduction of
the notion of a shuffle algebra. The shuffle algebra is an algebra based on the shuffle
product, which in its turn is defined via (k, l)-shuffles. Let us start with the definitions
of these shuffles.
Definition 2.1 ((k, l)-shuffle). A (k, l)-shuffle is a permutation P of the k + l letters, such that
P(1) < ⋅ ⋅ ⋅ < P(k)

and

P(k + 1) < ⋅ ⋅ ⋅ < P(k + l).

Exercise 2.2. How can one explain a (k, l)-shuffle using a deck of cards?

Using these (k, l)-shuffles we can introduce the shuffle multiplication, symbolically
represented by the symbol ∙.
Let us consider a set of arbitrary objects Zi .
Definition 2.3 (Shuffle multiplication). Using the notations
k

nk

Z1 ⋅ ⋅ ⋅ Zk = Z1 ⊗ ⋅ ⋅ ⋅ ⊗ Zk ∈ ⨂ ⋀ M,

k≥1

and, by convention,
Z1 ⋅ ⋅ ⋅ Zk = 0

for k = 0.

We write the shuffle multiplication as
Z1 ⋅ ⋅ ⋅ Zk ∙ Zk+1 ⋅ ⋅ ⋅ Zk+l = ∑ ZP(1) ⋅ ⋅ ⋅ ZP(k+l)
Pk,l

(2.1)

where ∑Pk,l denotes the sum over all (k, l)-shuffles and ⋀ M stands for the nk -th exterior power of the

exterior algebra ⋀ M over the manifold M, and nk the exterior algebra degree of the factor Zk .

Several examples will be instructive to make the shuffle multiplication clear.

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

2.1 Shuffle algebra and the idea of algebraic paths | 9

Example 2.4 (Shuffle multiplication).
– The situation with two objects is evident:
Z1 ∙ Z2 = Z1 Z2 + Z2 Z1
–

Shuffle multiplication of three objects reads:
Z1 ∙ Z2 Z3 = Z1 Z2 Z3 + Z2 Z1 Z3 + Z2 Z3 Z1

–

Four objects multiply as:
Z1 Z2 ∙ Z3 Z4 = Z1 Z2 Z3 Z4 + Z1 Z3 Z2 Z4 + Z1 Z3 Z4 Z2

+ Z3 Z1 Z2 Z4 + Z3 Z1 Z4 Z2 + Z3 Z4 Z1 Z2 .

(2.2)

If we consider the objects Z to be one-forms ω (or linear functionals) defined

on some manifold and compare their shuffle products with the usual antisymmetric
wedge products, then shuffle product can be treated as a symmetric counterpart to
the wedge product.
Let now M be a manifold and
1

Ω = ⋀M = ⋀M
be the set of one-forms on M. We interpret Ω as a K-module, where for the moment we
assume that K is a general ring of scalars with a multiplicative unity. Introducing the
shuffle product on a K-module Ω defines the shuffle K-algebra.2
Definition 2.5 (Shuffle K-algebra). Consider a K-module Ω and the regular tensor algebra over K
based on Ω, denoted by T (Ω). Then T r (Ω) represents the degree r components of the algebra. It is
easy to see that
T 0 (Ω) = K.
Replacing the tensor product by the shuffle multiplication generates a new algebra called the shuffle
K-algebra Sh(Ω) based on the K-module Ω.

In this algebra the shuffle product plays a role of the algebra multiplication m, so that
one can write
m ≡ ∙ : Sh → Sh,
and for the algebra unit map one has
u : K → Sh,

1K 󳨃→ 1Sh .

2 The shuffle product acts here like the vector product on the usual Euclidian vector space over ℝ.

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated

Download Date | 1/8/20 7:38 PM

10 | 2 Prolegomena to the mathematical theory of Wilson lines
It is now possible to extend the algebraic structure based on the shuffle product
by introducing the K-linear maps ϵ, Δ.
Definition 2.6 (Co-unit and co-multiplication).
–
Co-unit ϵ ∈ Alg(Sh(Ω), K) is defined by
{
–

ϵ(1Sh ) = 1K
ϵ(ω1 ⋅ ⋅ ⋅ ωn ) = 0

for n = 0
for n > 0.

(2.3)

Co-multiplication Δ : Sh(Ω) → Sh(Ω) ⊗ Sh(Ω) acts as
Δ(1) = 1 for n = 0
{
{
{
{ Δ(ω ⋅ ⋅ ⋅ ω ) =
1
n
{
n

{
{
{
∑ (ω1 ⋅ ⋅ ⋅ ωi ) ⊗ (ωi+1 ⋅ ⋅ ⋅ ωn )
i=0
{

for n > 0.

(2.4)

The map Δ can be considered as a K-module morphism and is also an associative comultiplication since
(1 ⊗ Δ)Δ = (Δ ⊗ 1)Δ.
Exercise 2.7. Prove the above statement.

The co-multiplication Δ and co-unit ϵ introduces a co-algebra structure on the shuffle
algebra, so that it becomes a bi-algebra. We can go a step further and show that the
shuffle algebra is also a Hopf algebra.3 For that reason we introduce the notion of an
antipode.
Definition 2.8 (Antipode). A K-linear map J
J : Sh → Sh,
is called the shuffle algebra antipode provided that
J(ω1 ⋅ ⋅ ⋅ ωn ) = (−1)n ωn ⋅ ⋅ ⋅ ω1 .
It is evident that
J(1) = 1,

J2 = 1.

Consider now the shuffle multiplication
ms : Sh ⊗ Sh → Sh

3 A Hopf algebra is at the same time an algebra and a co-algebra, see Appendix A.

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

(2.5)

2.1 Shuffle algebra and the idea of algebraic paths | 11

and the unit map
u : K → Sh.
Let the transposition map or flipping operation
T : Sh ⊗ Sh → Sh ⊗ Sh
be defined as
T(v1 ⊗ v2 ) = v2 ⊗ v1 .

(2.6)

Then, for all v1 , v2 ∈ Sh, the antipode can be shown to possess the following properties:
J(v1 ∙ v2 ) = J(v2 ) ∙ J(v1 )

ms ∘ (J ⊗ 1) ∘ Δ = ms ∘ (1 ⊗ J) ∘ Δ = u ∘ ϵ
T ∘ (J ⊗ J) ∘ Δ = Δ ∘ J,

ϵ ∘ J = ϵ.

(2.7)

Therefore, the following theorem holds:
Theorem 2.9 (Sh(Ω) is a Hopf algebra). The shuffle algebra Sh(Ω) is a Hopf K-algebra
provided that its co-multiplication Δ, co-unit ϵ and antipode J are defined as in equations
(2.3), (2.4), and (2.5).
Keeping in mind the algebraic structure of the shuffle algebra discussed above,
we can go on with the study of the algebra homomorphisms4 Alg(Sh(Ω), K).
Definition 2.10 (Group multiplication on Alg(Sh(Ω), K)). Consider the algebra homomorphisms
αi ∈ Alg(Sh(Ω), K).
Define the multiplication
α1 α2 ∈ Alg(Sh(Ω), K)
as
α1 α2 = (α1 ⊗ α2 )Δ.
For this multiplication we have
ϵα1 = α1 ϵ = α1
and
α1 (α2 α3 ) = (α1 ⊗ α2 ⊗ α3 )(1 ⊗ Δ)Δ = (α1 ⊗ α2 ⊗ α3 )(Δ ⊗ 1)Δ = (α1 α2 )α3 .

4 It suffices here to describe a homomorphism as a map between two sets which preserves their algebraic structures.

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

12 | 2 Prolegomena to the mathematical theory of Wilson lines

Figure 2.1: Multiplication of algebra morphisms.

The multiplication of algebra morphisms is depicted in Figure 2.1.
It is easy to observe that:
Proposition 2.11. The multiplication in Definition 2.10, defined on the algebra morphisms
Alg(Sh(Ω), K),
turns it into a group.
We can now study the properties of the algebra homomorphisms Alg(Sh(Ω),
Sh(Ω)). To this end, let us define an algebra morphism which might look a bit strange
at the moment, but will turn out to be valuable when considering the group structure
of algebraic paths and loops.
Definition 2.12 (L-operator). For
α ∈ Alg(Sh(Ω), K)
define
̃Lα = (α ⊗ 1)Δ ∈ Alg(Sh(Ω), Sh(Ω))

(2.8)

L̂α = ̃Lα ⊗ 1 ∈ Hom(Sh(Ω) ⊗ Ω, Sh(Ω) ⊗ Ω).

(2.9)

and

This operator has the following interesting property with respect to the products of
elements of Alg(Sh(Ω), K):
Property 2.13. If
α2 ∈ Alg(Sh(Ω), K),
then, by Definition 2.10,
̃ α = α1 α2
α2 L
1

www.pdfgrip.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 7:38 PM

Wilson lines in quantum field theory 2nd edition

Tài liệu liên quan

Tài liệu bạn tìm kiếm đã sẵn sàng tải về