# Classical Theory of Gauge Fields

Valery Rubakov
Translated by Stephen S Wilson
Pages: 456
https://www.jstor.org/stable/j.ctt7snx1

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Preface
(pp. ix-xii)
4. Part I
• Chapter 1 Gauge Principle in Electrodynamics
(pp. 3-10)

The electromagnetic field in a vacuum is described by two spatial vectors E(x) and H(x), the electric and the magnetic fields. They form the antisymmetric field strength tensorFμν:

Fi0= –F0i=Ei

Fij=εijkHk

(where${H_i}\; = \;{\textstyle{1 \over 2}}{\varepsilon _{ijk}}{F_{jk}}$).

Here and throughout the book, Greek indices refer to space–time and take valuesμ, ν, λ, p, . . .= 0, 1, 2, 3 (ind-dimensional space–timeμ, ν, λ, p, . . .= 0, 1, 2, . . . (d- 1). Latin indices refer to space and take valuesi, j, k, . . .= 0, 1, 2, 3(i, j, k, . . .= 0, 1, 2, . . . = 1, 2, . . . , (d- 1) ind-dimensional space–time). Repeated Greek indices are assumed to denote summation with the Minkowski metric tensor,${\eta _{\nu \,\mu }} = {\text{diag}}\;{\text{(}} + 1,\; - 1,\; - 1, - 1),$, where, we shall not usually distinguish between superscripts and subscripts (for example,FμνFμνwill denote FμνFλpημληvp). Repeated Latin indices will...

• Chapter 2 Scalar and Vector Fields
(pp. 11-32)

Throughout the book, we shall use the system of units$\hbar \; = \;c\; = \;1$. The only non-trivial dimension then is the dimension of mass. Length and time have dimension 1/M. Indeed

$[c]\; = \;\frac{L}{T},\quad [\hbar ]\; = \;M\frac{{{L^2}}}{T}$(2.1)

(the last equation is evident, for example, from the relation$E\; = \;\hbar {\kern 1pt} \omega$). From (2.1) and$\hbar \; = \;c\; = \;1$it follows that

$L\; = \;T\; = \;\frac{1}{M}$.

Physically, 1/Mis the Compton wavelength of a particle of massM.

Problem 1.Find, in conventional units, the length and time corresponding to 1/gram.

Problem 2.a) Find the dimension ofEandHin the system of units$\hbar \; = \;c\; = \;1$. Show that the electromagnetic field action is dimensionless. b) The same...

• Chapter 3 Elements of the Theory of Lie Groups and Algebras
(pp. 33-56)

Agroupis a setGin which a multiplication operation with the following properties is defined:

1. associativity: for alla, b, cG,(ab)c=a(bc);

2. existence of a unit elementeG, such that for allaG,ae=ea=a;

3. existence of an inverse elementa−1Gfor eachaGsuch thata−1a =aa−1=e.

If the multiplication operation is commutative (i.e.ab=bafor alla, bG), the group is said to beAbelian, otherwise it isnon-Abelian.

GroupsG1andG2areisomorphicif there exists a bijective mappingf:G1G2consistent with the multiplication operations

f(g1g2) =f(g1)f(g2),f(g−1) = [f(g)]−1.

In what follows, we shall write group isomorphisms asG1=G2and we...

• Chapter 4 Non-Abelian Gauge Fields
(pp. 57-84)

In the theory of the complex scalar field (Section 2.4), we encountered globalU(1) symmetry: the Lagrangian is invariant under transformations

ϕ(x) →gϕ(x),

whereg=eis an arbitrary element of the groupU(1), independent of the space–time coordinates. In this section, we consider the generalization ofU(1) symmetry (which is Abelian, sinceU(1) is an Abelian group) to non-Abelian cases.

The simplest model with global non-Abelian symmetry is the model ofNcomplex scalar fieldsϕiwith Lagrangian

${\cal L}\; = \;{\partial _\mu }\varphi _i^*{\partial _\mu }{\varphi _i}\; - \;{m^2}\varphi _i^*{\varphi _i}\; - \;\lambda {(\varphi _i^*{\varphi _i})^2}$(4.1)

(here and below, we shall assume summation over the indexi= 1, . . . ,N. This model clearly has an AbelianU(1) symmetry

ϕieϕi....

• Chapter 5 Spontaneous Breaking of Global Symmetry
(pp. 85-104)

In the situations considered up to now, symmetries (global and gauge) have led to certain properties of small perturbations of the fields. Namely, global symmetries in scalar theories have implied equality of the masses of all small linear waves for fields belonging to the same representation of the symmetry group, and also the same interaction properties of these fields (see Section 4.1). Gauge symmetries have led to the masslessness of vector gauge fields: indeed, expressions of the type

${m^2}A_\mu ^aA_\mu ^a$

are not invariant under gauge transformations (and generally, it is impossible to construct an invariant expression quadratic in$A_\mu ^a$which does...

• Chapter 6 Higgs Mechanism
(pp. 105-126)

In this chapter we consider the situation with a non-trivial ground state of a scalar field in models with gauge invariance (Anderson 1963, Englert and Brout 1964, Higgs 1964, Guralniket al. 1964). We shall sometimes use the term “spontaneous symmetry breaking” in this case also, although, unlike in the models with global symmetry considered in Chapter 5, breaking of gauge invariance does not occur in reality. As the simplest example, we consider a model with$U\left( 1 \right)$gauge symmetry. We choose the Lagrangian in the form

${\cal L}\; = - \frac{1}{4}{F_{\mu {\kern 1pt} \nu }}{F_{\mu {\kern 1pt} \nu }}\; + \;({D_{\mu {\kern 1pt} }}\varphi )*{D_\mu }\varphi \; - \;[ - {\mu ^2}\varphi {\kern 1pt} *\;\varphi \; + \;\lambda {(\varphi {\kern 1pt} *\;\varphi )^2}]$, (6.1)

whereϕis a complex scalar field,Fµν=µAννAµ,Dµϕ = (µieAµ)ϕ. From the beginning we...

• Supplementary Problems for Part I
(pp. 127-134)

Problem 1.Mixing of fields. Consider the theory of two real scalar fields${\varphi _1},\;{\varphi _2}$with Lagrangian

$\begin{array}{c}{\cal L}\; = \;\frac{1}{2}{({\partial _\mu }{\varphi _1})^2}\; + \;\frac{1}{2}{({\partial _\mu }{\varphi _2})^2} \\\;\; - \frac{{m_{11}^2}}{2}\varphi _1^2\; - \;m_{12}^2{\varphi _1}{\varphi _2}\; - \frac{{m_{22}^2}}{2}\varphi _2^2 \\\;\; - \frac{{{\lambda _{11}}}}{4}\varphi _1^4\; - \;\frac{{{\lambda _{12}}}}{2}\varphi _1^2\varphi _2^2\; - \;\frac{{{\lambda _{22}}}}{4}\varphi _2^4. \\\end{array}$. (S1.1)

Note that the mass term in this Lagrangian can be written in matrix form

ϕTMϕ,

where

$\varphi \; = \;\left( {\begin{array}{*{20}{c}}{{\varphi _1}} \\{{\varphi _2}} \\\end{array}} \right),\quad {\varphi ^T}\; = \;({\varphi _1},\;{\varphi _2})$

$M\: = \:\left( {\begin{array}{*{20}{c}} {m_{11}^2{\rm{ }}} & {m_{12}^2} \\ {m_{12}^2{\rm{ }}} & {m_{22}^2{\rm{ }}} \\ \end{array}} \right)$

(the matrixMis called the matrix of mass squares, or the mass matrix). The Lagrangian (S1.1) is invariant under the discrete symmetry$({\varphi _1}\; \to \; - {\varphi _1},\;{\varphi _2}\; \to - {\varphi _2})$.

1. What constraints on$m_{ij}^2$andλijare imposed by the requirement that the classical energy is bounded from below?

2. Find the range of values of$m_{11}^2,\;m_{12}^2$and$m_{22}^2$for which the discrete symmetry is not spontaneously broken.

3. In...

5. Part II
• Chapter 7 The Simplest Topological Solitons
(pp. 137-172)

Until now we have considered small linear perturbations of fields about the ground state (classical vacuum) and have been interested mainly in the mass spectrum. In quantum field theory these elementary excitations correspond topoint-likeparticles. Here and in subsequent chapters, we shall consider solitons; these are solutions of the classical field equations, which, in their own right, without quantization, are similar to particles. They are lumps of fields (and, hence, of energy) of finite size; more precisely, the fields decrease rapidly from the center of a lump.¹ The existence and stability of solitons is due, in the first place,...

• Chapter 8 Elements of Homotopy Theory
(pp. 173-192)

In Chapter 7, we met examples of mappings from one manifold to another.

There, the global properties of these mappings, which remain unchanged under continuous changes of the mappings, were important. In this chapter, we briefly discuss certain topological (i.e. global) properties of mappings in quite general form, together with some specific results which are useful for physics.

LetX,Ybe topological spaces (we shall often simply refer to spaces), i.e. sets in which the concept of the proximity of two points is defined. For us, the important cases will be those in which the topological spaces are domains...

• Chapter 9 Magnetic Monopoles
(pp. 193-214)

We end the discussion of topological solitons with an important example, the magnetic monopole of ’t Hooft and Polyakov. Interest in these solutions is, in the first place, due to the fact that they exist in four-dimensional space–time and are present in all models unifying the strong, weak and electromagnetic interactions in the framework of gauge theory with a compact simple or semi-simple gauge group. These models are called grand unified theories. Thus, the existence of magnetic monopoles is a very general prediction of grand unified theories, essentially independently of the choice of model (although some properties of monopoles,...

• Chapter 10 Non-Topological Solitons
(pp. 215-224)

Until now, we have considered solitons, whose existence is related to the topological properties of field configurations. Furthermore, we have limited ourselves to a discussion of static solutions, which was in fact justified for topological solitons (it is more or less clear that the absolute minimum of the energy in the sector with non-zero topological number is attained, if it is attained at all, at a static configuration). However, neither non-trivial topological properties, nor time independence of the fields are mandatory for solitons, which are conceived as stable (or metastable) particle-like solutions of the field equations for fields with finite...

• Chapter 11 Tunneling and Euclidean Classical Solutions in Quantum Mechanics
(pp. 225-248)

Localized solutions of classical field equations, whose existence is a result of the nonlinearity of these equations, are important, not only for the description of particle-like states, or solitons. Such solutions also arise in the study of a completely different class of problems, concerning tunneling processes in quantum field theory (and in quantum mechanics). In theories with a small coupling constant these processes can be studied within a semiclassical approach, where localized solutions of the field equations inEuclidean space–time, i.e. solutions of the instanton type, play a key role. They determine the leading semiclassical exponential in the tunneling...

• Chapter 12 Decay of a False Vacuum in Scalar Field Theory
(pp. 249-262)

In this chapter we consider a model of a real scalar field ind-dimensional space–time$(d\; \ge \;3)$with the action

$S\; = \;\int {{d^d}x} \;\left[ {\frac{1}{2}{{({\partial _\mu }\varphi )}^2}\; - \;V(\varphi )} \right]$, (12.1)

where the scalar potentialV(ϕ) has the form shown in Figure 12.1. We stress thatV(ϕ) represents the energydensityof a homogeneous and static scalar field, with value equal toϕ(everywhere in space).

At the classical level in the model there are two stable static states: the stateϕ=ϕ- (false vacuum), whose energy density is chosen to be zero, without loss of generality, and the stateϕ=ϕ0(true vacuum), where the energy density is negative,...

• Chapter 13 Instantons and Sphalerons in Gauge Theories
(pp. 263-286)

In Chapters 11 and 12 we saw that solutions of Euclidean equations of motion are useful for describing tunneling processes in quantum mechanics and scalar field theory. In this chapter, we consider Euclidean solutions in gauge theories and give interpretations of these in terms of tunneling processes. We also discuss unstable static solutions of field equations in gauge theories (sphalerons), which enable us to find the height of corresponding barriers.

First and foremost, we need to understand how to make the transition to Euclidean time in gauge theories. The most naive transition, involving the substitutiont= −, without any modification of...

• Supplementary Problems for Part II
(pp. 287-292)

Problem 1.Non-topological solitons

Let us consider the theory of two real scalar fieldsϕain (1 + 1)-dimensional space–time. We choose the Lagrangian in the form

${\cal L}\; = \;\frac{1}{2}\;\sum\limits_a {{\partial _\mu }{\varphi ^a}{\partial _\mu }{\varphi ^a}\; - \;{V_0}(\varphi )\; - \;{V_1}(\varphi )}$,

where

${V_0}(\varphi )\; = \;\frac{\lambda }{4}\left( {\sum\limits_a {{\varphi ^a}{\varphi ^a}\; - \;{\upsilon ^2}} } \right)$

${V_1}(\varphi )\; = \; - \varepsilon {\kern 1pt} {\varphi ^1}$.

We note that forε= 0 the Lagrangian is invariant under globalO(2) symmetry. The termV1(ϕ) in the potential breaks this symmetry explicitly.

1. Find the dimensions of the constantsλ,vandε.

2. Find the ground state and the spectrum of small perturbations about it for smallε.

3. Prove that for sufficiently smallεthere exists a soliton, a static local minimum of the energy functional which is a...

6. Part III
• Chapter 14 Fermions in Background Fields
(pp. 295-328)

Free particles with spin${1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}$, fermions, in four-dimensional space–time are described by wave functions with four components,${\psi _\alpha }({x^0},\;{\bf{x}}{\rm{),}}\;\alpha \; = \;1,\;2,\;3,\;4$. It is convenient to represent these in the form of columns with four components:

$\psi \; = \;\left( {\begin{array}{*{20}{c}} {{\psi _1}} \\ {{\psi _2}} \\ {{\psi _3}} \\ {{\psi _4}} \\\end{array}} \right)\;$.

In the absence of background fields, the wave functions of fermions with massmsatisfy the Dirac equation

$i{\gamma ^\mu }{\partial _\mu }\psi \; - \;m\psi \; = \;0$, (14.1)

where the${\gamma ^\mu }$are$4\; \times \;4$Dirac matrices. The relativistic relation${p^2}\; = \;{m^2}$will hold if the Dirac matrices satisfy the equation

$\{ {\gamma ^\mu },\;{\gamma ^\nu }\} \; = \;2{\eta ^{\mu {\kern 1pt} \nu }}$, (14.2)

where the parentheses denote the anticommutator, and${\eta ^{\mu {\kern 1pt} \nu }}$is the Minkowski metric tensor. Indeed, acting on equation (14.1) by the operator$( - i{\gamma ^\mu }{\partial _\mu } - \;m)$we obtain,...

• Chapter 15 Fermions and Topological External Fields in Two-dimensional Models
(pp. 329-350)

In this chapter, we shall consider two effects arising due to the interactions of fermions with topologically non-trivial external bosonic fields, namely the fractionalization of the fermion number and the non-conservation of the fermion quantum numbers due to level crossing. These two phenomena occur in both two- and four-dimensional models. Since two-dimensional theories are easier to analyze, in this chapter we shall consider precisely models in two-dimensional space–time.

Let us make one general remark. In many cases the interaction of fermions with bosonic fields can be analyzed by assuming that the scalar and vector fields are external (and classical)...

• Chapter 16 Fermions in Background Fields of Solitons and Strings in Four-Dimensional Space–Time
(pp. 351-372)

The purpose of this chapter is to discuss certain dynamical effects associated with the behavior of fermions in background fields of topological objects in four-dimensional space–time, namely ’t Hooft–Polyakov magnetic monopoles (Sections 16.1 and 16.2) and strings, i.e. Abrikosov–Nielsen–Olesen vortices (Section 16.3). These objects themselves were described in Sections 9.1, 9.2 and 7.3, and we shall use the notation introduced there. The interest in the matters considered in this chapter is primarily due to the fact that monopoles and strings occur in a natural way in theories of the grand unification of the strong, weak and...

• Chapter 17 Non-Conservation of Fermion Quantum Numbers in Four-dimensional Non-Abelian Theories
(pp. 373-396)

In Section 15.2, for the example of a two-dimensional Abelian model, we considered a mechanism for the non-conservation of fermion quantum numbers, associated with the occurrence of fermion level crossing. In this chapter, we shall see that this mechanism also works in non-Abelian four-dimensional theories. In the Standard Model, it leads to electroweak non-conservation of the baryon and lepton numbers (’t Hooft 1976a, b). Under the usual conditions (at low temperatures and densities or in collisions of particles whose energies are not too high) the probabilities of electroweak processes with non-conservation of the baryon number are extremely small, since they...

• Supplementary Problems for Part III
(pp. 397-402)

Problem 1.Symmetry restoration in cold dense fermionic matter (Kirzhnits and Linde 1976).

Consider the four-dimensional theory of a single real scalar field$\varphi$, interacting with a single type of Dirac fermion. The action of the scalar field itself is chosen in the form

${S_\varphi }\; = \;\int {{d^4}x} \;\left[ {\frac{1}{2}{{({\partial _\mu }\varphi )}^2}\; - \;\frac{\lambda }{4}\,{{({\varphi ^2}\; - \;{\upsilon ^2})}^2}} \right],$(S3.1)

and the fermion action in the form

${S_\psi }\; = \;\int {{d^4}x} \,(i\bar \psi {\kern 1pt} {\gamma ^\mu }{\partial _\mu }\psi \; - \;f\varphi {\kern 1pt} \bar \psi {\kern 1pt} \psi ).$

The coupling constants 𝜆 and 𝑓 are assumed to be small.

1. Show that the theory has a discrete symmetry$\varphi \; \to \; - {\kern 1pt} \varphi$. In the absence of real fermions this discrete symmetry is spontaneously broken: there are two ground states with$\varphi \; = \; \pm {\kern 1pt} \upsilon$. Consider next the system with a...

7. Appendix. Classical Solutions and the Functional Integral
(pp. 403-428)
8. Bibliography
(pp. 429-440)
9. Index
(pp. 441-444)