Gauge Field Theories: Spin One and Spin Two: 100 Years After General Relativity

by Gunter Scharf

Overview

One of the main problems of theoretical physics concerns the unification of gravity with quantum theory. This monograph examines unification by means of the appropriate formulation of quantum gauge invariance. Suitable for advanced undergraduates and graduate students of physics, the treatment requires a basic knowledge of quantum mechanics.
Opening chapters introduce the free quantum fields and prepare the field for the gauge structure, describing the inductive construction of the time-ordered products by causal perturbation theory. The analysis of causal gauge invariance follows, with considerations of massless and massive spin-1 gauge fields. Succeeding chapters explore the construction of spin-2 gauge theories, concluding with an examination of nongeometric general relativity that offers an innovate approach to gravity and cosmology.

Product Details

ISBN-13:	9780486815145
Publisher:	Dover Publications
Publication date:	07/20/2016
Series:	Dover Books on Physics
Sold by:	Barnes & Noble
Format:	eBook
File size:	71 MB
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About the Author

Günter Scharf is Professor Emeritus of Physics at the University of Zurich. His other Dover book is the third edition of Finite Quantum Electrodynamics: The Causal Approach.

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Gauge Field Theories: Spin One and Spin Two

100 Years After General Relativity

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By Günter Scharf

Dover Publications, Inc.

Copyright © 2016 Günter Scharf
All rights reserved.
ISBN: 978-0-486-81514-5

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CHAPTER 1

Free fields

Free fields are mathematical objects, they are not very physical. For example a free spin-1/2 Dirac field is a rather bad description of an electron because its charge and Coulomb field are ignored. In case of the photon the description by a free (transverse) vector field seems to be better, but still is not perfect. Elementary particles are complicated real objects, free fields are simple mathematical ones. Nevertheless, free fields are the basis of quantum field theory because the really interesting quantities like interacting fields, scattering matrix (S-matrix) etc. can be expanded in terms of free fields. We, therefore, first discuss all kinds of free fields which we will use later. Among them are some strange, but interesting guys called ghost fields. The German notion "spirit fields" (Geist-Felder instead of Gespenster-Felder) is more adequate. The reason is that these ghost fields define the infinitesimal gauge transformations of quantized gauge fields. That means they are at the heart of quantum gauge theory and so are never at any time negligible ghosts.

Our convention of the Minkowski metric is g[μ]v = diag(1, -1, -1, -1). If not explicitly written, we put the velocity of light and Planck's constant equal to 1, c = h = 1. We sometimes refer for further discussion to the previous book G. Scharf "Finite quantum electrodynamics", Springer Verlag 1995, which will be abbreviated by FQED.

1.1 Bosonic scalar fields

First let us consider a neutral or real massive scalar field which is a solution of the Klein-Gordon equation

[MATHEMATICAL EXPRESSION OMITTED] (1.1.1)

A real classical solution of this equation is given by

[MATHEMATICAL EXPRESSION OMITTED] (1.1.2)

where

[MATHEMATICAL EXPRESSION OMITTED] (1.1.3)

In quantum field theory a(p) and a*(p) become operator-valued distributions, that means

a(f) = ∫ d3 p f*(p)a(p) (1.1.4)

(f a test function) is an operator in some Hilbert space and

a+(f) = ∫ d3pf (p)a+(p) (1.1.5)

its adjoint. In the distribution a(p)+ we make no difference about the place of the superscript +, before or behind the argument. The properties of the unsmeared objects a(p), a+(p) are further analyzed in the problems 1.8-9 at the end of this chapter. In the following all equations between distributions mean that they become operator equations after smearing with test functions.

The crucial property of these operators is the fulfillment of the canonical commutation relations

[a(f),a+(g)] = ∫ d3p f*(p)g(p) = (f, g), (1.1.6)

the result is the L2 scalar product of the test functions. The relation can be written in distributional form as follows

[a(p), a+(q)] = δ3(p - q). (1.1.7)

all other commutators vanish. The quantized Bose field is now given by

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.8)

It is obviously hermitian

φ+(x) = φ(x). (1.1.9)

Let us call the second term in (1.1.8) involving a+ the creation part φ(+) and the first term with a(p) the absorption part φ(-). Then by (1.1.7) their commutator is equal to

[MATHEMATICAL EXPRESSION OMITTED] (1.1.10)

To write this in Lorentz-covariant form we add the integration over p0 and insert the one-dimensional δ-distribution

[MATHEMATICAL EXPRESSION OMITTED], (1.1.11)

note that E is positive (1.1.3). The commutator (1.1.10) is now equal to

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.12)

In the same way we get

[MATHEMATICAL EXPRESSION OMITTED].

Then the commutation relation for the total scalar field reads

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.13)

This is the so-called Jordan-Pauli distribution Dm. It has a causal support, that means its support lies in the forward and backward light cones (see problems 1.1-3 and FQED, Sect.2.3)

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.14)

This property is crucial for the causal method in Sect.2. We already remark that Dm can be split into retarded and advanced functions

[MATHEMATICAL EXPRESSION OMITTED], with [MATHEMATICAL EXPRESSION OMITTED] (1.1.15)

Our next task is to write the scalar field in Lorentz-invariant form, too. For this purpose we introduce the measure

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.16)

This is a Lorentz-invariant measure on the positive mass shell M+ = {p [member of] R4 | p2 = m2, p0 > 0}. But the scalar field (1.1.8)

[MATHEMATICAL EXPRESSION OMITTED] (1.1.17)

still does not look covariant. Obviously, the operators

[MATHEMATICAL EXPRESSION OMITTED] (1.1.18)

must be Lorentz scalars. According to (1.1.7) they obey the commutation relations

[[??](p), [??]+(q)] = 2E(p)δ3(p - q) (1.1.19)

and all other commutators vanish. To get the corresponding operator equations, we smear (1.1.17) with a real test function f(x) [member of] S(R4) in 4-dimensional Schwartz space (see any book on distributions, for example I.M. Gelfand et al., Generalized functions, Academic Press, New York 1964-68)

[MATHEMATICAL EXPRESSION OMITTED], (1.1.20)

where

[MATHEMATICAL EXPRESSION OMITTED] (1.1.21)

[MATHEMATICAL EXPRESSION OMITTED] (1.1.22)

and

[MATHEMATICAL EXPRESSION OMITTED] (1.1.23)

is the four-dimensional Fourier transform. Then

[MATHEMATICAL EXPRESSION OMITTED] (1.1.24)

is the Lorentz-invariant form of the commutation relation. The scalar product herein corresponds to the Hilbert space L2(M+,dμm).

To show that the whole procedure is well defined and free of contradictions, we have to construct a concrete representation of the various operators in the so-called Fock-Hilbert space. To construct the latter we start from a normalized vector Ω, |Ω| = 1 defined by

[MATHEMATICAL EXPRESSION OMITTED] (1.1.25)

This vector is assumed to be unique and called the vacuum. Then the a can be interpreted as absorption operators, because in Ω nothing can be absorbed according to (1.1.25). Next we consider the vectors a+(f)Ω and calculate their scalar products

[MATHEMATICAL EXPRESSION OMITTED] (1.1.26)

where the commutation relation (1.1.24) has been used to commute the absorption operator a(f) to the right, giving zero on Ω by (1.1.25). We see that these vectors form a Hilbert space which is isomorphic to

H1 = L2(M+, dμm) (1.1.27)

and consists of complex functions [MATHEMATICAL EXPRESSION OMITTED], with

[MATHEMATICAL EXPRESSION OMITTED] (1.1.28)

This is the one-particle space, so that a+ can indeed be interpreted as an emission operator. It generates a one-particle state from the vacuum. As mentioned before the notion "particle" does not mean that this is a real physical particle. At best we have an approximate description of some real particle in terms of the free scalar field.

The n-particle space is defined as the symmetric tensor product

Hn = SnH[cross product]n1 (1.1.29)

where Sn is the symmetrization operator

[MATHEMATICAL EXPRESSION OMITTED] (1.1.30)

the sum goes over all permutations of the momenta of the n particles. This space is spanned by the vectors

[MATHEMATICAL EXPRESSION OMITTED] (1.1.31)

As in (1.1.26) one can verify that the mapping (1.1.31) is a unitary correspondence. The direct sum

F = [cross product]∞n=0Hn

gives the Fock-Hilbert space where the scalar field operates.

The representation of the field operators just constructed, the so-called Fock representation, realizes a unitary representation of the proper Poincaré group at the same time. By definition the vacuum is invariant

U (a, Λ)Ω = Ω, (1.1.32)

where A denotes proper Lorentz transformations (i.e. without reflections) and a a [member of] R4 represents the translations. From (1.1.17) we then have

[MATHEMATICAL EXPRESSION OMITTED] (1.1.33)

where we have used the Lorentz invariance of the Minkowski scalar product in the last term. The transformed field (1.1.33) must be equal to

- U(a, Λ)φ(x)U(a,Λ)-1.

We smear the emission part φ(+) with f(x) and apply it to the vacuum, using (1.1.32),

[MATHEMATICAL EXPRESSION OMITTED].

By (1.1.33) this is equal to

[MATHEMATICAL EXPRESSION OMITTED] (1.1.34)

This implies

[MATHEMATICAL EXPRESSION OMITTED] (1.1.35)

which is an irreducible unitary representation of the Poincaré group in H1 (1.1.27). The representation in the n-particle sector Hn is the corresponding tensor representation

[MATHEMATICAL EXPRESSION OMITTED] (1.1.36)

It is no longer irreducible.

Next we want to find out how the emission and absorption operators operate in the Fock representation. From the correspondence (1.1.31) we immediately get

[MATHEMATICAL EXPRESSION OMITTED] (1.1.37)

By linearity this extends to

[MATHEMATICAL EXPRESSION OMITTED] (1.1.38)

where

Φ = ([??]n)Nn=0 [member of] FN (1.1.39)

[??]n = 0 for n >N, Φ is a general vector containing not more than N particles. For arbitrary N this is a dense set in Fock space which is in the domain of a+(f).

In case of the absorption operator we use the commutation relation

[MATHEMATICAL EXPRESSION OMITTED].

In the next step we commute a(f) with a+(f2) and so on. This leads to

[MATHEMATICAL EXPRESSION OMITTED] (1.1.40)

[MATHEMATICAL EXPRESSION OMITTED], (1.1.41)

where the overlined fj is lacking. Writing the scalar product as a p-integral and changing the symmetrization operator Sn into Sn+1 we finally get

[MATHEMATICAL EXPRESSION OMITTED]

By linearity this extends to

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.42)

For completeness let us determine the adjoint operator of a(f). Let Φ,ψ [member of] FN (1.1.39) then the scalar product in Fock space is given by

[MATHEMATICAL EXPRESSION OMITTED], (1.1.43)

where [bar.f](p) = f(-p)* and the star denotes the complex conjugate. The Fourier transform of [bar.f](p) in x-space is just the complex conjugate function f(x)*. From (1.1.43) we obtain the relation

a(f)+ = a+([bar.f]).

For later use we write down the operation of the hermitian scalar field

[MATHEMATICAL EXPRESSION OMITTED] (1.1.44)

in Fock space:

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.45)

Here the symmetrization has explicitly been written out. In the Fock representation the commutation relation (1.1.12) can be written in terms of vacuum expectation values in the following form

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.46)

because φ(-)(x)φ(+)(y) is the only term which has a non-vanishing vacuum expectation value. This will later be generalized to more than two factors.

The charged or complex scalar field is a slight generalization of the neutral one:

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.47)

It contains two different kinds of particles whose absorption and emission operators satisfy

[a(p), a+(q)] = δ(p - q) = [b(p), b+(q)] (1.1.48)

and all other commutators vanish. Then it follows

[φ(x), φ(y)+] = -iDm(x - y) (1.1.49)

but

[φ(x); φ(y)] = 0. (1.1.50)

The Lorentz-invariant form is given by

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.51)

[??]+([??])Ω spans the one-particle sector Ha1, but [??]+([??])Ω spans the one-anti - particle sector Hb1 which is different. The many-particle sectors are again obtained by tensor products and the total Fock space is the direct sum

[MATHEMATICAL EXPRESSION OMITTED]. (1.1.52)

1.2 Fermionic scalar (ghost) fields

"Fermionic" means that we now quantize a scalar field with anticommutators. These fields occur as so-called ghost fields in gauge theory. This terminology is somewhat misleading because the ghost fields are genuine dynamical fields which interact with other fields in the theory. Their ghost character only expresses the fact that the ghost particles cannot occur as asymptotic scattering states. There seems to be a contradiction to the well-known theorem of spin and statistics. This theorem tells us that fields with integer spin should be quantized with commutators and those with half-integer spin with anticommutators. We will return to this point in detail below, for the moment we remark that the 'wrong" commutation relation is possible here because the scalar field under consideration describes two different kinds of particles, similarly as the charged scalar field (1.1.46):

[MATHEMATICAL EXPRESSION OMITTED]. (1.2.1)

In addition, we introduce a second scalar field

[MATHEMATICAL EXPRESSION OMITTED] (1.2.2)

This is not the adjoint of u(x). The absorption and emission operators cj, c+k obey the anticommutation relations

{cj(p), ck(q)+} = δjkδ3 (p - q). (1.2.3)

The absorption and emission parts (with the adjoint operators) are again denoted by (-) and (+). They satisfy the following anticommutation relations

[MATHEMATICAL EXPRESSION OMITTED] (1.2.4)

[MATHEMATICAL EXPRESSION OMITTED]. (1.2.5)

All other anticommutators vanish. This implies

{u(x), [??](y)} = -iDm(x - y) (1.2.6)

and

{u(x), u(y)} = 0. (1.2.7)

As before the fields can be written in Lorentz-covariant form by introducing

[MATHEMATICAL EXPRESSION OMITTED] (1.2.8)

Then we have

[MATHEMATICAL EXPRESSION OMITTED] (1.2.9)

[MATHEMATICAL EXPRESSION OMITTED] (1.2.10)

The vectors [??]j([??])+Ω, j = 1, 2 generate the one-particle sectors H(j)1, and the n-particle sectors are obtained as antisymmetric tensor products

H(j)n = S-nH(j)[cross product]n1, (1.2.11)

where

[MATHEMATICAL EXPRESSION OMITTED] (1.2.12)

is the antisymmetrization operator. The total Fock space is the direct sum

[MATHEMATICAL EXPRESSION OMITTED]. (1.2.13)

Let us now discuss the relation to the theorem of spin and statistics. This theorem can be expressed in the following form (see R.F. Streater, A.S. Wightman, PCT, Spin and Statistics, and All That", Benjamin 1964):

Theorem 1.2.1.. In a quantum field theory with a Hilbert space with positive definite metric there cannot exist scalar fields different from zero which satisfy the anti-commutation relations

{u{x), u(y)} = 0 (1.2.14)

{u(x), u+(y)} = 0 (1.2.15)

for all (x - y)2< 0.

The first condition is fulfilled (1.2.7), but the second one is not:

[MATHEMATICAL EXPRESSION OMITTED] (1.2.16)

The causal Jordan-Pauli distribution Dm vanishes for space-like arguments (x-y)2< 0, but the D(-)m-distribution does not. For example, in the massless case m = 0 we have the simple expression

[MATHEMATICAL EXPRESSION OMITTED] (1.2.17)

and the principal value contribution does not vanish for x2< 0. The situation in the massive case is similar (see FQED, Sect.2.3). Consequently, there is no contradiction to the spin-statistics theorem. The point is the minus sign in front of c1 in (1.2.2) which implies [??] ≠ u+.

1.3 Massless vector fields

These fields obey the wave equation

[]Aμ(x) = 0. (1.3.1)

Examples of massless vector-particles are the photon and the gluons, so that these fields are the genuine gauge fields. The photon has only two physical transversal degrees of freedom. Therefore, two subsidiary conditions are necessary to eliminate the unphysical components. As one such condition we may choose the Lorentz condition

[partial derivative]Aμ(x) = 0 (1.3.2)

which is Lorentz-invariant. But the second condition, for example the temporal gauge condition

A0(x) = 0, (1.3.3)

cannot be chosen covariantly. This is the reason for the subtleties in the following. We recall that the free fields considered here are only the zeroth approximation to the real photon in the lab.

To start with we disregard the subsidiary conditions completely. We quantize Aμ(x) as four independent real scalar fields. Let

[MATHEMATICAL EXPRESSION OMITTED], (1.3.4)

be a real classical solution of the wave equation with

ω(k) = |k|def = k0, (1.3.5)

the star denotes the complex conjugate. After quantization aμ(k) become operator-valued distributions. Let us assume the usual commutation relations

[MATHEMATICAL EXPRESSION OMITTED] (1.3.6)

Then we know from Sect. 1.1 that aμ+ are emission operators and aμ absorption operators in Fock space.

There is, however, a serious difficulty with Lorentz covariance in this approach: If we retain the classical expression (1.3.4) in the form

[MATHEMATICAL EXPRESSION OMITTED] (1.3.7)

we obtain the following commutator

[MATHEMATICAL EXPRESSION OMITTED] (1.3.8)

The Lorentz invariant Jordan-Pauli distribution (1.1.13) for mass 0 appears here. However, the right-hand side is not a second rank Lorentz tensor of the same type as the left-hand side. We should have gμv instead of δμv. The simplest way to remedy this defect is to change the sign in

[a0(k), a0(k')+] = -δ3(k - k'). (1.3.9)

After 3-dimensional smearing this implies

[a0(f), a0(f)+] = -(f, f). (1.3.10)

But this contradicts a positive definite metric in Hilbert space

[MATHEMATICAL EXPRESSION OMITTED]. (1.3.11)

We, therefore, will proceed differently, the "indefinite metric" will appear in a more satisfactory way.

Another possibility to solve the problem is to change the classical definition (1.3.4) of A0 into

[MATHEMATICAL EXPRESSION OMITTED]. (1.3.12)

This makes A0 a skew-adjoint operator instead of self-adjoint. As we shall discuss below, the physical Hilbert space will be defined in such a way that all expectation values of A0 (and of any quantity derived from it) vanish. Then the non-self-adjointness of A0 (1.3.12) causes no problems. But the spatial components remain hermitian

(Aj)+ = Aj, j = 1, 2, 3, (1.3.13)

so that the adjoint operation is not Lorentz-invariant. We will introduce a second conjugation below, which is Lorentz-invariant.

(Continues...)

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Table of Contents

Contents

1. Free fields, 1,
1.1 Bosonic scalar fields, 1,
1.2 Fermionic scalar (ghost) fields, 8,
1.3 Massless vector fields, 10,
1.4 Operator gauge transformations, 18,
1.5 Massive vector fields, 23,
1.6 Fermionic vector (ghost) fields, 29,
1.7 Tensor fields, 30,
1.8 Spinor fields, 36,
1.9 Normally ordered products in free fields, 39,
1.10 Problems, 47,
2. Causal perturbation theory, 49,
2.1 The S-matrix in quantum mechanics, 50,
2.2 The method of Epstein and Glaser, 56,
2.3 Splitting of causal distributions in x-space, 67,
2.4 Splitting in momentum space, 74,
2.5 Calculation of tree graphs, 80,
2.6 Calculation of loop graphs, 87,
2.7 Normalizability, 93,
2.8 Problems, 96,
3. Spin-1 gauge theories: massless gauge fields, 98,
3.1 Causal gauge invariance, 98,
3.2 Self-coupled gauge fields to first order, 102,
3.3 Divergence- and co-boundary-couplings, 107,
3.4 Yang-Mills theory to second order, 112,
3.5 Reductive Lie algebras, 117,
3.6 Coupling to matter fields, 119,
3.7 Gauge invariance to all orders, 122,
3.8 Unitarity, 127,
3.9 Other gauges, 131,
3.10 Gauge independence, 137,
3.11 Appendix A: Cauchy problem for the iterated wave equation, 141,
3.11 Problems, 142,
4. Spin-1 gauge theories: massive gauge fields, 145,
4.1 Massive QED and Abelian Higgs model, 146,
4.2 General massive gauge theory, 150,
4.3 First order gauge invariance, 152,
4.4 Second order gauge invariance, 155,
4.5 Third order gauge invariance, 165,
4.6 Derivation of the electroweak gauge theory, 168,
4.7 Coupling to leptons, 172,
4.8 More fermionic families, 176,
4.9 Gauge invariance to third order: axial anomalies, 183,
4.10 Problems, 185,
5. Spin-2 gauge theories, 190,
5.1 Causal gauge invariance with massless tensor fields, 191,
5.2 First order gauge invariance and descent equations, 194,
5.3 Massive tensor fields, 197,
5.4 Massive gravity, 203,
5.5 Expansion of the Einstein-Hilbert Lagrangian, 205,
5.6 Expansion in the massive case, 210,
5.7 Second order gauge invariance: graviton sector, 210,
5.8 Second order gauge invariance: ghost sector, 221,
5.9 Coupling to matter, 224,
5.10 Radiative corrections, 231,
5.11 Yang-Mills fields in interaction with gravity, 235,
5.12 Massive gravity: second order, 239,
5.13 Problems, 245,
6. Non-geometric general relativity, 247,
6.1 Geodesic equation, 248,
6.2 Einstein's equations and Maxwell's equations, 250,
6.3 Spherically symmetric fields and the circular velocity, 252,
6.4 Solutions of the vacuum equations, 256,
6.5 Cosmology in the cosmic rest frame, 259,
6.6 Failure of homogeneous cosmology, 263,
6.7 An inhomogeneous universe, 265,
6.8 Next to leading order, 267,
6.9 Calculation of the energy-momentum tensor, 272,
6.10 Null geodesics, 276,
6.11 The redshift, 280,
6.12 Area and luminosity distances, 283,
6.13 The Riemann and Weyl tensors, 285,
Bibliographical notes, 287,
Subject index, 294,