This chapter is devoted to describing the elements of the theory of connections on principal bundles and a few of their physical applications, including the Yang-Mills theories, the Wu-Yang description of the monopole, and SU(2)-instantons. These, and further applications to gauge anomalies that will be made in chapter 10, motivate the sections dealing with characteristic classes. The chapter also includes an elementary discussion of some index theorems, of importance in the theory of gauge anomalies.

Connections on a principal bundle: an outline

A geometrical object such as a vector is moved by parallel transport on a manifold M if its components with respect to a parallelly transferred frame are kept constant. But how is the parallel transport of frames defined? We saw in section 1.3 that it is convenient to look jointly at the manifold M and at the set of all frames rx (bases of Tx(M)) at the different points x ∈ M as the bundle of linear frames L(M)(GL(n, R)(M). However, given the frame bundle L(M) there is no canonical way of relating a certain frame rx at a point x ∈ M to another frame rx′ at another point x′ ∈ M or, in other words, given a curve on the base manifold M, starting at x ∈ M, there is no intrinsic way of lifting it to a curve in L(M) starting at a certain p ∈ L(M) over x, π(p) = x.

This chapter is devoted to introducing infinite-dimensional Lie groups and algebras and cohomology (in contrast with the finite-dimensional case) as a preparation for chapter 10. Special attention is given to gauge groups and current algebras (an example of the same kind of generalization was already given in chapter 8 in the context of supersymmetric extended objects).

A second set of examples, in which the extension properties of the Lie algebras involved are studied, is provided by the Virasoro and Kac- Moody algebras, as well as the two-dimensional conformal group. It is also shown that Polyakov's induced two-dimensional gravity (which itself is not discussed) provides yet another example of a Wess-Zumino term, obtained from the group Diff S1 of the diffeomorphisms of the circle.

Introduction

Most of the infinite-dimensional groups that appear in physics are defined by endowing the space of mappings of a (finite-dimensional) manifold M into another finite-dimensional manifold Q with a group structure. For instance, this is the case for the groups Map(M,G) ≡ G(M) that correspond to the smooth mappings M → G and which (in physics) are sometimes referred to as ‘local’ groups; this is also the case of the gauge groups, as well as of the loop groups LG or Map(S1,G) given by the smooth mappings S1 → G. The loop groups can be generalized to groups of mappings Sn → G; these generalized loop groups or sphere groups may be denoted by LnG (with L1G ≡ LG and L0G = G), and were already encountered in section 8.8 in a particular case in which G was the (graded) supertranslation group.

This chapter contains some topics of differential geometry which are needed for the rest of the book. They have been selected to serve as a reference for readers already familiar with them or as a self-contained introduction for those who encounter them for the first time.

The central concepts of Lie group, Lie algebra and their relations are analysed. This involves the introduction of left and right-invariant vector fields and forms, which in turn motivates the study of various concepts of differential geometry, such as differential forms, differential operators and de Rham cohomology. In particular, the Maurer-Cartan equations will be extensively used in the text due to their relation with Lie algebra cohomology. A review on fibre bundles is also provided due to their importance in the problem of extensions and in the study of Yang-Mills theories which will be treated in later chapters.

A few comments on how to extend some of the above Lie group concepts to graded Lie groups are made by taking the supertranslation group as an example. This requires a small knowledge of supersymmetry; those not interested in ‘super’ objects may skip it since it is possible to read the book omitting the few sections that contain topics related with supersymmetry. The last two sections (Appendices) contain information and tables on the homotopy groups and the Poincare polynomials of the simple Lie groups and on the homotopy groups of spheres.

This chapter is devoted to the topological and cohomological properties of abelian and non-abelian chiral anomalies in Yang-Mills theories.

First, the Gribov ambiguity and the appearance of anomalies are related to the non-trivial topology of the configuration or Yang-Mills orbit space. This is followed by the explicit path integral calculation of the abelian chiral anomaly in D = 2p dimensions and the non-abelian gauge anomalies (for D = 2) by using Fujikawa's method. It is seen how these results may be interpreted in terms of suitable index theorems on spaces of adequate dimensions (D = 2p and (D + 2) respectively). The consistency conditions for the anomalies and the Schwinger terms are interpreted in terms of a cohomology in which the cocycles are valued in ℱ[A], the space of functionals of the gauge fields. Then it is shown how the cohomological descent procedure starting from the Chern character forms provides a method for obtaining non-trivial candidates for both the non-abelian anomalies and the Schwinger terms.

The question of the ambiguity of the cohomological descent procedure, which gives rise to different (but cohomologous) expressions for the Schwinger terms, the BRST formulation of the gauge cohomology and the Wess-Zumino-Witten terms are also discussed. At the end, some comments on the possible consistency of anomalous gauge theories are made.

The group of gauge transformations and the orbit space of Yang-Mills potentials

The requirement that field theories invariant under rigid transformations (gi ≠ gi(x)) of a group G should also be invariant under local (gi = gi(x)) transformations constitutes the gauge invariance principle.

This chapter is devoted to providing a physical motivation for group cohomology and group extensions by analysing some specific features of classical and quantum mechanics of non-relativistic particles. In particular the need of considering projective representations of the Galilei group and Bargmann's superselection rule are discussed. The projective representations lead to the introduction of two-cocycles; the three-cocycle appears as a result of a consistent breach of associativity. The consistency conditions that have to be fulfilled in both cases are analysed, and illustrated in the case of two-cocycles with the examples of the Weyl-Heisenberg group and the Galilei group. All these concepts will be mathematically defined in chapters 4, 5, and 6, and addressed again in a more elaborate way from the point of view of classical physics in chapter 8.

The relation between cohomology and quantization and in particular the topics of symplectic cohomology, dynamical groups and geometric quantization are briefly treated in this chapter. Again, the Galilei and the Weyl-Heisenberg groups will be used as illustrations of the theory.

Finally, it is shown how the group contraction procedure can generate non-trivial group cohomology.

Some known facts of ‘non-relativistic’ mechanics: two-cocycles

A theory is called ‘non-relativistic’ if its formulation is covariant under the Galilei group G, which is the relativity group of a ‘non-relativistic’ theory. It should be noticed, however, that theories covariant under the Poincare group and the Galilei group are both, strictly speaking, equally relativistic; only their relativity group is different.

The relation between mechanics and cohomology already sketched in chapter 3 is re-analysed here from the Lagrangian point of view, itself initially introduced in the 1-jet bundle framework. It is seen how quasi-invariance for Lagrangians is related to group (or Lie algebra) central extensions. This point of view is extended by considering J(J1(E))-valued cocycles in such a way that the problem of quasi-invariant Lagrangians is described in terms of one-cocycles as well as two-cocycles, and they are seen to be related by the cohomological descent procedure. This general scheme will appear again in chapter 10 in the context of non-abelian, consistent, chiral gauge anomalies.

The complementary aspect of obtaining physical actions (or terms in them) from non-trivial cohomology is also studied. This leads to the concept of Wess-Zumino term on a group manifold. The cohomological descent is then applied to this picture, which exhibits the different role of the left and right versions of the symmetry Lie algebra involved.

Examples of these two aspects are given by using the Galilei group (also studied in chapter 3) and the supersymmetric extended objects (which may be omitted by readers not interested in supersymmetry). Finally, a Lagrangian action for the monopole (chapter 2) is studied as a different kind of Wess-Zumino term associated with quasi-invariance under gauge (rather than rigid) transformations.

A short review of the variational principle and of the Noether theorem in Newtonian mechanics

In the variational formulation of dynamical systems the starting point is the definition of the Lagrangian.

Up to this point we have not had much to say about the detailed structure of the Hamiltonian operator H. This operator can be defined by giving all its matrix elements between states with arbitrary numbers of particles. Equivalently, as we shall show here, any such operator may be expressed as a function of certain operators that create and destroy single particles. We saw in Chapter 1 that such creation and annihilation operators were first encountered in the canonical quantization of the electromagnetic field and other fields in the early days of quantum mechanics. They provided a natural formalism for theories in which massive particles as well as photons can be produced and destroyed, beginning in the early 1930s with Fermi's theory of beta decay.

However, there is a deeper reason for constructing the Hamiltonian out of creation and annihilation operators, which goes beyond the need to quantize any pre-existing field theory like electrodynamics, and has nothing to do with whether particles can actually be produced or destroyed. The great advantage of this formalism is that if we express the Hamiltonian as a sum of products of creation and annihilation operators, with suitable non-singular coefficients, then the S-matrix will automatically satisfy a crucial physical requirement, the cluster decomposition principle, which says in effect that distant experiments yield uncorrelated results. Indeed, it is for this reason that the formalism of creation and annihilation operators is widely used in non-relativistic quantum statistical mechanics, where the number of particles is typically fixed.

In our calculations of radiative corrections in Chapter 11 we went just one step beyond the lowest order in perturbation theory. However, there is a very important class of problems where even the simplest calculation requires that from the beginning we consider classes of Feynman diagrams of arbitrarily high order in coupling constants like e. These problems are those involving bound states – in electrodynamics, either ordinary atoms and molecules, or such exotic atoms as positronium or muonium.

It is easy to see that such problems necessarily involve a breakdown of ordinary perturbation theory. Consider for instance the amplitude for electron–proton scattering as a function of the center-of-mass energy E. As shown in Section 10.3, the existence of a bound state like the ground state of hydrogen implies the existence of a pole in this amplitude at E = mp + me—13.6 eV. However, no single term in the perturbation series for electron-proton scattering has such a pole. The pole therefore can only arise from a divergence of the sum over all diagrams at center-of-mass energies near mp + me.

The reason for this divergence of the perturbation series is also easy to see, especially if for the moment we consider the time-ordered diagrams of old-fashioned perturbation theory instead of Feynman diagrams. Suppose that in the center-of-mass system the electron and proton both have momenta of magnitude q ≪ me, and consider an intermediate state in which the electron and proton momenta are different but also of order q.

In Chapters 7 and 8 we applied the canonical quantization operator formalism to derive the Feynman rules for a variety of theories. In many cases, such as the scalar field with derivative coupling or the vector field with zero or non-zero mass, the procedure though straightforward was rather awkward. The interaction Hamiltonian turned out to contain a covariant term, equal to the negative of the interaction term in the Lagrangian, plus a non-covariant term, which served to cancel non-covariant terms in the propagator. In the case of electrodynamics this non-covariant term (the Coulomb energy) turned out to be not even spatially local, though it is local in time. Yet the final results are quite simple: the Feynman rules are just those we should obtain with covariant propagators, and using the negative of the interaction term in the Lagrangian to calculate vertex contributions. The awkwardness in obtaining these simple results, which was bad enough for the theories considered in Chapters 7 and 8, becomes unbearable for more complicated theories, like the non-Abelian gauge theories to be discussed in Volume II, and also general relativity. One would very much prefer a method of calculation that goes directly from the Lagrangian to the Feynman rules in their final, Lorentz-covariant form.

Fortunately, such a method does exist. It is provided by the path-integral approach to quantum mechanics. This was first presented in the context of non-relativistic quantum mechanics in Feynman's Princeton Ph. D. thesis, as a means of working directly with a Lagrangian rather than a Hamiltonian. In this respect, it was inspired by earlier work of Dirac.

Ever since the birth of quantum field theory in the papers of Born, Dirac, Fermi, Heisenberg, Jordan, and Pauli in the late 1920s, its development has been historically linked to the canonical formalism, so much so that it seems natural to begin any treatment of the subject today by postulating a Lagrangian and applying to it the rules of canonical quantization. This is the approach used in most books on quantum field theory. Yet historical precedent is not a very convincing reason for using this formalism. If we discovered a quantum field theory that led to a physically satisfactory S-matrix, would it bother us if it could not be derived by the canonical quantization of some Lagrangian?

To some extent this question is moot because, as we shall see in Section 7.1, all of the most familiar quantum field theories furnish canonical systems, and these can easily be put in a Lagrangian form. However, there is no proof that every conceivable quantum field theory can be formulated in this way. And even if it can, this does not in itself explain why we should prefer to use the Lagrangian formalism as a starting point in constructing various quantum field theories.

The point of the Lagrangian formalism is that it makes it easy to satisfy Lorentz invariance and other symmetries: a classical theory with a Lorentz-invariant Lagrangian density will when canonically quantized lead to a Lorentz-invariant quantum theory.

Our immersion in the present state of physics makes it hard for us to understand the difficulties of physicists even a few years ago, or to profit from their experience. At the same time, a knowledge of our history is a mixed blessing — it can stand in the way of the logical reconstruction of physical theory that seems to be continually necessary.

I have tried in this book to present the quantum theory of fields in a logical manner, emphasizing the deductive trail that ascends from the physical principles of special relativity and quantum mechanics. This approach necessarily draws me away from the order in which the subject in fact developed. To take one example, it is historically correct that quantum field theory grew in part out of a study of relativistic wave equations, including the Maxwell, Klein–Gordon, and Dirac equations. For this reason it is natural that courses and treatises on quantum field theory introduce these wave equations early, and give them great weight. Nevertheless, it has long seemed to me that a much better starting point is Wigner's definition of particles as representations of the inhomogeneous Lorentz group, even though this work was not published until 1939 and did not have a great impact for many years after. In this book we start with particles and get to the wave equations later.

This is not to say that particles are necessarily more fundamental than fields. For many years after 1950 it was generally assumed that the laws of nature take the form of a quantum theory of fields.