Paul Dirac was one of the finest physicists of this century. The development of quantum mechanics began at the turn of the century, but it was Dirac who, in 1925 and 1926, brought the subject to its definitive form, creating a theory as compelling as Newton's mechanics had been.

Dirac immediately set about reconciling the quantum theory with Einstein's special theory of relativity (of 1905). The nature of the marriage between these two marvellous theories, and the fruits of that union, have been the constant preoccupation of fundamental physics from 1925 to the present day. Dirac contributed more than anyone else to this crucial enterprise, including in 1930 the prediction of the existence of antimatter.

Dirac died in 1984, and St John's College, Cambridge (Dirac's college), very generously endowed an annual lecture to be held at Cambridge University in Dirac's memory. The first two Dirac Lectures, printed here, are contrasting variations on Dirac's theme of the union of quantum theory and relativity.

Richard Feynman, in the years since the Second World War, did more than anyone else to evolve Dirac's relativistic quantum theory into a general and powerful method of making physical predictions about the interactions of particles and radiation. His work complements Dirac's in a remarkable way. His style of doing physics has been vastly influential. His lecture here, which gives some flavour of that style, expounds the physical reality underlying Dirac's prediction of antimatter.

The crowning achievement to date of the relativistic quantum theory has been the unification of electricity and magnetism on the one hand (themselves unified by Maxwell a century ago) with the weak forces of radioactive decay on the other. Steven Weinberg is one of the chief authors of this unification, in work which predicted the existence and properties of new particles (weighing as much as heavy atoms), which were subsequently triumphantly produced, precisely as predicted, at the European laboratory CERN in Geneva in 1983.

Let us recapitulate. Although the form (4.32) for the path integral was motivated by analogy with finite-dimensional integrals, in no way is it a definition. Rather, as we have stressed repeatedly, it is an algorithm for handling the Feynman perturbation series. Strong coupling apart, all the manipulations of the path integral that we have performed (integration by parts, the introduction of auxiliary fields, the inclusion of background fields) stand in one-to-one correspondence to rearrangements of the perturbation series. In fact, path integrals are unnecessary in the derivation of expressions like (4.50). See Fried (1972), for example. The path integrals are as rigorous – or non-rigorous – as the series, irrespective of whether they can be given proper mathematical definition or not.

At the same time, the analogy with finite-dimensional integrals suggests approaches that go beyond this purely operational stance. In particular, because of the dominant role played by the classical action in the path integral formalism we would like to use it to establish the relation of quantum dynamics to classical dynamics.

This is the most compelling reason for introducing the path integral formalism, requiring a distinct relaxation in our treatment of it. By formal analogy with finite-dimensional saddle-point and stationary-phase methods, elevating the classical action to the phase of the integrand promotes series expansions in ħ as the natural way to proceed.

Although we have been calculating an energy density (i.e. a static quantity) at thermal equilibrium, the underlying picture is intrinsically dynamical. Most simply, we divide the universe into two parts: a large box in which we perform calculations upon the fields; the rest of the universe, which plays the role of a thermal reservoir at temperature T. Equilibrium is achieved as a result of a dynamical two-way exchange of energy between the fields and the reservoir in which they are immersed, a real-time process.

In the calculations of the previous chapter the underlying dynamics were hidden by the imaginary-time prescription. Although there is no loss in this case, this evasion leaves us feeling somewhat cheated. For example, consider the Yukawa theory of (1.60) in which a B field interacts with A fields via the interaction ½gBA2. Suppose the mass of the B is more than twice that of the A. The population of B particles will change either by their decaying into As, or by real As sustained by the heat-bath colliding to produce them. These individual interactions in real-time will appear as discontinuities (dispersive parts) of the B-field propagator. It is advantageous to be able to calculate them directly, without recourse to analytic continuation from an Euclidean expression in which the heat bath has no role.

In this chapter we shall present a real-time formulation of scalar field theory in thermal equilibrium at finite-temperature to give a flavour of the method. The extension to more realistic theories causes no new conceptual problems.

Any approach to a real-time description has as its key ingredient the doubling of fields (and Hilbert spaces). In the heat-bath picture this can be anticipated canonically in the following way.

The functional approach that was adopted in the earlier sections for the scalar theory had two principal aims. The first was to write down equations (the Dyson–Schwinger equations) for the Green functions that were a consequence of canonical quantisation and directly reflected the nature of the particle interactions. The second was to find a reliable integral realisation for the generating functional Z[j] that manifestly satisfied these equations and thus embodied canonical quantisation. The integral form itself then suggested tactics for understanding the theory.

Both of these steps are essentially combinatoric and permit direct generalisation to more realistic theories. As a first move towards realism we shall sketch the extension of these ideas to Fermi fields. We face two separate problems. The first, and most important, is that of the need to accommodate Fermi statistics in the formalism. This will be our main task in this chapter. Secondly, we need to describe internal spin degrees of freedom, and for this we need n > 2 dimensions. Contemporary models often begin classically in large numbers of spatial dimensions. With immediate realism in mind, we are mainly interested in n = 4 dimensions from the start. (Whereas for scalar fields n = 4 dimensions was pathological in that quantum fluctuations were most likely to completely screen the bare charge g0, the argument does not naturally extend to Fermi fields (with scalars).)

When we first considered stationary-phase and saddle-point calculations in chapter 5, we assumed (correctly) that the solutions to δS[A]/δA = 0 and δSE[A]/δA = 0 of finite action were constant configurations Ac. However, there are circumstances when the extrema are non-trivial (i.e. non-constant).

In this chapter we shall examine some simple examples of saddle-point calculations, for the generating functional ZE, about non-trivial Euclidean configurations. (The saddle-point calculations of section 5.5 for large-order terms in the ħ expansion of ZE also involved non-trivial extrema. However, here we are talking about calculations of ZE itself.) Such configurations are termed instantons ('t Hooft, 1976) or pseudoparticles (e.g. see Polyakov (1977)). We shall use the former name since they play no role as classical precursors of particles. Rather, their main application is in quantum tunnelling in one aspect or another.

It is for this reason that we have introduced instantons now, rather than earlier. We have two applications in mind. The first, and a major use of these ideas, is in early universe calculations, for which the discussion of the previous two chapters has provided a background. Our second application, the θ-vacua of non-Abelian gauge theories, is not motivated by non-zero temperature effects, but is expressed through the same tunnelling tactics.

In section 6.5 we looked at the problem of evaluating quantum mechanical path integrals in curvilinear co-ordinates. For the model considered there (a point particle in two dimensions) the choice of curvilinear co-ordinates was capricious, but the use of curvilinear co-ordinates is natural in describing constrained systems. For example, suppose the particle in two dimensions was forced to move on a ring of radius R. The problem of path roughness would then go away, since the scalar curvature would be constant along the path. However, we acquire a new problem in that the configuration space for the particle (the circle) is no longer simply connected – a notion that we shall define later. This, in turn, can induce a new term in the action.

We stress that this term is in no way a replacement for that due to roughness. For one thing, it is of order ħ, rather than of order ħ2. Most importantly it is not unique, leading to an ambiguity in quantisation. This ambiguity is a general property of quantisation on non-simply-connected manifolds. In this chapter we shall sketch how such O(ħ) terms (topological ‘charges’) arise naturally in the path integral formalism which, being an ‘integral’ over paths in the configuration space of the system, is ideally suited for handling non-trivial configuration spaces.

So far our approximations have been wholly analytic, essentially based on perturbation series. The calculation of (g − 2) for the electron has shown just how successful this approach can be.

However, in many ways (g − 2) is an exception. Most quantities that we would like to calculate (e.g. the pion mass in quantum chromodynamics, the field theory of the strong interactions) cannot be approached analytically. Although analytic methods do give us a lot of information (that is, after all, what this book is about) it is often of a qualitative nature. For this reason a variety of numerical methods have been developed for approximating the path integral directly. Bypassing problems of measure, they typically involve putting the theory on a Euclidean lattice, as in the case of the regularised ZΛ(j) of (6.74). The resulting integrals can then be calculated directly, although a sizeable lattice requires considerable computing resources. All the uncertainties of the method are displaced into the problem of recovering the continuum limit.

It is often difficult to link such tactics to the analytic ideas of continuum field theory. Before turning to more realistic theories we shall give a brief description of an alternative interpretation of formal Euclidean path integrals, proposed by Parisi and Wu (1981).

Non-Abelian gauge theories play a dual role in the construction of realistic field theories. On the one hand, quantum chromodynamics (QCD) deals with the SU(3) ‘colour’ interaction between quarks and gluons in the construction of hadrons. Technically it provides a generalisation of the original non-Abelian SU(2) gauge theory proposed by Yang & Mills (1954), although the context is very different. On the other hand, the massive gauge particles that mediate the electroweak interactions are explained through a non-Abelian gauge theory in which symmetry breaking induces gauge field masses without interfering with renormalisability.

We shall turn to this latter use in later chapters. In this chapter we shall just highlight a few of the properties of a pure unbroken non-Abelian gauge theory like QCD (in n = 4 dimensions). The emphasis will be on using the path integral to quantise the theory, developing ideas that have already been introduced in QED (and scalar theories). In particular, the path integral is ideal for displaying gauge identities, which are necessarily more complicated because of the non-Abelian nature of the gauge group. Further consequences of having a non-Abelian group e.g. the existence of θ-vacua, will be postponed until later chapters. For a fuller discussion of non-Abelian gauge theories the reader is referred to the standard texts (e.g. Abers & Lee, 1973).

In this book we shall be almost exclusively concerned with the interactions of relativistic particles that are the quanta of elementary fields. There is some ambiguity in the definition of ‘elementary’, but by it we mean local fields whose propagation and interactions can be described by a local Hamiltonian, or Lagrangian, density. Individual terms in these densities describe the basic transformations that the quanta can undergo. For example, if the classical Lagrangian density for a field A has a quartic gA4 interaction we assume that, quantum mechanically, one A-particle can turn directly into three (virtual) A-particles. The way in which these virtual particles further split or recombine determines the way in which A-particle interactions take place.

The aim of this first chapter is to indicate how canonical quantisation (i.e. the Hamiltonian formulation) can be reformulated as statements about how particles interact. The quantification of the qualitative statement that one A-particle can turn into three, or whatever, will occur through a set of relations termed the Dyson–Schwinger equations. In our approach these equations will play a critical role in formulating an alternative quantisation of field theory through path integrals. The path integral formulation, rather than the canonical approach, will be at the centre of all our calculational methods.

The discussion of the previous chapter has given us some understanding of the effective potential for the standard GSW model. A similar analysis can be performed for any grander unified theory of elementary fields.

However, the pattern of symmetry breaking that we now observe was not always present. Standard cosmology predicts that in the early states of the universe there was a large matter and radiation density at high temperature. On the basis of simple arguments from statistical mechanics we would not expect the symmetries of such a system to be those we experience now.

The existence of different phases may seem of marginal interest, since the cooling down of the universe to what is effectively absolute zero occurred in the distant past. This is not so. The reason is that different field theories for current (zero-temperature) unification will, in general, have different cosmological implications if used to describe the early evolution of the universe. Given that the mass scales introduced by grand unification are much higher than those accessible to accelerator physics, cosmological predictions provide an important means of discrimination between candidate theories.

We shall not attempt early-universe calculations. In this chapter we only consider the first step, the calculation of temperature-dependent quantum effects in field theories at non-zero temperature.

In the previous chapters we have only quantised classical fields, in that we have assumed that the quanta of the theory can be identified with fields already present in the classical action.

We know that this is untrue for hadrons, which form the overwhelming majority of any particle listing. QCD tells us that, in some complicated way, all hadrons are composite entities made from gluons and quarks. For this reason we have avoided applications that have involved them. However, arguments can be made for a much greater degree of compositeness than hadrons alone. As we have seen, the number of quarks and leptons in grand unified models is considerable. We have already mentioned three families. There may be more. It is quite possible that leptons and quarks are not elementary entities but are themselves composite.

Secondly, we have noted in passing that the breaking of symmetry by elementary Higgs fields requires incredibly fine tuning of parameters. This is a pointer towards taking the classical field theory as an effective low-energy approximation for a theory with composite Higgs particles. Further limitations to the classical models exist that might by improved by extending compositeness to the heavy gauge bosons and even to photons and gluons.

The literature on compositeness is extensive. As an example, we cite Peccei (1983) for a summary of recent ideas.