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.

The use of functional in the quantisation of relativistic local field theories has a long history, going back to the work of Schwinger and Symanzik in the 1950s. As exemplified by the generating functionals for Green functions, they can embody the canonical Hamiltonian results in a very convenient way (the Dyson–Schwinger equations). By the end of the 1960s the use of functionals had become standard practice (see Fried's book of 1972 with essentially the same title as this) but it was by no means obligatory. Straightforward manipulation of Feynman diagrams was often sufficient.

The renaissance of field theory in the 1970s after the failure of the pure S-matrix approach was based on models that were much more complicated than those used hitherto. Non-Abelian gauge theories, spontaneous symmetry breaking, supersymmetry (to name but three essential steps) required a reappraisal of tactics. At the same time the quantities that needed to be calculated changed in character. Rather than S-matrix elements it was important to determine free energies, tunnelling decay-rates, critical temperatures, Wilson loops, etc.

These quantities, with a natural definition via the standard functionals of the theory, required an approach based on them. The ingredient that gave the functional approach the additional power to cope with the new complexity was the side-stepping of canonical Hamiltonian methods by the use of path integrals to represent the functionals.

So far everything has been extremely formal, a euphemism for the fact that almost any diagram that we write down will be represented by an integral that does not exist. This is because products of the distributions ΔF with coincident space-time points are not defined. Moreover, the presence of singularities arising from such formal products is not peculiar to the scalar theories that we have considered so far, but to all local field theories.

Before considering tactics for handling this problem there is a question of philosophy. One fact that has been hammered home by models that unify the different forces of nature is that, in each step towards a common unity, a new and yet higher mass scale is introduced. With each breakthrough in our understanding the ‘best’ current model has been seen to be a ‘low’-energy effective model for a yet more complete theory that contains it as a subsector. Further, a field that is ‘elementary’ at one level may be composite (e.g. a bound state of new elementary fields) at the next.

It is true that models giving rise to unavoidable infinities may still permit useful calculations at ‘tree’ level (e.g. the four-Fermi model of weak interactions). Nonetheless, the desire for a consistent theory with no infinities at each step has been a crucial force in model-making.

Introduction

The interacting boson model-1 originated from early ideas of Feshbach and Iachello (Iachello, 1969; Feshbach and Iachello, 1973, 1974), who in 1969 described some properties of light nuclei in terms of interacting bosons, and from the work of Janssen, Jolos and Dönau (1974), who in 1974 suggested a description of collective quadrupole states in nuclei in terms of a SU(6) group. The latter description was subsequently cast into a different mathematical form by us (Arima and Iachello, 1975) with the introduction of an s-boson, which made the SU(6), or rather U(6), structure more apparent. The success of this phenomenological approach to the structure of nuclei has led to major developments in the understanding of nuclear structure.

The major new development was the realization that the bosons could be interpreted as nucleon pairs (Arima et al., 1977) in much the same way as Cooper pairs in the electron gas (Cooper, 1956). This provided a framework for a microscopic description of collective quadrupole states in nuclei and stimulated a large number of theoretical investigations. An immediate consequence of this interpretation was that, since one expected both neutron and proton pairs, one was led to consider a model with two types of bosons, proton bosons and neutron bosons. In order to make the distinction between proton and neutron bosons more apparent, the resulting model was called the interacting boson model-2, while the original version retained the name of interacting boson model-1.