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.
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.
In this chapter, the concept of path integration is generalized to integration over quantum fields. Specifically we will develop an approach to quantum field theory that takes as its starting point an integration over all configurations of a given field, weighted by an appropriate action. To emphasize the importance of the formulation that, methodologically, represents the backbone of the remainder of the text, we have pruned the discussion to focus only on the essential elements. This being so, conceptual aspects stand in the foreground and the discussion of applications is postponed to the following chapters. In this chapter, the concept of path integration is extended from quantum mechanics to quantum field theory. Our starting point is a situation very much analogous to that outlined at the beginning of the previous chapter. Just as there are two different approaches to quantum mechanics, quantum field theory can also be formulated in two different ways: the formalism of canonically quantized field operators, and functional integration. As to the former, although much of the technology needed to efficiently implement this framework – essentially Feynman diagrams – originated in high-energy physics, it was with the development of condensed matter physics through the 1950s, 1960s, and 1970s that this approach was driven to unprecedented sophistication. The reason is that, almost as a rule, problems in condensed matter investigated at that time necessitated perturbative summations to infinite order in the non-trivial content of the theory (typically interactions).
In this chapter, the concept of path integration is generalized to integration over quantum fields. Specifically we will develop an approach to quantum field theory that takes as its starting point an integration over all configurations of a given field, weighted by an appropriate action. To emphasize the importance of the formulation that, methodologically, represents the backbone of the remainder of the text, we have pruned the discussion to focus only on the essential elements. This being so, conceptual aspects stand in the foreground and the discussion of applications is postponed to the following chapters. In this chapter, the concept of path integration will be extended from quantum mechanics to quantum field theory. Our starting point will be from a situation very much analogous to that outlined at the beginning of the previous chapter. Just as there are two different approaches to quantum mechanics, quantum field theory can also be formulated in two different ways: the formalism of canonically quantized field operators, and functional integration. As to the former, although much of the technology needed to efficiently implement this framework – essentially Feynman diagrams – originated in high-energy physics, it was with the development of condensed matter physics through the 1950s, 1960s, and 1970s that this approach was driven to unprecedented sophistication. The reason is that, almost as a rule, problems in condensed matter investigated at that time necessitated perturbative summations to infinite order in the non-trivial content of the theory (typically interactions).
The integrals in the path integral are strongly oscillating and can only be defined by analytic continuation. It is important to note that the continuous expression is just a notation for the discrete version of the path integral, but formal manipulations will be much easier to perform in this continuous formulation. Furthermore, the integral is only defined through the analytic continuation in time. As long as the Hamiltonian is quadratic in the momenta, the stationary phase approximation for the momentum integral is exact. In practical situations one splits from the action the quadratic part in the coordinates and velocities and considers the rest as a perturbation.
Introduction Preliminary remarks The path integral formalism is one of the most important tools in modern theoretical physics. Integration in functional spaces was considered for the first time by Norbert Wiener in his studies of diffusion processes [60, 61]. The corresponding integration measure is now known as the Wiener measure, i.e. the so-called Gaussian path integrals were studied. In quantum mechanics, functional integrals were introduced by R. Feynmann [62, 63] who was inspired by important observations due to P. A. M. Dirac concerning a relation between wave function and classical action [64, 65]. In modern physics the path integral formalism is applied not only to developing a quantum description for systems with infinitely many degrees of freedom as, for example, in quantum field theories or condensed matter physics, but also to ordinary quantum mechanical problems. Moreover there is a tendency to reformulate standard quantum mechanical problems (such as a particle in a “box” or potential well [66, 67], or a particle in the Coulomb field [68]) within the framework of the path integral method. The reason for doing this is rather obvious: physicists develop a new mathematical tool, a new language and, in order to comprehend all of its subtleties, it is necessary to investigate as many concrete examples as possible. Use of the path integral formalism is widespread in quantum field theory because it is the most adequate mathematical tool for a wide range of problems which allows, not only for elegant formulation of problems, but also for a great deal of simplification of their theoretical analysis.
In this chapter we discuss theories which are rigorous in their formulation but which in order to be useful need to be modified by introducing approximations of some kind. The approximations we are interested in are those which involve introduction of classical mechanical concepts, that is, the classical picture and/or classical mechanical equations of motion in part of the system. At this point, we wish to distinguish between “the classical picture,” which is obtained by taking the classical limit ħ → 0 and the appearance of “classical equations of motion.” The latter may be extracted from the quantum mechanical formulation without taking the classical limit—but, as we shall see later by introducing a certain parametrization of quantum mechanics. Thus there are two ways of introducing classical mechanical concepts in quantum mechanics. In the first method, the classical limit is defined by taking the limit ħ → 0 either in all degrees of freedom (complete classical limit) or in some degrees of freedom (semi-classical theories). We note in passing that the word semi-classical has been used to cover a wide variety of approaches which have also been referred to as classical S-matrix theories, quantum-classical theories, classical path theory, hemi-quantal theory, Wentzel Kramer-Brillouin (WKB) theories, and so on. It is not the purpose of this book to define precisely what is behind these various acronyms. We shall rather focus on methods which we think have been successful as far as practical applications are concerned and discuss the approximations and philosophy behind these. In the other approach, the ħ-limit is not taken—at least not explicitly— but here one introduces “classical” quantities, such as, trajectories and momenta as parameters, and derives equations of motion for these parameters. The latter method is therefore one particular way of parameterizing quantum mechanics. We discuss both of these approaches in this chapter. The Feynman path-integral formulation is one way of formulating quantum mechanics such that the classical limit is immediately visible [3]. Formally, the approach involves the introduction of a quantity S, which has a definition resembling that of an action integral [101].
These lectures are intended as an introduction to the technique of path integrals and their applications in physics. The audience is mainly first-year graduate students, and it is assumed that the reader has a good foundation in quantum mechanics. No prior exposure to path integrals is assumed, however. The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuitive than the usual approaches. Applications of path integrals are as vast as those of quantum mechanics itself, including the quantum mechanics of a single particle, statistical mechanics, condensed matter physics and quantum field theory. After an introduction including a very brief historical overview of the subject, we derive a path integral expression for the propagator in quantum mechanics, including the free particle and harmonic oscillator as examples. We then discuss a variety of applications, including path integrals in multiply-connected spaces, Euclidean path integrals and statistical mechanics, perturbation theory in quantum mechanics and in quantum field theory, and instantons via path integrals. For the most part, the emphasis is on explicit calculations in the familiar setting of quantum mechanics, with some discussion (often brief and schematic) of how these ideas can be applied to more complicated situations such as field theory.