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).
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).
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).
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics, and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications. As well as the canonical examples of a quantum particle in a single and a double potential well, we discuss the generalization of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various concepts of field theory, namely stationary phase analysis, the Euclidean formulation of field theory, instanton techniques, and the role of topology in field theory, are introduced in this chapter. The path integral: general formalism Broadly speaking, there are two approaches to the formulation of quantum mechanics: the “operator approach” based on the canonical quantization of physical observables and the associated operator algebra, and the Feynman path integral.
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics, and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications. As well as the canonical examples of a quantum particle in a single and a double potential well, we discuss the generalization of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we will temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of functional field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various techniques and concepts of field theory, namely stationary phase analyses of functional integrals, the Euclidean formulation of field theory, instanton techniques, and the role of topological concepts in field theory, will be motivated and introduced in this chapter.
To introduce the formalism of the functional integral in quantum mechanics, this chapter starts from the definition of the temporal evolution operator. It is important to emphasize that in the classical limit is not only the classic trajectory that have the grater contribution, but all the paths close to this also contribute. The chapter shows how integral formulation is perfectly consistent with Schrödinger's time-dependent differential equation. It retrieves this equation from definition of transition amplitude as a path integral. The chapter discusses functional derivative generalized to the case of the ordinary functional derivative. A very useful tool to solve problems related to functional integrals is the Wick rotation or Euclidean rotation. In quantum mechanics there are few problems we can solve exactly, and very often, when possible, we use perturbation theory to find an approximate solution. The chapter follows this approach with formalism of functional integration and construct a perturbation theory from which can be obtained a physical interpretation.
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.
To describe physics properly, we need to take into account quantum effects. Thus, for every non-quantum physical theory, we must come up with an appropriate quantum theory. A traditional approach is to replace all the scalars in the classical description of this theory by the corresponding operators. The problem with the above approach is that due to non-commutativity of the quantum operators, two mathematically equivalent formulations of the classical theory can lead to different (non-equivalent) quantum theories. An alternative quantization approach that directly transforms the non-quantum action functional into the appropriate quantum theory, was indeed proposed by the Nobelist Richard Feynman, under the name of path integration. Feynman path integration is not just a foundational idea, it is actually an efficient computing tool (Feynman diagrams).\nFrom the pragmatic viewpoint, Feynman path integral is a great success. However, from the foundational viewpoint, we still face an important question: why the Feynman's path integration formula? In this paper, we provide a natural explanation for Feynman's path integration formula.