Chapter 3
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 double potential well, we discuss the generalisation 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 quantisation 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, viz. 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.
3.1 The Path Integral: General Formalism
Broadly speaking, there are two basic approaches to the formulation of quantum mechan- ics: the ‘operator approach’ based on the canonical quantisation of physical observables
Concepts in Theoretical Physics 64 CHAPTER 3. FEYNMAN PATH INTEGRAL
together with the associated operator algebra, and the Feynman1 path integral.2 Whereas canonical quantisation is usually taught first in elementary courses on quantum mechan- ics, path integrals seem to have acquired the reputation of being a sophisticated concept that is better reserved for advanced courses. Yet this treatment is hardly justified! In fact, the path integral formulation has many advantages most of which explicitly support an intuitive understanding of quantum mechanics. Moreover, integrals — even the infinite dimensional ones encountered below — are hardly more abstract than infinite dimensional linear operators. Further merits of the path integral include the following: