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Oxford University Press eBooks

INTRODUCTION TO PATH INTEGRALS

Dante Gatteschi, Roberta Sessoli, Jacques Villain

Published Mar 2, 2006• 1 citation
Publisher

Abstract

This chapter introduces the formalism of field theory applied to tunnelling effects, including imaginary time and path integrals. The theory is then specialized by applying the path integral method to spins, and the concept of instanton is introduced. For spins, two paths are possible which lead to interference effects akin to Berry phase.

DOI
10.1093/acprof:oso/9780198567530.003.0007

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This chapter introduces the formalism of field theory applied to tunnelling effects, including imaginary time and path integrals. The theory is then specialized by applying the path integral method to spins, and the concept of instanton is introduced. For spins, two paths are possible which lead to interference effects akin to Berry phase.
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.
In this chapter we show that it is possible to describe quantum mechanics by using path integrals. This idea was depeloped by Richard Feynman on the basis of an earlier intuition of Paul Dirac. We discuss the path integral approach starting with the problem of a single quantum particle in the presence of an external potential. We introduce the concept of the saddle point and the stationary phase approximation, which is solved by using Fresnel integrals. We show that Gaussian integrals are crucial in the imaginary time formulation of the path integral, which is the appropriate framework for studying quantum systems at thermal equilibrium. Finally, we analyze the quantum tunneling problem, deriving an elegant formula for the tunneling probability based on the simple, but meaningful, saddle point approximation.
Quantum field theory (QFT) provides us with one and almost only suitable language (or mathematical tool) for describing not only the motion and interaction of particles but also their “annihilation” and “creation” out of a field considered a priori in a sophisticated way, whose view seems to be suited for describing dislocations, as a particle or a string embedded within a crystalline ordered field. This chapter concisely overviews the method of QFT, emphasizing distinction from the quantum mechanics, conventionally used for a single and/or many particle problems, and its equivalence to the statistical mechanics. The alternative formalism based on Feynman path integral and its imaginary time representation are reviewed, as the foundation for our use in Chapter 10.