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CRC Press eBooks

Propagators and Feynman Path Integral

Nandita Rudra

Published Aug 19, 2019• 0 citations
Publisher

Abstract

The path integral approach to quatum mechanics was developed by Feynman when he was a graduate student at Princeton University. The method provides a deep insight into quantum dynamics. Due to computational complexity the path integral formalism is not convenient to deal problems in non-relativistic quantum mechanics. It, however, provides an excellent method for quantizing quantum fields and has become a powerful tool in quantum field theory, statistical physics and numerical computation.

DOI
10.1201/9780429325762-5

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The path integral approach to quatum mechanics was developed by Feynman when he was a graduate student at Princeton University. The method provides a deep insight into quantum dynamics. Due to computational complexity the path integral formalism is not convenient to deal problems in non-relativistic quantum mechanics. It, however, provides an excellent method for quantizing quantum fields and has become a powerful tool in quantum field theory, statistical physics and numerical computation.
The path integral is a powerful tool for studying quantum mechanics because it has the merit of establishing the connection between classical mechanics and quantum mechanics. This formalism quickly gained prominence in various fields of theoretical physics, including its generalization to quantum field theory, quantum mechanics, and statistical physics. Using the Feynman propagator, we can calculate the partition function, the free energy, wave functions, and the energy spectrum of the considered physical system. Moreover, the Feynman formalism finds broad applications in geophysics and in the field of financial sciences.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman's approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.
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.
These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions, quantization of gauge theories etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.