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Institution of Engineering and Technology eBooks

Feynman path integral

Alireza R. Baghai-Wadji

Published Dec 18, 2023• 0 citations
Publisher

Abstract

This chapter introduced the concept of the wavenumber operator rather than employing the standard notion of the momentum operator. The introduction and usage of the wavenumber operator were hailed as a significant achievement. Indeed the introduction of the wavenumber operator significantly simplifies the discussion. The Feynman path integral was derived using the Trotter product formula. The chapter presented schemes for the symmetrization and then the generalization of the standard Trotter product rule, which is the quintessential tool in establishing the Feynman path integral. The proposed symmetrization improves the convergence rate of the calculations by one order. The standard Trotter product rule merely involves two operators. The proposed generalization admits the involvement of any finite number of operators. The introduction of the wavenumber operator and the symmetrization and generalization of the Trotter product formula have been presented in this chapter for the first time. The reader may consult the references [1-16] further below to deepen their understanding and hopefully appreciate the simplifications and generalizations achieved in this chapter.

DOI
10.1049/sbew569g_ch5

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Feynman path integral

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This chapter introduced the concept of the wavenumber operator rather than employing the standard notion of the momentum operator. The introduction and usage of the wavenumber operator were hailed as a significant achievement. Indeed the introduction of the wavenumber operator significantly simplifies the discussion. The Feynman path integral was derived using the Trotter product formula. The chapter presented schemes for the symmetrization and then the generalization of the standard Trotter product rule, which is the quintessential tool in establishing the Feynman path integral. The proposed symmetrization improves the convergence rate of the calculations by one order. The standard Trotter product rule merely involves two operators. The proposed generalization admits the involvement of any finite number of operators. The introduction of the wavenumber operator and the symmetrization and generalization of the Trotter product formula have been presented in this chapter for the first time. The reader may consult the references [1-16] further below to deepen their understanding and hopefully appreciate the simplifications and generalizations achieved in this chapter.
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.
Abstract The path integral method discussed up to the preceding section is based on Trotter’s formula (1.10) and its accuracy is of the order of Δt for the time slice Δt. This is sufficient for discussing quantum systems qualitatively. Higher accuracy of the Quantum Monte method is required for studying quantum systems numerically. In the present chapter, we explain the algebraic generalization of the path integral method. By applying the integral representations given in chapters 1−4 to such generalized product formulae, some new kinds of path integral of higher accuracy can be obtained. This is crucial when the essence of a phenomenon can be found only by increasing the accuracy of calculations. It should be emphasized that such higher order decomposition formulae to be explained in the present chapter can also be applied not only to the path integral method but to many other problems such as non-linear dynamics in Hamiltonian systems.
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.
Operator calculusThe operator calculus developed by Feynman [Fey51] makes it possible to represent functions of (noncommuting) operators as path integrals, with the integrand being the path-ordered exponential of operators, the order of which is controlled by a parameter that varies along the trajectory.This procedure is termed Feynman disentangling.It is also applicable to functions of matrices (say, γ-matrices which are associated with a spinor particle).When applied to the evolution operator, this procedure results in the standard path-integral representation of quantum mechanics.In this chapter we first demonstrate the general technique using the simplest example, a free propagator in Euclidean space, and then consider the path-integral representation of quantum mechanics, as well as propagators in an external electromagnetic field. Free propagatorLet us first consider the simplest propagator of a free scalar field which is given in the operator formalism by the vacuum expectation value of thewith ϕ being the field-operator.The T -product (1.1) obeys the equationwhere d = 4 is the dimension of space-time, however the formulas are applicable at any value of d.In the operator formalism, Eq. (1.2) is a * The ordered products of operators were introduced by Dyson [Dys49].This paper and other classical papers on quantum electrodynamics are collected in the book edited by Schwinger [Sch58].