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Birkhäuser Boston eBooks

Time Maps in the Study of Feynman’s Operational Calculus via Wiener and Feynman Path Integrals

G. W. Johnson, Lisa Marie Johnson

Published Jan 1, 2001• 16 citations
Publisher

Abstract

It is known that Wiener and Feynman path integrals provide one way of making Feynman’s heuristic operational calculus for noncommuting operators mathematically rigorous. The disentangling process and associated operator orderings are central to Feynman’s ideas. We begin here to study the effects of time maps in clarifying the disentangling process and in altering the operator orderings in certain prescribed ways. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-0167-0_10

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It is known that Wiener and Feynman path integrals provide one way of making Feynman’s heuristic operational calculus for noncommuting operators mathematically rigorous. The disentangling process and associated operator orderings are central to Feynman’s ideas. We begin here to study the effects of time maps in clarifying the disentangling process and in altering the operator orderings in certain prescribed ways. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Disentangling is the essential operation of Feynman's operational calculus for noncommuting operators. Thus formulas which simplify this operation are central to the subject. In a recent paper the procedure for 'extracting a linear factor' has been established in the setting of Feynman's operational calculus for time independent operators A1,...,AnA_1, ... , A_nA1​,...,An​ and associated probability measures μ1,...,μn{\mu}_1,..., {\mu}_nμ1​,...,μn​ . While the setting just described is natural in many circumstances, it is not natural for evolution problems. There the measures should not be restricted to probability measures and it is worthwhile to allow the operators to depend on time. The main purpose for this paper is to extend the procedure for extracting a linear factor to this latter setting. We should mention that Feynman's primary motivation for developing an operational calculus for noncommuting operators came from a desire to describe the evolution of certain quantum systems.m systems.
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].
We investigate what happens if in the Feynman‐Kac functional, we perform the time integration with respect to a Borel measureηrather than ordinary Lebesgue measurel. Letu(t) be the operator associated with this functional through path integration. We show thatu(t), considered as a function of timet, satisfies a certain Volterra‐Stieltjes integral equation. This result establishes a “FeynmanKac formula with Lebesgue‐Stieltjes measureη.” One recovers the classical Feynman‐Kac formula by lettingη=l. We deduce from the integral equation thatu(t) satisfies a differential equation associated with the continuous partμofηwhenη=μ=l, this differential equation reduces to the heat or the Schrödinger equation in the probabilistic or quantum‐mechanical case, respectively. Moreover, we observe a new phenomenon, due to the discrete partvofη: the functionu(t) undergoes a discontinuity at every point in the support ofv, assumed here to be finite. Further, one obtains an explicit expression foru(t) in terms of operators alternatively associated withμandv. Our results are new even in the probabilistic or “imaginary time” case and allow us to unify various concepts. The derivation of our integral equation has an interesting combinatorial structure and makes essential use of the “generalized Dyson series”— recently introduced by G. W. Johnson and the author—that “disentangle” the operatoru(t). We provide natural physical interpretations of our results in both the diffusion and quantum‐mechanical cases. We also suggest further connections with FeynmanȈs operational calculus for noncommuting operators.
Up to this point we have focused on temporal lattice regularizations in order to give meaning to the traditional configuration and phase space path integrals including coherent state phase space path integrals. In this chapter, we take up the question of introducing a continuous-time regularization procedure into a formal phase space path integral. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The map from the commutative algebra to the noncommutative algebra of operators is the essential operation of Feyn- man's operational calculus for noncommuting operators. Thus formulas which simplify this operation are meaningful to the subject. In a recent paper the procedure for for iterative disentangling has been es- tablished in the setting of Feynman's operationalcalculus for time indepen- dent operatorsA1, · · · ,An and associatedprobabilitymeasures µ1, · · · ,µn. The main purpose for this paper is to extend the procedure for methods for iterative to time dependent operators.
Bernstein diffusions belong to a new class of time symmetric (but not time homogeneous) stochastic processes associated with the quantum dynamics of nonrelativistic particles in potentials. It is shown in which sense Feynman’s formal path integral method can be interpreted in terms of those processes, specially for the subset of Gaussian Bernstein diffusions. The familiar Ornstein-Uhlenbeck process becomes, in this framework, a particular Gaussian process in a large class of Bernstein diffusions, all associated with the same dynamics. The method is also illustrated for some Hilbert space valued Gaussian Diffusions These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.