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Russian Journal of Mathematical Physics

Generalized measure permutation formulas in Feynman’s operational calculi

Byung-Moo Ahn, B. S. Kim, I. Yoo

Published Apr 1, 2014• 1 citation
Publisher

Abstract

Measure permutation formulas in Feynman’s operational calculi for noncommuting operators give relationships between the two operators $$\mathcal{T}_{\mu 1,\mu 2} f\left( {\tilde A,\tilde B} \right)$$ and $$\mathcal{T}_{\mu 2,\mu 1} f\left( {\tilde A,\tilde B} \right)$$ . We develop generalized and iterated measure permutation formulas in the Jefferies-Johnson theory of Feynman’s operational calculi. In particular, we apply our formulas to derive an identity for a function of the Pauli matrices.

DOI
10.1134/s1061920814020010

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Measure permutation formulas in Feynman’s operational calculi for noncommuting operators give relationships between the two operators Tμ1,μ2f(A~,B~)\mathcal{T}_{\mu 1,\mu 2} f\left( {\tilde A,\tilde B} \right)Tμ1,μ2​f(A~,B~) and Tμ2,μ1f(A~,B~)\mathcal{T}_{\mu 2,\mu 1} f\left( {\tilde A,\tilde B} \right)Tμ2,μ1​f(A~,B~) . We develop generalized and iterated measure permutation formulas in the Jefferies-Johnson theory of Feynman’s operational calculi. In particular, we apply our formulas to derive an identity for a function of the Pauli matrices.
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.
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.
Abstract The existence of a real linear space structure on the set of observables of a quantum system—i.e., the requirement that the linear combination of two generally non-commuting observables A, B is an observable as well—is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form aA+bBaA+bBaA+bB a A + b B ( a,b∈Ra,b\in {{\mathbb {R}}}a,b∈R a , b ∈ R ) if such measuring instruments are given for the addends observables A and B when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of f(aA+bB)f(aA+bB)f(aA+bB) f ( a A + b B ) out of the spectral measures of A and B. We present such a construction with a formula which is valid for general unbounded self-adjoint operators A and B, whose spectral measures may not commute, and a wide class of functions f:R→Cf: {{\mathbb {R}}}\rightarrow {{\mathbb {C}}}f:R→C f : R → C . In the bounded case, we prove that the Jordan product of A and B (and suitably symmetrized polynomials of A and B) can be constructed with the same procedure out of the spectral measures of A and B. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman–Kac formula.
This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n -fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.
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].