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Journal of applied mathematics & informatics

METHODS FOR ITERATIVE DISENTANGLING IN FEYNMAN'S OPERATIONAL CALCULI : THE CASE OF TIME DEPENDENT NONCOMMUTING OPERATORS

Byung Moo Ahn

Published May 1, 2010• 0 citations
OpenAlex

Abstract

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.

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Russian Journal of Mathematical Physics•Apr 1, 2014•1 citation
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.
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.
Feynman's method for disentangling noncommuting operators is discussed and applied to nonstationary problems in quantum mechanics, including the excitation of a harmonic oscillator by an external force and/or by time-varying frequency; spin rotation in a time-varying magnetic field; the disentangling of an atom (ion) Hamiltonian in a laser field; a model with the hidden symmetry group of the hydrogen atom; and the theory of coherent states. The Feynman operator calculus combined with simple group-theoretical considerations allows disentangling the Hamiltonian and obtaining exact transition probabilities between the initial and final states of a quantum oscillator in analytic form without cumbersome calculations. The case of a D-dimensional oscillator is briefly discussed, in particular, in application to the problem of vacuum pair creation in an intense electric field.
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.