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Physics-Uspekhi

Feynman disentangling of noncommuting operators and group representation theory

V. S. Popov

Published Dec 31, 2007• 19 citations
Publisher

Abstract

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.

DOI
10.1070/pu2007v050n12abeh006401

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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.
Feynman’s disentangling theorem is applied to noncommuting operators in the problem of quantum parametric oscillator, which is mathematically equivalent to the problem of SU (1, 1) pseudospin rotation. The number states of the oscillator correspond to unitary irreducible representations of the SU (1, 1) group. Feynman disentangling is combined with group-theoretic arguments to obtain simple analytical formulas for the matrix elements and transition probabilities between the initial and final states of the oscillator. Feynman disentangling of time evolution operators is also discussed for an atom or ion interacting with a laser field and for a model Hamiltonian possessing the “ hidden” symmetry of the hydrogen atom.
The notational ambiguities in Feynman's calculus are all remedied here by setting a more natural foundation of the ordered exponential operators, which will be called briefly “expansional” operators in this paper. The essential point to be stressed is than the pure exponential operator is merely a special case of the more wider class of operators, i.e., expansional operators, and the latter type of operators generally appears in quantum mechanics. A clear-cut distinction between between these two types of operators is all important. The so-called disentangling process is a device to decompose any one expansional operator into a product of some simple exponential operators. In the first place the general view-points concerning the expansional operators are presented, in which a rigorous organization of disentangling procedure is accomplished. Next are given some examples treating a generalized forced harmonic oscillator, where the transformation operator is completely disentangled and its representative yields automatically the classical action in its compact form, which process constitutes the original intention of this work. No accounts are given here concerning the quantized field theory.
This book covers the entire span of quantum mechanics whose developments have taken place during the early part of the twentieth century up till the present day. We start with the Rutherford-Bohr model of the atom followed by Schrodinger's wave mechanics with its application to the solution of calculating the energy spectrum of a particle in a box, the harmonic oscillator and finally the hydrogen atom. Heisenberg's matrix mechanics and its duality with Schrodinger's wave mechanics, quantum mechanics in the interaction picture. Dirac's relativistic theory of the electron exhibiting the spin of the electron as a relativistic effect when it interacts with an external electromagnetic field. Feynman's path integral approach to non-relativistic quantum mechanics with is a marvellous intuitive interpretation as a sum over paths and how classical mechanics is obtained from its limit as Planck' constant tends to zero, methods for computing the spectra of the Dirac Hamiltonian in a radial potential, quantum field theory as developed by Feynman, Schwinger, Tomonaga and Dyson for describing the interaction between electrons, positrons, and photons via propagators using both the operator theoretic expansions and Feynman's path integral. We also introduce time independent and time dependent perturbation theory in quantum mechanics with applications to quantum gate design for quantum computers forming a major part of the research conducted by the author's research group, Quantum noise introduced into the Schrodinger and Dirac's equation based on the Hudson-Parthasarathy quantum stochastic calculus in Boson Fock space, scattering theory and wave operators with applications to quantum gate design, some aspects of second quantization like the interpretation of Boson Fock space in terms of harmonic oscillator algebras and the BCS theory of superconductivity, Wigner-Mackey-Frobenius theory of induced representations of a group with applications to Wigner's theory of particle classification, Dirac's equation in a gravitational field and Yang-Mills non-Abelian gauge theories with application to the construction of unified quantum field theories and finally, the more recent theory of super-symmetry which is a Boson-Fermion unification theory. We have discussed the statistics of Boson's, Fermions and Maxwell-Boltzmann based on entropy maximization. The book is written in problem-solution format and it would be of use to physicists and engineers interested respectively in developing unified field theories and in the design of quantum gates. Note: T&F does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.
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.