Feynman disentangling оf noncommuting operators and group representation theory

Vladimir S Popov

V. S. Popov

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.

Vladimir S Popov

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.

Vladimir S Popov, M. A. Trusov

The problem of the evolution of a singular quantum oscillator with a frequency exhibiting an arbitrary time dependence has been solved. The probabilities w mn of transitions in the oscillator spectrum and generating functions have been calculated, and the sum rules for w mn have been derived. The calculations are based on the Feynman disentangling method and the theory of representations of the SU(1, 1) group.

Masahito Hayashi