Phase space propagators for quantum operators

• Published in: Annals of physics Vol. 321; no. 8; pp. 1790 - 1813
• Main Authors: Ozorio de Almeida, A.M., Brodier, O.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.08.2006; Elsevier BV
• Subjects: CANONICAL TRANSFORMATIONS; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DEGREES OF FREEDOM; FOURIER TRANSFORMATION; Fourier transforms; FUNCTIONS; MATHEMATICAL EVOLUTION; PHASE SPACE; Phase transitions; Physics; PROPAGATOR; QUANTUM MECHANICS; QUANTUM OPERATORS; Quantum theory; SEMICLASSICAL APPROXIMATION; WIGNER THEORY
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2006.03.007

The Heisenberg evolution of a given unitary operator corresponds classically to a fixed canonical transformation that is viewed through a moving coordinate system. The operators that form the bases of the Weyl representation and its Fourier transform, the chord representation are, respectively, unitary reflection and translation operators. Thus, the general semiclassical study of unitary operators allows us to propagate arbitrary operators, including density operators, i.e., the Wigner function. The various propagation kernels are different representations of the super-operators which act on the space of operators of a closed quantum system. We here present the mixed semiclassical propagator, that takes translation chords to reflection centres, or vice versa. In contrast to the centre–centre propagator that directly evolves Wigner functions, they are guaranteed to be caustic free, having a simple WKB-like universal form for a finite time, whatever the number of degrees of freedom. Special attention is given to the near-classical region of small chords, since this dominates the averages of observables evaluated through the Wigner function.