Propagators and Canonical Transformations (original) (raw)
Related papers
Quantum Mechanics in Phase Space: An Overview with Selected Papers (World Scientific)
Wigner's quasi-probability distribution function in phase space is a special (Weyl-Wigner) representation of the density matrix. It has been useful in describing transport in quantum optics; nuclear physics; and quantum computing, decoherence, and chaos. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter century: It furnishes a third, alternative formulation of quantum mechanics, independent of the conventional Hilbert space or path integral formulations. In this logically complete and self-standing formulation, one need not choose sides between coordinate and momentum space. It works in full phase-space while accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but which compose together in novel algebraic ways. This volume is a selection of 23 classic and/or useful papers about the phase-space formulation, with an introductory overview that provides a trail-map to these papers, and with an extensive bibliography. The overview collects often-used formulas and simple illustrations, suitable for applications to a broad range of physics problems, as well as teaching. It thereby provides supplementary material that may be used for a beginning graduate course in quantum mechanics.
ANL-HEP-PR-01-095 Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space
2016
Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides-coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.
Quantum mathematics: Backgrounds and some applications to nonlinear dynamical systems
Nonlinear Oscillations, 2008
UDC 517.9 The backgrounds of quantum mathematics, a new discipline in mathematical physics, are discussed and analyzed from both historical and analytical points of view. The magic properties of the second quantization method, invented by Fock in 1934, are demonstrated, and an impressive application to the theory of nonlinear dynamical systems is considered. Von Neumann first applied the spectral theory of self-adjoint operators on Hilbert spaces to explain the radiation spectra of atoms and the related matter stability [2] (1926); Fock was the first to introduce the notion of many-particle Hilbert space, named a Fock space, and introduced related creation and annihilation operators acting on it [3] (1932); Weyl understood the fundamental role of the notion of symmetry in physics and developed a physics-oriented group theory; moreover, he showed the importance of different representations of classical matrix groups for physics and studied unitary representations of the Heisenberg-Weyl group related to creation and annihilation operators on a Fock space [4] (1931). At the end of the 20th century, new developments were due to Faddeev with co-workers (quantum inverse spectral theory transform [5], 1978); Drinfeld, Donaldson, and Witten (quantum groups and algebras, quantum topology, and quantum superanalysis [6-8], 1982-1994);
2005
A complete solution to the problem of setting up Wigner distribution for N-level quantum systems is presented. The scheme makes use of some of the ideas introduced by Dirac in the course of defining functions of noncommuting observables and works uniformly for all N. Further, the construction developed here has the virtue of being essentially input-free in that it merely requires finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task which, as is shown, can always be accomplished analytically. As an illustration, the case of a single qubit is considered in some detail and it is shown that one recovers the result of Feynman and Wootters for this case without recourse to any auxiliary constructs.