Zachos C.K., Fairlie D.B., Curtright T.L. (eds.) Quantum Mechanics in Phase Space: An Overview with Selected Papers

World Scientific, 2005. - 551 pp.
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, quantum computing, and quantum chaos. It is also important 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 — coordinate or momentum space. It works in full phase space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. This invaluable book is a collection of the seminal papers on the formulation, with an introductory overview which provides a trail map for those papers; an extensive bibliography; and simple illustrations, suitable for applications to a broad range of physics problems. It can provide supplementary material for a beginning graduate course in quantum mechanics.Overview of Phase-Space Quantization
Selected papers
H. Weyl, Z. Phys 46 (1927) 1-46, "Quantenmechanik und Gruppentheorie".
J. v Neumann, Math. Ann. 104 (1931) 570-578, "Die Eindeutigkeit der Schrodingerschen Operatoren".
E. Wigner, Phys. Rev. 40 (1932) 749-759, "On the Quantum Correction for Thermodynamic Equilibrium".
H. Groenewold, Physica 12 (1946) 405-460, "On the Principles of Elementary Quantum Mechanics".
J. Moyal, Proc. Camb. Phil. Soc. 45 (1949) 99-124, "Quantum Mechanics as a Statistical Theory".
M. Bartlett and J. Moyal, Proc. Camb. Phil. Soc. 45 (1949) 545-553, "The Exact Transition Probabilities of Quantum-Mechanical Oscillators Calculated by the Phase-Space Method".
T. Takabayasi, Prog. Theor. Phys. 11 (1954) 341-373, "The Formulation of Quantum Mechanics in Terms of Ensemble in Phase Space".
G. Baker, Phys. Rev. 109 (1958) 2198-2206, "Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space".
D. Fairlie, Proc. Camb. Phil. Soc. 60 (1964) 581-586, "The Formulation of Quantum Mechanics in Terms of Phase Space Functions".
N. Cartwright, Physica 83A (1976) 210-212, "A Non-Negative Wigner-Type Distribution".
A. Royer, Phys. Rev. A15 (1977) 449-450, "Wigner Function as the Expectation Value of a Parity Operator".
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Ann. Phys. 111 (1978) 61-110; ibid. 111-151, "Deformation Theory and Quantization: I. Deformations of Symplectic Structures; II. Physical Applications".
G. Garcia-Calderon and M. Moshinsky, J. Phys. A13 (1980) L185-L188, "Wigner Distribution Functions and the Representation of Canonical Transformations in Quantum Mechanics".
J. Dahl and M. Springborg, Mol. Phys. 47 (1982) 1001-1019, "Wigner's Phase Space Function and Atomic Structure. I. The Hydrogen Atom Ground State".
M. Hillery, R. O'Connell, M. Scully, and E. Wigner, Phys. Rep. 106 (1984) 121-167, "Distribution Functions in Physics: Fundamentals".
Y. Kim and E. Wigner, Am. J. Phys. 58 (1990) 439-448, "Canonical Transformation in Quantum Mechanics".
R. Feynman, in Essays in Honor of David Bohm, B. Hiley and F. Peat, eds. (Routledge and Kegan Paul, London,1987) pp. 235-248, "Negative Probability".
M. De Wilde and P. Lecomte, Lett. Math. Phys. 7 (1983) 487-496, "Existence of Star-Products and of Formal Deformations of the Poisson Lie Algebra of Arbitrary Symplectic Manifolds".
B. Fedosov, J. Diff. Geom. 40 (1994) 213-238, "A Simple Geometrical Construction of Deformation Quantization".
T. Curtright, D. Fairlie, and C. Zachos, Phys. Rev. D58 (1998) 025002 1-14, "Features of Time-Independent Wigner Functions".
T. Curtright and C. Zachos, Mod. Phys. Lett. A16 (2001) 2381-2385, "Negative Probability and Uncertainty Relations".
T. Curtright, T. Uematsu, and C. Zachos, J. Math. Phys. 42 (2001) 2396-2415, "Generating All Wigner Functions".
M. Hug, C. Menke, and W. Schleich, Phys. Rev. A57 (1998) 3188-3205; ibid 3206-3224, "Modified Spectral Method in Phase Space: Calculation of the Wigner Function: I. Fundamentals; II. Generalizations".