Admissible states in quantum phase space

Abstract We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wave functions in terms of time-dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time-dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.

[1]  Moyal–Nahm equations , 1999, hep-th/9901072.

[2]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[3]  Hai-Woong Lee,et al.  Theory and application of the quantum phase-space distribution functions , 1995 .

[4]  J. Neumann.  Die Eindeutigkeit der Schrödingerschen Operatoren , 1931 .

[5]  C. Zachos,et al.  Features of time-independent Wigner functions , 1997, hep-th/9711183.

[6]  A. Pinzul,et al.  Absence of the holographic principle in noncommutative Chern-Simons theory , 2001, hep-th/0107179.

[7]  M. Scully,et al.  The Wigner phase-space description of collision processes , 1983 .

[8]  J. E. Moyal.  Quantum mechanics as a statistical theory , 1949, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  Nicolaas P. Landsman,et al.  Mathematical Topics Between Classical and Quantum Mechanics , 1998 .

[10]  Wigner functions with boundaries , 2000, quant-ph/0012140.

[11]  F. Bayen,et al.  Deformation theory and quantization. I. Deformations of symplectic structures , 1978 .

[12]  B. K. Jennings,et al.  Wigner's function and other distribution functions in mock phase spaces , 1984 .

[13]  Negative Probability and Uncertainty Relations , 2001, hep-th/0105226.

[14]  E. Wigner.  On the quantum correction for thermodynamic equilibrium , 1932 .

[15]  D. Fairlie.  The formulation of quantum mechanics in terms of phase space functions , 1964, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  Bohmian trajectories and quantum phase space distributions , 2002, quant-ph/0208156.

[17]  Formal solutions of stargenvalue equations , 2001, quant-ph/0110035.

[18]  P. Holland.  The Quantum Theory of Motion , 1993 .

[19]  J. Linnett,et al.  Quantum mechanics , 1975, Nature.

[20]  C. Zachos,et al.  Phase-space quantization of field theory. , 1999, hep-th/9903254.

[21]  I. M. Pyshik,et al.  Table of integrals, series, and products , 1965 .

[22]  V. I. Tatarskii,et al.  The Wigner representation of quantum mechanics , 1983 .

[23]  P. Carruthers,et al.  Quantum collision theory with phase-space distributions , 1983 .

[24]  Shin,et al.  Floquet analysis of quantum resonance in a driven nonlinear system. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  H. Weyl.  Quantenmechanik und Gruppentheorie , 1927 .

[26]  G. A. Baker,et al.  Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space , 1958 .

[27]  Causal interpretation and quantum phase space , 2001, quant-ph/0110062.

[28]  Half Quantization , 1999, quant-ph/9912071.

[29]  L. Cohen.  Generalized Phase-Space Distribution Functions , 1966 .

[30]  MOYAL BRACKETS IN M-THEORY , 1997, hep-th/9707190.

[31]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[32]  Burst process of stretched fiber bundles. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[33]  Quantum theory in curved spacetime using the Wigner function , 1997, hep-th/9701182.

[34]  T. B. Smith.  Semiclassical approximation in the Weyl picture by path summation , 1978 .

[35]  Nathan Seiberg,et al.  String theory and noncommutative geometry , 1999 .

[36]  Kin’ya Takahashi.  Distribution Functions in Classical and Quantum Mechanics , 1989 .