We present a compact, systematic formulation of the dynamics of the Husimi Q- and GlauberSudarshan P-phase space distribution functions expressed in terms of their complementary Hamiltonian symbols: Anti-Wick for Q and Wick for P. The resulting evolution equations have a universal leading structure, the classical Liouvillian drift plus terms with higher-order derivatives of the Hamiltonian. For Hamiltonians no higher than quartic in the moduli of the complex phase space variables αi, the higher-order terms reduce to a second-order Fokker-Planck type term with a traceless diffusion matrix, thereby clarifying and recovering recent results for such Hamiltonians within a simple star-product framework. We further derive a transparent Ehrenfest theorem for Wick/Anti-Wick symbols of the operators representing dynamical observables. Using these results, we show that a previously reported nonclassical contribution to the Q-function drift for the anharmonic oscillator is an artifact of the quantization scheme used. Our paper consolidates the formulation of the dynamics of the phase space distribution functions using complementary symbols and provides an efficient route to compute and interpret quantum phase space evolution.
We present a compact, systematic formulation of the dynamics of the Husimi Q- and GlauberSudarshan P-phase space distribution functions expressed in terms of their complementary Hamiltonian symbols: Anti-Wick for Q and Wick for P. The resulting evolution equations have a universal leading structure, the classical Liouvillian drift plus terms with higher-order derivatives of the Hamiltonian. For Hamiltonians no higher than quartic in the moduli of the complex phase space variables αi, the higher-order terms reduce to a second-order Fokker-Planck type term with a traceless diffusion matrix, thereby clarifying and recovering recent results for such Hamiltonians within a simple star-product framework. We further derive a transparent Ehrenfest theorem for Wick/Anti-Wick symbols of the operators representing dynamical observables. Using these results, we show that a previously reported nonclassical contribution to the Q-function drift for the anharmonic oscillator is an artifact of the quantization scheme used. Our paper consolidates the formulation of the dynamics of the phase space distribution functions using complementary symbols and provides an efficient route to compute and interpret quantum phase space evolution.
Time evolution is considered in phase space in terms of evolution kernels for various phase-space quasi-distributions. The propagators for the Wigner function, the standard-ordered function, the Kirkwood (antistandard-ordered) function, the Glauber P and Q functions, and the Husimi function are explicitly written as bilinear transforms of the evolution operator. Free propagation, propagation in dispersive media, and scattering, are studied, and manifestations of causality and interference are analyzed. It is shown that free propagation and scattering in the Husimi, Glauber, and Kirkwood representations with the underlying dynamics of the Schr\"odinger equation involve divergent evolution kernels connecting distant phase-space points at all times. The time evolution is much simpler in the Wigner representation where (i) free propagation is a simple classical translation involving no interference, and (ii) analytical properties of the scattering matrix restrict the velocities of propagation so that no information can travel due to scattering faster than free motion. As an example, a correlation is found between the coordinate and momentum of particles detected after they are released from a box. Propagators with relativistic dispersion relations of free photons or Klein-Gordon particles are briefly discussed in an Appendix.
This paper proposes an approach to interpreting quantum expectation values that may help address the quantum measurement problem. Quantum expectation values are usually calculated via Hilbert space inner products and, thereby, differently from expectation values in classical mechanics, which are weighted phase-space integrals. It is shown that, by using Anti-Wick quantization to associate dynamical variables with self-adjoint linear operators, quantum expectation values can be interpreted as genuine weighted averages over phase space, paralleling their classical counterparts. This interpretation arises naturally in the Segal-Bargmann space, where creation and annihilation operators act as simple multiplication and differentiation operators. In this setting, the Husimi Q-function - the coherent-state representation of the quantum state - can be seen as a true probability density in phase space. Unlike Bohmian mechanics, the present approach retains the standard correspondence between dynamical variables and self-adjoint operators while paving the way for a classical-like probabilistic interpretation of quantum statistics.
The new c-number calculus for functions of noncommuting operators, developed in Paper I and employed in Paper II to formulate a general phase-space description of boson systems, deals with situations involving equal-time operators only. In the present paper extensions are presented for the treatment of problems involving boson operators at two or more instants of time. The mapping of time-ordered products onto c-number functions is studied in detail. The results make it possible to evaluate time-ordered products of boson operators by phase-space techniques. The usual Wick theorem for boson systems is obtained as a special case of a much more general theorem on time ordering. Our method of derivation appears to provide the first direct proof of Wick's theorem as well as a clear insight into its true meaning. A closed expression is also obtained for the time-evolution operator in terms of the solution of the c-number differential equation for the phase-space equivalent of this operator. The new calculus is also applied to the problem of evaluating normally ordered time-ordered, and also the antinormally ordered time-ordered, correlation functions.
The Monte Carlo trajectory sampling of stochastic differential equations based on the quasiprobability distribution functions, such as the Glauber–Sudarshan P, Wigner, and Husimi Q functions, enables us to investigate bosonic open quantum many-body dynamics described by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation. In this method, the Monte Carlo samplings for the initial distribution and stochastic noises incorporate quantum fluctuations, and thus, we can go beyond the mean-field approximation. However, description using stochastic differential equations is possible only when the corresponding Fokker–Planck equation has a positive-semidefinite diffusion matrix. In this work, we analytically derive the stochastic differential equations for arbitrary Hamiltonian and jump operators based on the path-integral formula, independently of the derivation of the Fokker–Planck equation. In the course of the derivation, we formulate the path-integral representation of the GKSL equation by using the -ordered quasiprobability distribution function, which systematically describes the aforementioned quasiprobability distribution functions by changing the real parameter . The essential point of this derivation is that we employ the Hubbard–Stratonovich transformation in the path integral, and its application is not always feasible. We find that the feasible condition of the Hubbard–Stratonovich transformation is identical to the positive-semidefiniteness condition of the diffusion matrix in the Fokker–Planck equation. In the benchmark calculations, we confirm that the Monte Carlo simulations of the obtained stochastic differential equations well reproduce the exact dynamics of physical quantities and non-equal time correlation functions of numerically solvable models, including the Bose–Hubbard model. This work clarifies the applicability of the approximation and gives systematic and simplified procedures to obtain the stochastic differential equations to be numerically solved.
A model for the evolution of a quantum particle in the phase plane is derived. The quasi-distribution function is decomposed into a suitable over-complete basis of coherent states. In this approach, a coarse grain length scale is introduced by using the spread of the coherent states. The obtained evolution system is completely equivalent to the Wigner formulation of the quantum mechanics. It shows some analogy with the von Neumann approach and the particle in the cell method in classical plasma physics. Differing from the usual formulation of the quantum phase space, given in terms of infinite-order partial differential equations, in this model, the evolution equations of the second differential order are expressed by a hierarchy of coupled functions (the first term being the Husimi function). The resulting formulation reveals itself to be particularly close to the classical description of the particles motion. This formal analogy is useful to gain new physical insights and to profit from numerical methods developed for classical systems.
We derive a continuity equation for the Husimi function evolving under a general non-Hermitian Hamiltonian and identify the phase space flow associated with it. For the case of unitary evolution we obtain explicit formulas for the quantum flow, which can be written as a classical term plus quantum corrections. The quantum terms can be expanded in powers of Planck's constant providing a series of semiclassical corrections to the classical flow. We test the exact and semiclassical formulas for a particle in a double well potential and find numerical evidence that the zeros of the Husimi function are always saddle points of the flow. Merging or splitting of stagnation points, reported for the Wigner flow, was not observed.