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Journal of Physics A Mathematical and General

On dynamical properties in a Moyal quantization

G. Braunss, Dirk Rompf

Published Aug 21, 1993• 6 citations
Publisher

Abstract

Dynamical properties of classical Hamiltonian systems and their Moyal quantizations are related.
DOI
10.1088/0305-4470/26/16/026

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This paper describes the states of quantum versions of finite dimensional Hamilton–Dirac systems in terms of (analogues of) the Wigner function (various definitions of Wigner functions for classical Hamiltonian systems can be found in [8]). We give a system of equations, which we call the Moyal–Dirac system, describing the evolution of this Wigner function. The passage from the Moyal equation describing the evolution of the quantum version of a classical Hamiltonian system to the Moyal–Dirac system describing the evolution of the quantum version of the corresponding Hamilton–Dirac system is similar to the passage from the Liouville equation describing the evolution of a classical Hamiltonian system to the Liouville–Dirac system of equations describing the evolution of the corresponding Hamilton–Dirac system. Thus, our Moyal–Dirac system is a generalization of both the Liouville–Dirac system and the Moyal equation
The quantum time of arrival (TOA) problem requires the statistics of measured arrival times given only the initial state of a particle. Following the standard framework of quantum theory, the problem translates into finding an appropriate quantum image of the classical arrival time TC(q,p), usually in operator form T̂. In this paper, we consider the problem anew within the phase space formulation of quantum mechanics. The resulting quantum image is a real-valued and time-reversal symmetric function TM(q,p) in formal series of ℏ2 with the classical arrival time as the leading term. It is obtained directly from the Moyal bracket relation with the system Hamiltonian and is hence interpreted as a Moyal deformation of the classical TOA. We investigate its properties and discuss how it bypasses the known obstructions to quantization by showing the isomorphism between TM(q,p) and the rigged Hilbert space TOA operator constructed in Pablico and Galapon [Eur. Phys. J. Plus 138, 153 (2023)], which always satisfy the time-energy canonical commutation relation for arbitrary analytic potentials. We then examine TOA problems for a free particle and a quartic oscillator potential as examples.
Quantization of constraint systems within the Weyl-Wigner-Groenewold-Moyal framework is discussed. Constraint dynamics of classical and quantum systems is reformulated using the skew-gradient projection formalism. The quantum deformation of the Dirac bracket is generalized to match smoothly the classical Dirac bracket in and outside of the constraint submanifold in the limit ℏ→0\hbar \to 0ℏ→0.