Wigner Quantization of Hamilton–Dirac Systems

T. S. Ratiu (1) , O. G. Smolyanov (2)

1 CIB - Bernouilli Center [Lausanne]
2 Mechanics and Mathematics Faculty, Moscow State University, Moscow

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

Domaines: Physique mathématique [math-ph]

T. S. Ratiu, O. G. Smolyanov. Wigner Quantization of Hamilton–Dirac Systems. Доклады Академии Наук / Doklady Mathematics, 2015, 91 (1), pp.114-116. ⟨10.1134/S1064562415010287⟩. ⟨hal-01398473⟩