ar X iv : h ep - t h / 05 04 16 6 v 1 2 0 A pr 2 00 5 Deformation Quantization and Wigner Functions ∗

We review the Weyl-Wigner formulation of quantum mechanics in phase space. We discuss the concept of Narcowich-Wigner spectrum and use it to state necessary and sufficient conditions for a phase space function to be a Wigner distribution. Based on this formalism we analize the modifications introduced by the presence of boundaries. Finally, we discuss the concept of environment-induced decoherence in the context of the Weyl-Wigner approach. 1 The Weyl-Wigner formulation of quantum mechanics The Wigner function was introduced in the 1930’s [1]. Wigner was trying to derive quantum corrections to Boltzmann’s kinetic equation. With this aim in mind he had to define some quantum counterpart to the Liouville density ρcl. Let us recall some basic facts about this distribution. ρcl is assumed to be a real, normalized, non-negative, C function in phase-space. Here, we assume a flat 2n-dimensional phase-space T M ≃ R. The expectation value of some observable A ∈ C (T M ;R) is then given by: < A > N = ∫ dx ∫ dp A(x, p)ρcl(x, p). (1) Moreover, the probability distribution that a measurement of A yield the value a ∈ R is evaluated according to: P (A = a) = ∫ dx ∫ dp δ (A(x, p) − a) ρcl(x, p). (2) Talk presented by the second author at the Workshop on Quantum Gravity and Noncommutative Geometry, 20-23 July 2004, Universidade Lusófona, Lisbon, Portugal. [email protected] [email protected]