Geodesic
Perturbation Theory in Terms of a Generalized Phase-Space Quantization Procedure
Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 1, pp. 1-18

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A new approach to perturbation theory in the quantum phase-space formalism is proposed, in order to devise some approximated description of the quantum phase-space dynamics, which is not directly related to the usual semi-classical approximation. A general class of equivalent quasi-distribution functions based on the Wigner-Moyal formalism is considered and a first-order invariant formulation of the dynamics is obtained. The relationship between the various phase-space representations is expressed in term of a pseudo-differential operator defined by the Moyal product. In particular, our theory is applied to the sub-class of representations obtained by a first order perturbation of the Wigner representation. Finally the connection of our approach with some well established gauge-invariant formulations of the Wigner dynamics in the presence of an external magnetic field is investigated. 
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Morandi, Omar; Barletti, Luigi; Frosali, Giovanni. Perturbation Theory in Terms of a Generalized Phase-Space Quantization Procedure. Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 1, pp. 1-18. http://geodesic.mathdoc.fr/item/BUMI_2011_9_4_1_a0/