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Fermionic Coherent States
Combescure M., Robert D.
Journal of Physics A: Mathematical and Theoretical 45 (2012) 244005 - http://hal.in2p3.fr/in2p3-00721798
Mathématiques/Physique mathématique
Physique/Physique Nucléaire Théorique
Fermionic Coherent States
M. Combescure1, D. Robert2
1 :  IPNL - Institut de Physique Nucléaire de Lyon
http://www.ipnl.in2p3.fr/
CNRS : UMR5822 – IN2P3 – Université Claude Bernard - Lyon I (UCBL)
France
2 :  LMJL - Laboratoire de Mathématiques Jean Leray
http://www.math.sciences.univ-nantes.fr/jeanleray/
CNRS : UMR6629 – Université de Nantes – École Centrale de Nantes
France
The aim of this paper is to give a self-contained and unified presentation of a fermionic coherent state theory with the necessary mathematical details, discussing their definition, properties and some applications. After defining Grassmann algebras, it is possible to get a classical analog for the fermionic degrees of freedom in a quantum system. Following the basic work of Berezin (1966 The Method of Second Quantization (New York: Academic); 1987 Introduction to Superanalysis (Dordrecht: Reidel Publishing Company)), we show that we can compute with Grassmann numbers as we do with complex numbers: derivation, integration, Fourier transform. After that we show that we have quantization formulas for fermionic observables. In particular, there exists a Moyal product formula. As an application, we consider explicit computations for propagators with quadratic Hamiltonians in annihilation and creation operators. We prove a Mehler formula for the propagator and Mehlig-Wilkinson-type formulas for the covariant and contravariant symbols of 'metaplectic' transformations for fermionic states.

Articles dans des revues avec comité de lecture
2012
Journal of Physics A: Mathematical and Theoretical
Publisher Institute of Physics: Hybrid Open Access
ISSN 1751-8113 (eISSN : 1751-8121)
45
244005