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Journal of Physics A: Mathematical and Theoretical 45 (2012) 244005
Fermionic Coherent States
M. Combescure1, D. Robert2

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.
1:  IPNL - Institut de Physique Nucléaire de Lyon
2:  LMJL - Laboratoire de Mathématiques Jean Leray
Mathematics/Mathematical Physics

Physics/Nuclear Theory