Grassmann phase space methods for fermions. I. Mode theory
Dalton, B.J. and Jeffers, J. and Barnett, S.M. (2016) Grassmann phase space methods for fermions. I. Mode theory. Annals of Physics, 370. pp. 1266. ISSN 00034916

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Abstract
In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by cnumber phase space variables, with the density operator equivalent to a distribution function of these variables. The anticommutation rules for fermion annihilation, creation operators suggest the possibility of using anticommuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum–atom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics. The theory of Grassmann phase space methods for fermions based on separate modes is developed, showing how the distribution function is defined and used to determine quantum correlation functions, Fock state populations and coherences via Grassmann phase space integrals, how the Fokker–Planck equations are obtained and then converted into equivalent Ito equations for stochastic Grassmann variables. The fermion distribution function is an even Grassmann function, and is unique. The number of cnumber Wiener increments involved is 2n2, if there are n modes. The situation is somewhat different to the bosonic cnumber case where only 2n Wiener increments are involved, the sign of the drift term in the Ito equation is reversed and the diffusion matrix in the Fokker–Planck equation is antisymmetric rather than symmetric. The unnormalisedB distribution is of particular importance for determining Fock state populations and coherences, and as pointed out by Plimak, Collett and Olsen, the drift vector in its Fokker–Planck equation only depends linearly on the Grassmann variables. Using this key feature we show how the Ito stochastic equations can be solved numerically for finite times in terms of cnumber stochastic quantities. Averages of products of Grassmann stochastic variables at the initial time are also involved, but these are determined from the initial conditions for the quantum state. The detailed approach to the numerics is outlined, showing that (apart from standard issues in such numerics) numerical calculations for Grassmann phase space theories of fermion systems could be carried out without needing to represent Grassmann phase space variables on the computer, and only involving processes using cnumbers. We compare our approach to that of Plimak, Collett and Olsen and show that the two approaches differ. As a simple test case we apply the B distribution theory and solve the Ito stochastic equations to demonstrate coupling between degenerate Cooper pairs in a four mode fermionic system involving spin conserving interactions between the spin 1/2 fermions, where modes with momenta −k,+k—each associated with spin up, spin down states, are involved.
Author(s):  Dalton, B.J., Jeffers, J. and Barnett, S.M. 

Item type:  Article 
ID code:  57786 
Keywords:  fermion phase space theory, fock state populations, FokkerPlanck equations, Grassmann distribution functions, Ito stochastic equations, quantum correlation functions, Physics, Physics and Astronomy(all) 
Subjects:  Science > Physics 
Department:  Faculty of Science > Physics 
Depositing user:  Pure Administrator 
Date deposited:  14 Sep 2016 09:29 
Last modified:  05 Jun 2019 02:35 
Related URLs:  
URI:  https://strathprints.strath.ac.uk/id/eprint/57786 
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