Grassmann phase space methods for fermions. II. Field theory
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 suggests 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 quantumatom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics. This paper presents a phase space theory for fermion systems based on distribution functionals, which replace the density operator and involve Grassmann fields representing anticommuting fermion field annihilation, creation operators. It is an extension of a previous phase space theory paper for fermions (Paper I) based on separate modes, in which the density operator is replaced by a distribution function depending on Grassmann phase space variables which represent the mode annihilation and creation operators. This further development of the theory is important for the situation when large numbersmore »
 Authors:
 Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne, Victoria 3122 (Australia)
 Department of Physics, University of Strathclyde, Glasgow G4ONG (United Kingdom)
 School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ (United Kingdom)
 Publication Date:
 OSTI Identifier:
 22617473
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Annals of Physics; Journal Volume: 377; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANNIHILATION OPERATORS; CREATION OPERATORS; DISTRIBUTION FUNCTIONS; FERMI GAS; FERMIONS; FIELD EQUATIONS; FIELD THEORIES; FOKKERPLANCK EQUATION; PHASE SPACE; QUANTUM OPTICS
Citation Formats
Dalton, B.J., Email: bdalton@swin.edu.au, Jeffers, J., and Barnett, S.M. Grassmann phase space methods for fermions. II. Field theory. United States: N. p., 2017.
Web. doi:10.1016/J.AOP.2016.12.026.
Dalton, B.J., Email: bdalton@swin.edu.au, Jeffers, J., & Barnett, S.M. Grassmann phase space methods for fermions. II. Field theory. United States. doi:10.1016/J.AOP.2016.12.026.
Dalton, B.J., Email: bdalton@swin.edu.au, Jeffers, J., and Barnett, S.M. Wed .
"Grassmann phase space methods for fermions. II. Field theory". United States.
doi:10.1016/J.AOP.2016.12.026.
@article{osti_22617473,
title = {Grassmann phase space methods for fermions. II. Field theory},
author = {Dalton, B.J., Email: bdalton@swin.edu.au and Jeffers, J. and Barnett, S.M.},
abstractNote = {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 suggests 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 quantumatom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics. This paper presents a phase space theory for fermion systems based on distribution functionals, which replace the density operator and involve Grassmann fields representing anticommuting fermion field annihilation, creation operators. It is an extension of a previous phase space theory paper for fermions (Paper I) based on separate modes, in which the density operator is replaced by a distribution function depending on Grassmann phase space variables which represent the mode annihilation and creation operators. This further development of the theory is important for the situation when large numbers of fermions are involved, resulting in too many modes to treat separately. Here Grassmann fields, distribution functionals, functional Fokker–Planck equations and Ito stochastic field equations are involved. Typical applications to a trapped Fermi gas of interacting spin 1/2 fermionic atoms and to multicomponent Fermi gases with nonzero range interactions are presented, showing that the Ito stochastic field equations are local in these cases. For the spin 1/2 case we also show how simple solutions can be obtained both for the untrapped case and for an optical lattice trapping potential.},
doi = {10.1016/J.AOP.2016.12.026},
journal = {Annals of Physics},
number = ,
volume = 377,
place = {United States},
year = {Wed Feb 15 00:00:00 EST 2017},
month = {Wed Feb 15 00:00:00 EST 2017}
}

The Jaynes–Cummings model of a twolevel atom in a single mode cavity is of fundamental importance both in quantum optics and in quantum physics generally, involving the interaction of two simple quantum systems—one fermionic system (the TLA), the other bosonic (the cavity mode). Depending on the initial conditions a variety of interesting effects occur, ranging from ongoing oscillations of the atomic population difference at the Rabi frequency when the atom is excited and the cavity is in an nphoton Fock state, to collapses and revivals of these oscillations starting with the atom unexcited and the cavity mode in a coherentmore »

Grassmann algebra and fermions in the lattice: A simple example
Using the exact results for integrals of exponentials of polynomials of Grassmann variables and the Feynman construction of path integrals, an approximate method is presented to calculate the thermodynamic quantities of a fermionic system in equilibrium with a reservoir at temperature {ital T} and chemical potential {mu}. To exemplify, the quantum anharmonic fermionic oscillator is considered in a lattice with {ital N} points, where {ital N}=2, 3, and 4.