# 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 c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggests the possibility of using anti-commuting 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. This paper presents a phase space theory for fermion systems based on distribution functionals, which replace the density operator and involve Grassmann fields representing anti-commuting 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; FOKKER-PLANCK EQUATION; PHASE SPACE; QUANTUM OPTICS

### Citation Formats

```
Dalton, B.J., E-mail: 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., E-mail: 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., E-mail: 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., E-mail: 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 c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggests the possibility of using anti-commuting 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. This paper presents a phase space theory for fermion systems based on distribution functionals, which replace the density operator and involve Grassmann fields representing anti-commuting 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 multi-component Fermi gases with non-zero 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}

}