# Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions

## Abstract

We study the Maxwell–Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwell–Dirac equations, without any reference to gauge dependent quantities. We show how demanding invariance of tensor fields under the action of a fixed (but arbitrary) Lie subgroup of the Poincaré group leads to symmetry reduced equations. The procedure is illustrated, and the reduced equations worked out explicitly for standard spherical and cylindrical cases, which are coupled third order nonlinear PDEs. Spherical symmetry necessitates the existence of magnetic monopoles, which do not affect the coupled Maxwell–Dirac system due to magnetic terms cancelling. In this paper we do not take up numerical computations. As a demonstration of the power of our approach, we also work out the symmetry reduced equations for two distinctmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22403386

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 348; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CYLINDRICAL CONFIGURATION; DIRAC EQUATION; EXACT SOLUTIONS; GAUGE INVARIANCE; MAGNETIC MONOPOLES; MAXWELL EQUATIONS; POINCARE GROUPS; QUANTIZATION; QUANTUM MECHANICS; SPHERICAL CONFIGURATION; SYMMETRY; TENSOR FIELDS; VECTOR FIELDS; VECTORS

### Citation Formats

```
Inglis, Shaun, E-mail: sminglis@utas.edu.au, and Jarvis, Peter, E-mail: Peter.Jarvis@utas.edu.au.
```*Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions*. United States: N. p., 2014.
Web. doi:10.1016/J.AOP.2014.05.017.

```
Inglis, Shaun, E-mail: sminglis@utas.edu.au, & Jarvis, Peter, E-mail: Peter.Jarvis@utas.edu.au.
```*Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions*. United States. doi:10.1016/J.AOP.2014.05.017.

```
Inglis, Shaun, E-mail: sminglis@utas.edu.au, and Jarvis, Peter, E-mail: Peter.Jarvis@utas.edu.au. Mon .
"Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions". United States. doi:10.1016/J.AOP.2014.05.017.
```

```
@article{osti_22403386,
```

title = {Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions},

author = {Inglis, Shaun, E-mail: sminglis@utas.edu.au and Jarvis, Peter, E-mail: Peter.Jarvis@utas.edu.au},

abstractNote = {We study the Maxwell–Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwell–Dirac equations, without any reference to gauge dependent quantities. We show how demanding invariance of tensor fields under the action of a fixed (but arbitrary) Lie subgroup of the Poincaré group leads to symmetry reduced equations. The procedure is illustrated, and the reduced equations worked out explicitly for standard spherical and cylindrical cases, which are coupled third order nonlinear PDEs. Spherical symmetry necessitates the existence of magnetic monopoles, which do not affect the coupled Maxwell–Dirac system due to magnetic terms cancelling. In this paper we do not take up numerical computations. As a demonstration of the power of our approach, we also work out the symmetry reduced equations for two distinct classes of dimension 4 one-parameter families of Poincaré subgroups, one splitting and one non-splitting. The splitting class yields no solutions, whereas for the non-splitting class we find a family of formal exact solutions in closed form. - Highlights: • Maxwell–Dirac equations derived in manifestly gauge invariant tensor form. • Invariant scalar and four vector fields for four Poincaré subgroups derived, including two unusual cases. • Symmetry reduction imposed on Maxwell–Dirac equations under example subgroups. • Magnetic monopole arises for spherically symmetric case, consistent with charge quantization condition.},

doi = {10.1016/J.AOP.2014.05.017},

journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = ,

volume = 348,

place = {United States},

year = {2014},

month = {9}

}