Fang, Y-L;
Vassiliev, D;
(2015)
Analysis as a source of geometry: a non-geometric representation of the Dirac equation.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
, 48
(16)
, Article ARTN 165203. 10.1088/1751-8113/48/16/165203.
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Abstract
Consider a formally self-adjoint first order linear differential operator acting on pairs (two-columns) of complex-valued scalar fields over a four-manifold without boundary. We examine the geometric content of such an operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients for spinor fields and an electromagnetic covector potential. This observation allows us to give a simple representation of the massive Dirac equation as a system of four scalar equations involving an arbitrary two-by-two matrix operator as above and its adjugate. The point of the paper is that in order to write down the Dirac equation in the physically meaningful four-dimensional hyperbolic setting one does not need any geometric constructs. All the geometry required is contained in a single analytic object—an abstract formally self-adjoint first order linear differential operator acting on pairs of complex-valued scalar fields.
| Type: | Article |
|---|---|
| Title: | Analysis as a source of geometry: a non-geometric representation of the Dirac equation |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1088/1751-8113/48/16/165203 |
| Publisher version: | http://dx.doi.org/10.1088/1751-8113/48/16/165203 |
| Additional information: | © 2015 IOP Publishing Ltd Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. |
| Keywords: | analysis of partial differential equations, gauge theory, Dirac equation |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1421894 |
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