Analysis as a source of geometry: a non-geometric representation of the Dirac equation
| Autor: | Fang, Yan-Long, Vassiliev, Dmitri |
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| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | J. Phys. A: Math. Theor. 48 (2015) 165203 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1751-8113/48/16/165203 |
| Popis: | Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-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 4-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. Comment: Edited in accordance with referees' recommendations |
| Databáze: | arXiv |
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