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arXiv · 1401.3160

Analysis as a source of geometry: a non-geometric representation of the Dirac equation

Yan-Long Fang·Dmitri Vassiliev
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Abstract

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.

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Yan-Long Fang, Dmitri Vassiliev. 2015-03-08. Analysis as a source of geometry: a non-geometric representation of the Dirac equation. https://doi.org/10.1088/1751-8113%2F48%2F16%2F165203

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