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A tensor interpretation of the 2D Dirac equation

Dmitri Vassiliev
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Abstract

We consider the Dirac equation in flat Minkowski 3-space and rewrite it as the Maxwell equation in Minkowski 4-space with torsion. The torsion tensor is defined as the dual of the electromagnetic vector potential. Our model clearly distinguishes the electron and the positron without resorting to "negative frequencies": we produce a real scalar invariant (charge) which indicates whether we are looking at an electron or a positron. Another interesting feature of our model is that the free electron and positron are identified with gradient type solutions of the standard (torsion free) Maxwell equation; such solutions have traditionally been disregarded on the grounds of gauge invariance.

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Dmitri Vassiliev. 2000-06-17. A tensor interpretation of the 2D Dirac equation. https://arxiv.org/abs/math-ph/0006019

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