The conservation laws in the field theoretical representation of Dirac's theory

TL;DR

This paper reinterprets Dirac's current as a stress-energy tensor, revealing conservation laws for energy and momentum, and resolving contradictions in electron dynamics through a new tensorial framework.

Contribution

It introduces a tensor-based description of Dirac's theory, connecting electromagnetic and mechanical components to address longstanding contradictions.

Findings

Dirac's current relates to a stress-energy tensor, not a vector.

The stress-energy tensor has electromagnetic and mechanical parts.

This framework resolves known contradictions in electron dynamics.

Abstract

We show that in the new description, Dirac's ``current vector'' is not related to a vector but to a tensor: the ``stress-energy tensor.'' Corresponding to Dirac's conservation law, we have the conservation laws of momentum and energy. The stress-energy tensor consists of two parts: an ``electromagnetic'' part, which has the same structure as the stress-energy tensor of the Maxwell theory, and a ``mechanical'' part, as suggested by hydrodynamics. The connection between these two tensors, which appears organically here, eliminates the well-known contradictions inherent in the dynamics of electron theory. (Editorial note: In this paper Lanczos continues to discuss his ``fundamental equation,'' from which he consistently derives Proca's equation and its stress-energy tensor.)