The Dirac Equation and General Linear Transformations of Coordinate Systems

TL;DR

This paper explores extending the Dirac equation's covariance to general linear transformations by embedding spinors into a larger algebraic framework, potentially broadening its geometric and physical applicability.

Contribution

It proposes a novel approach to treat Dirac spinors within a larger space where GL(4,R) acts covariantly, using Grassmann and Clifford algebras, extending the equation's symmetry properties.

Findings

Dirac equation can be formulated covariantly under general linear transformations in an extended algebraic space.

The 16-dimensional space of anti-symmetric forms and Dirac matrices can host a broader class of transformations.

The approach aligns with the idea of Dirac spinors as elements of a Clifford algebra ideal.

Abstract

The spinor representation of the Lorentz group does not accept simple generalization with the group GL(4,R) of general linear coordinate transformations. The Dirac equation may be written for an arbitrary choice of a coordinate system and a metric, but the covariant linear transformations of the four-component Dirac spinor exist only for isometries. For usual diagonal Minkowski metric the isometry is the Lorentz transformation. On the other hand, it is possible to define the Dirac operator on the space of anti-symmetric (exterior) forms, and in such a case the equation is covariant for an arbitrary general linear transformation. The space of the exterior forms is 16-dimensional, but usual Dirac equation is defined for four-dimensional complex space of Dirac spinors. Using suggested analogy, in present paper is discussed possibility to consider the space of Dirac spinors as some "subsystem" of a bigger space, where the group GL(4,R) of general relativity acts in a covariant way. For such purposes in this article is considered both Grassmann algebra of complex anti-symmetric forms and Clifford algebra of Dirac matrices. Both algebras have same dimension as linear spaces, but different structure of multiplication. The underlying sixteen-dimensional linear space also may be considered either as space of complex 4 x 4 matrices, or as space of states of two particles: the initial Dirac spinor and some auxiliary system. It is shown also, that such approach is in good agreement with well known idea to consider Dirac spinor as some ideal of Clifford algebra. Some other possible implications of given model are also discussed.