From Poincare to affine invariance: How does the Dirac equation generalize?

TL;DR

This paper extends the Dirac equation to affine symmetry using SL(4,R), analyzing its implications for fields of any spin, and explores symmetry breaking scenarios that preserve Poincare invariance.

Contribution

It introduces a generalized Dirac equation with affine symmetry, embedding Lorentz fields into infinite-component SL(4,R) fermionic fields, and studies their coupling to affine gravity.

Findings

Development of a Dirac-type Poincare-covariant equation for any spin

Embedding Lorentz fields into SL(4,R) fermionic fields

A symmetry breaking scenario preserving Poincare symmetry

Abstract

A generalization of the Dirac equation to the case of affine symmetry, with SL(4,R) replacing SO(1,3), is considered. A detailed analysis of a Dirac-type Poincare-covariant equation for any spin j is carried out, and the related general interlocking scheme fulfilling all physical requirements is established. Embedding of the corresponding Lorentz fields into infinite-component SL(4,R) fermionic fields, the constraints on the SL(4,R) vector-operator generalizing Dirac's gamma matrices, as well as the minimal coupling to (Metric-)Affine gravity are studied. Finally, a symmetry breaking scenario for SA(4,R) is presented which preserves the Poincare symmetry.