$\bar{SL}(4,R)$ Embedding for a 3D World Spinor Equation

TL;DR

This paper develops a 3D curved spacetime Dirac-like equation using $ar{SL}(4,R)$ embedding, ensuring unitarity and correct particle properties, by constructing explicit infinite matrix representations of the embedding group.

Contribution

It introduces a novel embedding of $ar{SL}(3,R)$ into $ar{SL}(4,R)$ to formulate a Dirac-like equation with proper unitarity in 3D curved spacetime.

Findings

Constructed a Dirac-like equation with correct unitarity.

Embedded $ar{SL}(3,R)$ into $ar{SL}(4,R)$ for explicit matrix representations.

Achieved a consistent description of spinor particles in 3D curved spacetime.

Abstract

A generic-curved spacetime Dirac-like equation in 3D is constructed. It has, owing to the $\bar{SL}(n,R)$ group deunitarizing automorphism, a physically correct unitarity and flat spacetime particle properties. The construction is achieved by embedding $\bar{SL}(3,R)$ vector operator $X_{\mu}$, that plays a role of Dirac's $\gamma_{\mu}$ matrices, into $\bar{SL}(4,R)$. Decomposition of the unitary irreducible spinorial $\bar{SL}(4,R)$ representations gives rise to an explicit form of the infinite $X_{\mu}$ matrices.