On Spinors Transformations

TL;DR

This paper explores how orthogonal transformations, including space and time reversal, act on vectors and spinors within Clifford algebras, revealing that individual P or T transformations can change the spinor space and chirality.

Contribution

It provides a detailed analysis of automorphisms of Clifford algebras and their effects on spinors under P, T, and PT transformations, including how these transformations can alter spinor spaces.

Findings

All automorphisms of Clifford algebras in even dimensions are inner.

Under PT, spinors remain in their original space; under P or T alone, they may switch spaces and chirality.

Inner automorphisms can change the spinor space, affecting transformation properties.

Abstract

We begin showing that for even dimensional vector spaces $V$ all automorphisms of their Clifford algebras are inner. So all orthogonal transformations of $V$ are restrictions to $V$ of inner automorphisms of the algebra. Thus under orthogonal transformations $P$ and $T$ - space and time reversal - all algebra elements, including vectors $v$ and spinors $\varphi$, transform as $v \to x v x^{-1}$ and $\varphi \to x \varphi x^{-1}$ for some algebra element $x$. We show that while under combined $PT$ spinor $\varphi \to x \varphi x^{-1}$ remain in its spinor space, under $P$ or $T$ separately $\varphi$ goes to a 'different' spinor space and may have opposite chirality. We conclude with a preliminary characterization of inner automorphisms with respect to their property to change, or not, spinor spaces.