On Spinors Transformations

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 $\phi$, transform as $v\to xv{x}^{-1}$ and $\phi \to x\phi x^{-1}$ for some algebra element $x$. We show that while under combined $PT$ spinor $\phi \to x\phi x^{-1}$ remain in its spinor space, under $P$ or $T$ separately $\phi$ 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.

Minor changes to propositions 1 and 3

References in corpus (1)

• On Computational Complexity of Clifford Algebra

Cited by in corpus (7)

• The Boolean SATisfiability Problem in Clifford algebra
• On Clifford Algebras and binary integers
• On complex representations of Clifford algebra
• The Boolean SATisfiability Problem and the orthogonal group $O(n)$
• On Simple Spinors, Null Vectors and the Orthogonal Group O(n)
• The Clifford algebra of $R^{n,n}$ and the Boolean Satisfiability Problem
• On algebraic structure of matter spectrum