On Spinors Transformations

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 $\varphi \to x \varphi x^{-1}$remaininitsspinorspace,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.