Invariant systems under the space-time Inversion— existence of two kinds of inversions

Abstract

Restricting ourselves to the scale and conformai invariant systems, we investigate their invariance (or noninvariance) properties with respect to the inversions at the classical level. By employing some rules, for some dynamical variables we obtain the representation of the inversions with the radius |a| and encounter the two kinds of representations corresponding to a>0 and a 0 and the NI for a<0, is decisive in the spinor systems. Especially, the massless spinor QED and the QCD with vanishing quark masses are not invariant under the PI but invariant under the NI with some conditions. Furthermore, the Weinberg-Salam theory in the scale (or conformai) symmetric limit is not invariant under either of them. It is seen that the requirement of the inversion symmetry of a dynamics is more restrictive than that of the scale or conformal symmetry.