Possible Polarization Experiments in Fermion-Fermion Scattering as a Test of Invariance Principles

Abstract

The word "fermion" here means "spin-½ fermion." Discussion is restricted to processes in which two arbitrary incoming fermions are scattered into two arbitrary fermions (not necessarily the same) in the final state. Invariance under space inversion ($P_{\sigma }$), time-reversal ($P_T$) invariance, and charge-conjugation ($P_C$) invariance is not assumed to hold necessarily. Formulas are derived which can be applied to any polarization experiment, including correlation experiments. Necessary conditions are obtained which can be tested experimentally to check whether a certain invariance principle holds in nature. In addition to $P_{\sigma }$, $P_T$, $P_C$ invariance and their combinations, certain helicity invariance principles (related to ${\gamma }^5$ invariance principles) are discussed. The treatment is relativistic. By a suitable choice of rest frames for each particle, the appearance of some trigonometric functions of the scattering angle is avoided, and $\sigma$ matrices can be used instead of $\gamma$ matrices. Experiments in which the two incoming fermions are uncorrelated and one of them is unpolarized, are treated in great detail. It is shown to what extent such experiments can determine the $S$ matrix. A graphical method is introduced which can often be used to simplify discussion of multiple scattering experiments. Fermion—spin-zero-boson scattering is treated as a special case of fermion-fermion scattering. Those consequences of invariance principles which can be checked most easily in general polarization experiments are discussed.