Nondynamical Formalism and Tests of Time-Reversal Invariance

In this paper we consider those consequences of time-reversal invariance (or, briefly, $T$ invariance) which are independent of dynamics. A general prescription is developed to find all those tests of $T$ invariance which are valid for arbitrary values of the form factors. Our considerations are independent of whether or not any of the other usual conservation laws (such as parity conservation) hold. The spins of the particles may be arbitrary. The results and methods of earlier papers dealing with the nondynamical properties of particle reactions are used. For elastic processes it is shown that $T$ invariance eliminates some of the product sets and curtails others, but that no entire subclasses are eliminated. The restrictions on observables imposed by $T$ invariance are of two types: There are so-called "mirror relations" between pairs of observable components, and there are relations which are not of the mirror type and involve a number of observable components. It is shown that the number of relations of the latter type is always nonzero whenever $T$ invariance is not implied by other assumed conservation laws. Therefore, the mirror relations do not form a complete set of tests of $T$ invariance. They do not even form a sufficient set of such tests, as they can be satisfied by a symmetric as well as an antisymmetric $M$ matrix. A proof is given that in any non-mirror-type relation no particle is unpolarized in all the observable components which appear in the relation. It is also shown that the only reaction in which all mirror relations follow from parity conservation alone is the reaction involving two spin-½ and two spin-0 particles. A number of examples of elastic reactions are worked out in detail. For inelastic reactions the results are less interesting, since the restrictions imposed by $T$ invariance can all be written as mirror-type relations between observable components of the direct and time-reversed reactions.