Mirror Relations as Nondynamical Tests of Conservation Laws

Abstract

A mirror relation is defined as $L(x_a, y_b; z_c, w_d)=\eta L(z_c, w_d; x_a, y_b)$, with $\eta =+1\mathrm{} or -1\mathrm{}$, where $L(x_a, y_b; z_c, w_d)$ is an observable for an arbitrary four-particle reaction $a+b\to c+d$, and $x_a$ denotes the polarization state of particle $a$, etc. Invariances under the transformations $P$ (space reflection), $T$ (time reversal), $C$ (charge conjugation), $B$ (detailed balancing), and products of these, are considered. First, the types of reactions are listed which transform into themselves under any one of these transformations. The $M$ matrix of these self-transforming reactions will therefore be restricted by the requirement of invariance under these transformations. This allows a study of the validity of a particular conservation law using one single reaction. The restriction on the $M$ matrices of one kind of self-transforming reactions can be expressed very simply in terms of the requirement that the number of occurrences of each of the three unit vectors used to span the space of momenta be either even or odd throughout each term of the $M$ matrix. Under the transformations $T$, $\mathrm{PT}$, $\mathrm{TB}$, $\mathrm{TC}$, $\mathrm{PTC}$, $\mathrm{TCB}$, $\mathrm{PTB}$, and $\mathrm{PTCB}$, these restrictions are expressed in terms of the number of one of the unit vectors or in terms of the sum of the numbers of all three unit vectors. We call these transformations of the odd type. On the other hand, under transformations $P$, $B$, $\mathrm{PB}$, $\mathrm{PC}$, $\mathrm{CB}$, and $\mathrm{PCB}$, the restrictions are expressed in terms of the sum of the numbers of two unit vectors. We call these transformations of the even type. Under transformations of the even type, about half of the observables identically vanish. It is then shown that mirror relations among all observables arise under any of the transformations of the odd type. On the other hand, the only reaction for which mirror relations hold for all observables as a result of any of the transformations of the even type is the reaction 0+½→0+½. The factor $\eta$ for mirror relations for the odd type is shown to be ${\mathrm{}(-1\mathrm{})\mathrm{}}^{a_S}$, where $a_S$ is the sum of the numbers of those types of unit vectors appearing in the observable that are involved in characterizing the restriction of that particular transformation on the $M$ matrix of its self-transforming reaction. It is shown that in order to distinguish among invariances under the various transformations of the odd type, observables in the $\mathrm{eee}$ or $\mathrm{ooo}$ subclasses are useless, and the observables in at least two other subclasses must be measured in order to pick out unambiguously the transformation under which the reaction is invariant. As a byproduct, the number of terms in the $M$ matrix of a self-transforming, but otherwise arbitrary, reaction under any of the various transformations is derived, thus generalizing the results of a previous paper.