부산대학교 도서관

That specifying the phase factors which appear in the transformation law of fields under inversions can lead to selection rules was pointed out by Yang and Tiomno [Phys. Rev. (2) {\bf 79} (1950), 495--498; MR0037221 (12,227f)] and Žarkov [Ž. Èksper. Teoret. Fiz. {\bf 20} (1950), 492--496]. The subject was further discussed by Wick, the reviewer and Wigner [Phys. Rev. (2) {\bf 88} (1952), 101--105; MR0053796 (14,827e)] who paid special attention to the observability of the distinctions introduced by such phases. (The fields in whose transformation laws the phases are specified need not be observable; two theories with fields having different phases can predict the same results for all observables. Then the distinction introduced by the difference in phases is unobservable.) There is a wealth of literature applying these ideas to elementary particles of which only P. T. Matthews [Nuovo Cimento (10) {\bf 6} (1957), 642--649; MR0092635 (19,1138b)] will be mentioned. The present paper discusses the arbitrariness of the phases of the transformation law of fields under $P$, $C$, $T$ and their products within the context of standard relativistic local quantum field theory. The new information it contains is indispensable for a proper understanding of the subject. First, the relation between these transformations and multiplicative symmetries is studied. (A multiplicative symmetry is a unitary operator $U$ with the property that for some complex numbers $n_j(U)$ of absolute value one $U\varphi_j(x)U^{-1}=n_j(U)\varphi_j(x)$ for all fields $\varphi_j$.) It is shown that for every multiplicative symmetry $U$, $UP=PU$, $UC=CU^{-1}$, $UT=TU^{-1}$. Then, for any theory which is invariant with respect to an inversion $I$ satisfying $UI=IU^{-1}$ (e.g., $C$, $T$, $PC$, $PT$), it is shown that there exists a corresponding multiplicative symmetry, $U_I$. This $U_I$ can in turn be used to construct a new theory in which the phase in the transformation law of the complex fields under $I$ is one, but all observables are unchanged. \par Conclusion: The phases appearing in the transformation laws of complex fields under $C$, $T$, $PC$, $PT$ are unobservable. However, they cannot be simultaneously changed in an arbitrary way because such operators as $P$, $CT$, and $PCT$ commute with multiplicative operators. The detailed analysis of the possibilities is too complicated to summarize here. Several examples are given, including one which is invariant under $P$ only if non-real phases are used. \{If taken literally, the proofs of the paper are not markedly convincing, but they can all easily be made rigorous. Example: From $$ UP\varphi_j(x)P^{-1}U^{-1}=PU\varphi_j(x)U^{-1}P^{-1}, $$ the authors conclude without further ado $UP=PU$. The reader has to supply the assumptions and proof necessary for this conclusion.\}