부산대학교 도서관

That specifying the phase factors which appear in the transformation law offields under inversions can lead to selection rules was pointed out by Yangand Tiomno [Phys. Rev. (2) {\bf 79} (1950), 495--498; MR0037221 (12,227f)] andŽarkov [Ž. Èksper. Teoret. Fiz. {\bf 20} (1950), 492--496]. Thesubject was further discussed by Wick, the reviewer and Wigner [Phys. Rev.(2) {\bf 88} (1952), 101--105; MR0053796 (14,827e)] who paid special attentionto the observability of the distinctions introduced by such phases. (Thefields in whose transformation laws the phases are specified need not beobservable; two theories with fields having different phases can predictthe same results for all observables. Then the distinction introduced bythe difference in phases is unobservable.) There is a wealth of literatureapplying these ideas to elementary particles of which only P. T. Matthews[Nuovo Cimento (10) {\bf 6} (1957), 642--649; MR0092635 (19,1138b)] will bementioned. The present paper discusses the arbitrariness of the phases ofthe transformation lawof fields under $P$, $C$, $T$ and their products within the context ofstandard relativistic local quantum field theory. The new information itcontains is indispensable for a proper understanding of the subject. First,the relation between these transformations and multiplicative symmetries isstudied. (A multiplicative symmetry is a unitary operator $U$ with theproperty 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 isshown that for every multiplicative symmetry $U$, $UP=PU$, $UC=CU^{-1}$,$UT=TU^{-1}$. Then, for any theory which is invariant with respect to aninversion $I$ satisfying $UI=IU^{-1}$ (e.g., $C$, $T$, $PC$, $PT$), it isshown 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 phasein the transformation law of the complex fields under $I$ is one, but allobservables are unchanged.\parConclusion: The phases appearing in the transformation laws of complexfields under $C$, $T$, $PC$, $PT$ are unobservable. However, they cannot besimultaneously changed in an arbitrary way because such operators as $P$,$CT$, and $PCT$ commute with multiplicative operators. The detailedanalysis of the possibilities is too complicated to summarize here. Severalexamples are given, including one which is invariant under $P$ only ifnon-real phases are used. \{If taken literally, the proofs of the paper arenot 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 supplythe assumptions and proof necessary for this conclusion.\}