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For purely hadronic processes, the standard connection between spin and statistics as well as separate invariances under charge conjugation, parity, and time reversal are shown to be consequences of self-consistency in topological particle theory. I. INTRODUCTION In the preceding work' (referred to hereafter as FJ) the phases for zero-entropy terms in the topological expansion were determined through systematic enforcement of the requirements of pole factorization, Hermitian analyticity and crossing symmetry. In this paper we investigate the spin and statistics properties of these zero-entropy terms and also their invariance properties under parity, timereversal, and charge-conjugation transformations. All these properties are shown to follow from self-consistency considerations and a knowledge of the amplitude phases derived in FJ. In Sec. II the theory is shown to obey the standard spin and statistics relations. The proof differs considerably from an S-matrix proof of spin and statistics given a number of years ago by Stapp, since these topological amplitudes are not unitary at the zero-entropy level. We also show (Secs. III, IV, and V) that the zero-entropy sum is separately invariant under parity, time-reversal, and charge-conjugation transformations. The zero-entropy level and higher corrections arising from connected sums of these zero-entropy terms represent hadronic processes so we have been able to prove the existence of the standard discrete symmetries for strong interactions as well as the standard statistics. It should be emphasized that these symmetries were not built into the theory. Only requirements of self-consistency, pole factorization, crossing symmetry and Hermitian analyticity are used together with a standard discontinuity equation which embodies the dynamics. Again these results should be contrasted with Stapp's original work where the product of these three symmetries was shown to hold for the full unitary S matrix. The discussion of the discrete invariances is carried out in the framework of the topological bootstrap theory. After defining the invariances we simply investigate whether the amplitude for a process has the same absolute value as the appropriately transformed process. It is not therefore necessary to define the action of these symmetry transformations on individual incoming or outgoing state vectors. Thus many tiresome phase questions are avoided and yet any physically observable phases (such as those associated with intrinsic parities ) are unambiguously evident. The study of electroweak processes within the framework of the topological theory shows that the amplitudes for such processes appear above the zero-entropy level where the discrete symmetries no longer hold separately. The topological theory thus provides a basis for understanding the presence of the separate discrete symmetries in certain (strong) processes as well as their absence in other (weak) processes.

Derivation of discrete invariances (T,C, andP) and the connection between spin and statistics in topological particle theory