Unitarity, Causality, and Fermi Statistics

Conventionally, Poincaré-invariant $S$-matrix elements are constructed from auxiliary field operators, which transform like representations of an auxiliary group. Invariance with respect to the index transformations of this group may be extended to couple spin to internal symmetry properties in a covariant manner, as in $\tilde{U}\mathrm{}\left(12\right)\mathrm{}$ and $\mathrm{SL}(6,C)\mathrm{}$ theories. It is shown that such an index invariance of the $S$ matrix is compatible with unitarity only if the auxiliary operators are unitary representations of the auxiliary group. It is shown further that local fields transforming as such unitary representations can be made causal only if they satisfy commutation (not anticommutation) relations. Thus for index-invariant theories, we establish a direct incompatability between unitarity and causality for Fermi particles.