Vacuum polarization in a nonsimply connected spacetime

The vacuum polarization tensor of spinor electrodynamics is calculated to order $e^2$ in a nonsimply connected spacetime with the topology of $S^1\times R^3$, flat spacetime with periodicity in one spatial dimension. In contrast to the situation in a simply connected spacetime, there are two distinct types of spinors in this spacetime, the untwisted spinors and the twisted spinors. The vacuum polarization due to each type of spinor is calculated; it is found that the two types of spinors give rise to different vacuum polarization effects and hence must be regarded as physically inequivalent. The effect of the vacuum polarization upon photon propagation is considered, and it is shown that untwisted spinors give rise to noncausal effects whereas twisted spinors do not. The vacuum energy for a free spinor field in $S^1\times R^3$ is also calculated, and it is found that the twisted configuration gives rise to a vacuum state of lower energy than does the untwisted configuration. Various methods of incorporating twisted and untwisted spinors into the same theory, either as distinct fields or as different sectors of the same field theory, are discussed. No tractable method of incorporating both types of spinors is found which avoids the difficulty of noncausal photon propagation. It is therefore proposed that only the twisted spinors should be regarded as having physical significance.