Abstract
The general relationship between twisted, automorphic fields and the modern spectral theory of automorphic functions is outlined. For the purposes of illustrating the common features, the theory of a massless real scalar field with self interaction lambda phi 4/4! is considered in space-times with topologies Mn=(S1)n*R4-n, n=2, 3, and flat metrics. The vacuum energy density in the state phi =0 is calculated to order lambda for all topologically inequivalent configurations of real scalar fields with the help of Jackiw's results (1974) on the evaluation of the two-loop effective potential for the given theory in Minkowski space-time. It is found that taking the interaction into account does not change the rule of De Witt et al. (1979), the vacuum energy density of a real scalar field increases with increasing Mobiosity of the field.
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