Renormalization of self-interacting scalar field theories in a nonsimply connected spacetime

A discussion of the renormalization of λφ⁴ theory in nonsimply connected spacetime with topology S¹×R³ (periodic in one spatial direction) is given. The presence of either nontrivial topology or local spacetime curvature gives rise to nonlocal divergences which cannot be removed by local counterterms. The second-order self-energy is calculated, and it is shown that the nonlocal infinities cancel so that the theory is renormalizable to this order. This cancellation is shown to occur when the spacetime metric is conformally flat as well as when it is flat. In a nonsimply connected spacetime such as S¹×R³ it is possible to have a new field configuration (the twisted scalar field). It is shown that the second-order self-energy for a twisted, self-interacting scalar field is also free of nonlocal infinities.