Abstract
In this paper, we study lambda phi ^4 scalar field theory defined on the unramified extension of p-adic numbers {mathbb {Q}}_{p^n}. For different “space-time” dimensions n, we compute one-loop quantum corrections to the effective potential. Surprisingly, despite the unusual properties of non-Archimedean geometry, the Coleman–Weinberg potential of p-adic field theory has structure very similar to that of its real cousin. We also study two formal limits of the effective potential, p rightarrow 1 and p rightarrow infty . We show that the prightarrow 1 limit allows to reconstruct the canonical result for real field theory from the p-adic effective potential and provide an explanation of this fact. On the other hand, in the prightarrow infty limit, the theory exhibits very peculiar behavior with emerging logarithmic terms in the effective potential, which has no analogue in real theories.
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