Acausal quantum theory for non-Archimedean scalar fields

We construct a family of quantum scalar fields over a p-adic spacetime which satisfy p-adic analogues of the Gårding-Wightman axioms. Most of the axioms can be formulated in the same way for both the Archimedean and non-Archimedean frameworks; however, the axioms depending on the ordering of the background field must be reformulated, reflecting the acausality of p-adic spacetime. The p-adic scalar fields satisfy certain p-adic Klein-Gordon pseudo-differential equations. The second quantization of the solutions of these Klein-Gordon equations corresponds exactly to the scalar fields introduced here. The main conclusion is that there seems to be no obstruction to the existence of a mathematically rigorous quantum field theory (QFT) for free fields in the p-adic framework, based on an acausal spacetime. © 2019 World Scientific Publishing Company.

VIVO