Acausal quantum theory for non-Archimedean scalar fields

TL;DR

This paper develops a new quantum scalar field theory over $p$-adic spacetime, adapting axioms and equations to reflect the non-Archimedean, acausal nature of the setting, and connects it to $p$-adic Klein-Gordon equations.

Contribution

It introduces a $p$-adic quantum scalar field framework satisfying adapted Gårding--Wightman axioms and links second quantization to $p$-adic Klein-Gordon equations.

Findings

Constructed $p$-adic quantum scalar fields satisfying modified axioms.

Established the connection between second quantization and $p$-adic Klein-Gordon equations.

Reformulated axioms to account for acausality in $p$-adic spacetime.

Abstract

We construct a family of quantum scalar fields over a $p-$adic spacetime which satisfy $p-$adic analogues of the G\aa rding--Wightman axioms. Most of the axioms can be formulated the same way in 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.