Interaction Measures on the Space of Distributions over the Field of p-Adic Numbers

TL;DR

This paper constructs measures on the space of distributions over p-adic numbers, modeling finite volume polynomial interactions in a p-adic quantum field theory using Gaussian measures related to elliptic pseudo-differential operators.

Contribution

It introduces a novel approach by using Gaussian measures associated with elliptic pseudo-differential operators for p-adic quantum field theory.

Findings

Constructed measures on distribution spaces over Q_p^n for n ≤ 4.

Modeled polynomial interactions in p-adic quantum field theory.

Extended to Euclidean P(φ)-theories with boundary conditions.

Abstract

We construct measures on the space $\mathcal D'(Q_p^n)$, $n\le 4$, of Bruhat-Schwartz distributions over the field of $p$-adic numbers, corresponding to finite volume polynomial interactions in a $p$-adic analog of the Euclidean quantum field theory. In contrast to earlier results in this direction, our choice of the free measure is the Gaussian measure corresponding to an elliptic pseudo-differential operator over $Q_p^n$. Analogs of the Euclidean $P(\phi)$-theories with free and half-Dirichlet boundary conditions are considered.