Acausal quantum theory for non-Archimedean scalar fields

Abstract

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 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 pseudodifferential equations. The second quantization of the solutions of these Klein-Gordon equations corresponds exactly to the scalar fields introduced here.

Regularization of p-adic string amplitudes, and multivariate local zeta functions

“…It is important to mention here, that in the p-adic framework, the existence of fundamental solutions for pseudodifferential operators is also a consequence of the fact that the Igusa local zeta functions admit a meromorphic continuations, see [50,Chapter 5], [30,Chapter 10]. This analogy turns out to be very important in the rigorous construction of quantum scalar fields in the p-adic setting, see [36] and the references therein.…”

Construction of p-Adic Covariant Quantum Fields in the Framework of White Noise Analysis

“…In the last 35 years p-adic QFT has attracted a lot of attention of physicists and mathematicians, see e.g. [1], [7]- [10], [14], [18]- [19], [24]- [30], [33]- [36], [41]- [42], [46]- [53], and the references therein.…”

“…Since Q p is not an ordered field, the notions of past and future do not exist, then any p-adic QFT is an acausal theory. The reader may consult the introduction of [33] for an in-depth discussion of this matter. Consequently, the reflection positivity, if it exists in the p-adic framework, requires a particular formulation, that we do not know at the moment.…”

An Introduction to the Theory of Local Zeta Functions from Scratch

“…Let us mention that the works of Speer [50] and Bollini, Giambiagi and González Domínguez [11] on regularization of Feynman amplitudes in quantum field theory are based on the analytic continuation of distributions attached to complex powers of polynomial functions in the sense of Gel'fand and Shilov [21] (see also [5], [7], [10] and [42], among others). This analogy turns out to be very important in the rigorous construction of quantum scalar fields in the p-adic setting (see [43] and the references therein).…”

“…Some general references for differential equations over non-Archimedean fields are [1], [33], [56], [62]. Finally, the reader interested in the relations between p-adic analysis and mathematical physics may enjoy [12], [13], [19], [20], [22]- [24], [27], [33], [37]- [39], [43], [48], [49], [52], [54], [56], [57] and [62].…”