11 - Non-Archimedean White Noise, Pseudodifferential Stochastic Equations, and Massive Euclidean Fields

Summary

Introduction

There are general arguments that suggest that one cannot make measurements in regions of extent smaller than the Planck length ≈ 10−³³ cm, see e.g. [413] and the references therein. The construction of physical models at the level of the Planck scale is a relevant scientific problem and a very important area of mathematical research. In [436]–[438], I. Volovich conjectured the non-Archimedean nature of space-time at the level of the Planck scale. This conjecture has given rise to a lot of research, for instance, in quantum mechanics, see e.g. [222], [430], [433], [456], [457], in string theory, see e.g. [91], [163], [175], [431], [425], [427], and in quantum field theory, see e.g. [280], [344], [396]. On the other hand, the interaction between quantum field theory and mathematics is very fruitful and deep, see e.g. [167], [171], [213], [220], [219], [444], [445], among several articles. Let us mention explicitly the connection with arithmetic, see e.g. [213], [308], [371]. From this perspective the investigation of quantum fields in a non-Archimedean setting is quite a natural problem.

In this chapter we present a class of non-Archimedean Euclidean fields, in arbitrary dimension, which are constructed as solutions of certain covariant p-adic stochastic pseudodifferential equations, by using techniques involving white-noise calculus. This chapter is based on [472]. The connection between quantum fields and SPDEs has been studied intensively in the Archimedean setting, see e.g. [9]–[30] and the references therein. A massive non-Archimedean field ϕ is a random field parametrized by, the nuclear countably Hilbert spaces introduced in Chapter 10, which depends on (q, l,m, α), where q is an elliptic quadratic form, l is an elliptic polynomial, and m and α are positive numbers. Here m is the mass of ϕ. Heuristically, is the solution of (L_α + m²), where is a generalized Levy noise. This type of noise is introduced in this chapter. Here, where Fq := F is the Fourier transform defined using the bilinear symmetric form corresponding to the quadratic form q. However, in this chapter we work with Fourier transforms defined by using arbitrary bilinear forms.