The Gauss, Lebesgue and Feynman Distributions Over Non-Archimedean Fields

Abstract

The definitions of the Gauss and Feynman integrals with respect to the number line ℝ,

$${I_G} = \int\limits_R {\varphi (x){{\text{e}}^{ - \frac{{{x^2}}}{{2b}}}}} \frac{{{\text{d}}x}}{{\sqrt {2\pi b} }},{I_F} = \int\limits_R {\varphi (x){{\text{e}}^{ - \frac{{{x^2}}}{{2bi}}}}} \frac{{{\text{d}}x}}{{\sqrt {2\pi b} i}},$$

are based on the Lebesgue measure dx on ℝ. Gauss integral I _G is an integral with respect to the Gauss’ measure on ℝ which is absolutely continuous relative to the Lebesgue measure dx. The situation is more complicated for an oscillating Feynman integral. If the function φ(x) is summable with respect to the measure dx on ℝ, the integral I _F is a Lebesgue integral (since the exponent is bounded). Now if φ is not Lebesgue summable, then the integral is regarded as a generalized integral in the sense of the theory of distributions on ℝ, i.e., I _F=(μ _F ,φ) where µ _F is the corresponding distribution (= a generalized function) on ℝ.