Physics over a finite field and Wick rotation

Abstract

The paper develops an earlier proposition that the physical universe is a finite system co-ordinatised by a very large finite field ${\mathrm{F}}_{\mathfrak{p}}$ which looks like the field of complex numbers to an observer. We construct a place (homomorphism) $\mathrm{lm}$ from a pseudo-finite field ${\mathrm{F}}_{\mathfrak{p}}$ onto the compactified field of complex numbers in such a way that certain multiplicative subgroups ${}^{\prime }{\mathbb{R}}_{+}^{\prime }$ and ${}^{\prime }{\mathbb{S}}^{\prime }$ correspond to the polar coordinate system ${\mathbb{R}}_{+}$ and $\mathbb{S}$ of $\mathbb{C}.$ Thus ${\mathrm{F}}_{\mathfrak{p}},$ ${}^{\prime }{\mathbb{R}}_{+}^{\prime }$ and ${}^{\prime }{\mathbb{S}}^{\prime }$ provide co-ordinates for physical universe. We show that the passage from the scale of units in ${}^{\prime }{\mathbb{R}}_{+}^{\prime }$ to the scale of units of ${}^{\prime }{\mathbb{S}}^{\prime }$ corresponds to a multiplication (on the logarithmic scale) by a very large integer $\mathfrak{i}$ equal approximately to $\sqrt{\mathfrak{p}}.$ This provides an explanation to the phenomenon of Wick rotation. In the same model we explain the phenomenon of phase transition in a large finite system