Physics over a finite field and Wick rotation

TL;DR

This paper models the physical universe using a large finite field that resembles complex numbers, providing a novel algebraic explanation for Wick rotation and phase transitions in finite systems.

Contribution

It constructs a finite field framework that mimics complex analysis and explains Wick rotation and phase transitions through algebraic and number-theoretic methods.

Findings

Finite field models can replicate complex number structures.

Wick rotation corresponds to a large integer multiplication in the model.

Phase transitions are explained via algebraic properties of the finite system.

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 $'\mathbb{R}'_+$ and $'\mathbb{S}'$ correspond to the polar coordinate system $\mathbb{R}_+$ and $\mathbb{S}$ of $\mathbb{C}.$ Thus $\mathrm{F}_\mathfrak{p},$ $'\mathbb{R}'_+$ and $'\mathbb{S}'$ provide co-ordinates for physical universe. We show that the passage from the scale of units in $'\mathbb{R}'_+$ to the scale of units of $'\mathbb{S}'$ 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