Parcels of Universe or why Schr\"odinger and Fourier are so relatives?

TL;DR

This paper explores the deep mathematical and physical connection between the Fourier heat equation and the Schr"odinger wave equation, highlighting their relation through a Wick rotation and discussing implications for quantum and stochastic processes.

Contribution

It provides a new perspective on the relationship between heat diffusion and quantum mechanics, linking noncommutative geometry and stochastic processes to explain their connection.

Findings

The Schr"odinger equation can be derived from the Fourier heat equation via a Wick rotation.

The connection suggests underlying stochastic processes in quantum mechanics.

Volume quantization in noncommutative geometry relates to the Fourier-Schr"odinger relationship.

Abstract

This paper is about the surprising connection between the Fourier heat equation and the Schr\"odinger wave equation. In fact, if the independent "time" variable in the heat equation is replaced by the time variable multiplied by $i=\sqrt{-1}$, the heat equation becomes the Schr\"odinger equation. Two quite different physical phenomena are put in close connection: the heat diffusion in a material and the probability amplitude of particles in an atom. It is a fact of life that the movements of a small particle floating randomly in a fluid, the well-known Brownian motion, is regulated by the Fourier equation while the probabilistic behavior of the matter around us, the quantum world, is driven by the Schr\"odinger equation but no known stochastic process seems at work here. The apparent simplicity of the formal connection by a "time-rotation", a Wick rotation as it is commonly known, seems to point otherwise. Why this connection? Is there any physical intuitive explanation? Is there any practical value? In this paper, the authors attempt to shed some light on the above questions. The recent concept of volume quantization in noncommutative geometry, due to Connes, Chamseddine and Mukhanov, points again to stochastic processes also underlying the quantum world making Fourier and Schr\"oodinger strict relatives.