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

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