  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.
