Real-space diffusion theory from quantum mechanics using analytic continuation.

We show that a physical correspondence between Brownian motion and quantum mechanics can be established by formal analytic continuation if Wick rotation ( ) is replaced by the mirror symmetric, time transformation . Invariance of the square modulus of the wave function under this transformation reveals a tight connection between Born's interpretation of the probability density and the time , as being both the result of a time symmetry breaking in the informational content of the microscopic (quantum) world, which takes place as we move to a macroscopic level. We find that under the transformation , the Schrödinger equation and its complex conjugate conforms with a stochastic-mechanical model of the Brownian motion of a particle.