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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi> <mml:mo>→</mml:mo> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi></mml:math> ) is replaced by the mirror symmetric, <i>complex conjugate</i> time transformation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi> <mml:mo>→</mml:mo> <mml:mo>±</mml:mo> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi></mml:math> . 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 <i>proper</i> time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>(</mml:mo> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo></mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi> <mml:mo>→</mml:mo> <mml:mo>±</mml:mo> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi></mml:math> , the Schrödinger equation and its complex conjugate conforms with a stochastic-mechanical model of the Brownian motion of a particle. From the present analysis, Nelson's osmotic velocity arises naturally and the quantum phase function admits a classical interpretation in terms of a continuous (scalar) potential field that influences the motion of the particle. The maximum phase variation defines the direction of the osmotic and current velocity. Whereas the paper focuses on the correlation between quantum mechanics and Ficks's second law for a constant diffusion coefficient, a discussion is also provided on the possibility of a quantum mechanical formulation for anomalous diffusion, where the diffusion coefficient is a function of space and/or time.