Real-space diffusion theory from quantum mechanics using analytic continuation

Area of Science:

• Physics
• Quantum Mechanics
• Statistical Mechanics

Background:

• Brownian motion and quantum mechanics are typically described by distinct frameworks.
• Formal analytic continuation and Wick rotation are standard tools in theoretical physics.
• Nelson's stochastic mechanics provides a link between quantum mechanics and Brownian motion.

Purpose of the Study:

• To establish a physical correspondence between Brownian motion and quantum mechanics.
• To explore the implications of a complex conjugate time transformation.
• To provide classical interpretations for quantum mechanical concepts.

Main Methods:

• Formal analytic continuation with a complex conjugate time transformation (replacing Wick rotation).
• Analysis of the invariance of the wave function's square modulus.
• Conforming the Schrödinger equation with a stochastic-mechanical model.

Main Results:

• A direct physical link between quantum mechanics and Brownian motion is established.
• Born's probability density interpretation is connected to proper time via time symmetry breaking.
• Nelson's osmotic velocity and a classical potential interpretation of the quantum phase emerge naturally.
• The Schrödinger equation aligns with stochastic mechanics under the transformation.

Conclusions:

• The study reveals a deep connection between quantum mechanics and classical diffusion processes.
• Time symmetry breaking plays a crucial role in linking microscopic quantum phenomena to macroscopic classical behavior.
• The framework allows for classical interpretations of quantum phase and osmotic velocity.
• Potential for extending the model to anomalous diffusion is discussed.