[Paper Review] Quantum mechanics is the square root of a stochastic process

Marco Frasca|arXiv (Cornell University)|Jan 24, 2012

Quantum Mechanics and Applications3 references3 citations

TL;DR

This paper proposes that quantum mechanics arises as the square root of a stochastic process, specifically a combination of a Wiener (Brownian) process and a symmetric Bernoulli process (coin toss). By generalizing Itō calculus to fractional powers of stochastic differentials, the authors derive the Schrödinger equation from a complex-valued diffusion process, showing that the wave function emerges naturally from the square root of a classical stochastic process, thereby providing a novel stochastic foundation for quantum mechanics.

ABSTRACT

We prove a theorem showing that quantum mechanics is not directly a stochastic process characterizing Brownian motion but rather its square root. This implies that a complex-valued stochastic process is involved. Schrödinger equation is immediately derived without further assumptions using Itō integrals that are properly generalized. Fluctuations in space arise from a Brownian motion and the combined effect of a stochastic process with a symmetric Bernoulli distribution typical of tossing a coin.

Motivation & Objective

• To establish a mathematical foundation linking quantum mechanics to classical stochastic processes.
• To resolve long-standing objections that no classical stochastic process underlies quantum mechanics.
• To show that the Schrödinger equation emerges naturally from a square root of a Wiener process combined with a Bernoulli process.
• To generalize Itō calculus to handle non-integer powers of stochastic differentials, particularly (dW)^{1/2}.
• To demonstrate that the wave function is mathematically equivalent to a complex probability distribution derived from a stochastic process with null-measure components.

Proposed method

• Define a generalized Itō integral for non-integer powers of Wiener increments, such as (dW)^{1/2}, using regularization techniques to ensure convergence in the root mean square sense.
• Decompose the Wiener increment dW into sign(dW) and |dW|, treating sign(dW) as a null-measure process in Itō calculus but allowing its product with |dW| to yield non-trivial results.
• Introduce a Bernoulli process (representing a coin toss) as a fundamental stochastic component that, when combined with Brownian motion, induces the complex nature of the wave function.
• Derive the Schrödinger equation by applying the square root operation to a stochastic diffusion process, showing that [dX(t)]^2 yields terms proportional to the imaginary unit i.
• Generalize the stochastic process to include a potential via dX(t) = [dW(t) + V(X,t)dt]^{1/2}, leading to a modified diffusion equation that matches the time-dependent Schrödinger equation.
• Use regularization methods such as zeta function summation to handle divergent Riemann sums arising from absolute values of Wiener increments.

Experimental results

Research questions

• RQ1 Can the Schrödinger equation be derived from a classical stochastic process without additional assumptions?
• RQ2 What is the mathematical nature of (dW)^{1/2} in the context of stochastic calculus, and does it yield a well-defined limit?
• RQ3 How does the inclusion of a Bernoulli process (coin toss) transform a real-valued diffusion into a complex-valued quantum evolution?
• RQ4 Can the square root of a Wiener process reproduce the full structure of quantum mechanics, including time evolution and potential terms?
• RQ5 Does this formulation evade the objections raised by Grabert, Hänggi, and Talkner (GHT) regarding the failure of classical master equations to reproduce quantum transition probabilities?

Key findings

• The Schrödinger equation is derived as the square root of a stochastic process involving a Wiener process and a symmetric Bernoulli process, with no additional assumptions.
• The wave function emerges as a complex-valued probability amplitude from the square root of a real stochastic process, with the imaginary unit arising from the phase structure of the Bernoulli process.
• The square root process [dW]^{1/2} leads to a diffusion equation with a complex diffusion coefficient, yielding the Schrödinger equation in the form ∂ψ/∂t = -i(1/4)∂²ψ/∂X² + ...
• The generalized Itō integral for (dW)^{1/2} is well-defined only after regularization, such as using ζ(0) = -1/2, to handle divergent Riemann sums of |dW|.
• The inclusion of a potential V(X,t) in the stochastic process dX(t) = [dW(t) + V(X,t)dt]^{1/2} leads to the time-dependent Schrödinger equation, confirming the formalism's consistency with interacting systems.
• The formalism evades the GHT objection by showing that the Fokker-Planck operator reduces to the Schrödinger operator, thus preserving exact quantum transition probabilities.