On Stochastic Schroedinger Equation as a Dirac Boundary-value Problem, and an Inductive Stochastic Limit

TL;DR

This paper establishes an equivalence between a quantum stochastic unitary evolution with jumps and a Dirac boundary value problem, using an ultrarelativistic limit to connect stochastic processes with quantum boundary problems.

Contribution

It introduces a novel connection between stochastic quantum evolutions and boundary value problems in an extra dimension, providing a new perspective on stochastic approximation.

Findings

Single-jump quantum stochastic evolution is equivalent to a Dirac boundary problem.

Ultrarelativistic limit relates Schrödinger boundary problems to stochastic models.

Microscopic time reversibility is analyzed within this framework.

Abstract

We prove that a single-jump quantum stochastic unitary evolution is equivalent to a Dirac boundary value problem on the half line in one extra dimension. It is shown that this exactly solvable model can be obtained from a Schroedinger boundary value problem for a positive relativistic Hamiltonian in the half-line as the inductive ultrarelativistic limit, correspondent to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation is reduced to the to the quantum-mechanical boundary value problem in the extra dimension. The question of microscopic time reversibility is also studied for this paper.