From Dirac Notation to Probability Bracket Notation: Time Evolution and Path Integral under Wick Rotations

TL;DR

This paper introduces the Probability Bracket Notation (PBN), a formalism inspired by Dirac notation, to unify quantum mechanics and stochastic processes through transformations like Wick rotations, enabling new insights into path integrals and diffusion equations.

Contribution

It extends PBN with Wick rotations and anti-Hermitian operators, deriving path integrals and Euclidean Lagrangians for diffusions and oscillators, bridging quantum and stochastic frameworks.

Findings

PBN can transform Schrödinger equations into master equations under Wick rotations.

Derived Euclidean Lagrangian for induced diffusion processes.

Demonstrated PBN's versatility in analyzing quantum and stochastic systems.

Abstract

In this work, we advance the development of the Probability Bracket Notation (PBN), a formalism inspired by Dirac's notation in quantum mechanics, to provide a unified framework for probability modeling. We demonstrate that under a Special Wick Rotation (SWR), an imaginary-time map, the Schr\"odinger equation, the transition amplitude, and its associated path integral in Dirac notation transform into the master equation, the transition probability, and its Euclidean path integral in the PBN, from which we can reproduce the master equation, representing induced micro-diffusion processes. By extending this approach through a General Wick Rotation (GWR) and employing an anti-Hermitian wave-number operator, we perform parallel derivations of path integrals in both the Dirac and PBN frameworks. This leads to the formulation of the Euclidean Lagrangian for induced diffusions and the strong-damping harmonic oscillator (described by the Smoluchowski diffusion equation). Our findings highlight the versatility of the PBN in bridging quantum mechanics and stochastic processes, offering a coherent notation system for analyzing time evolution and path integrals across these domains.