2009/01/30 by Xing Wang, Wang, Xing M. · 1 citation
Computer Science · Physics and Astronomy · #81S40 #82C10 #82C31 #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Dynamics and Pattern Formation #Quantum Mechanics and Applications #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.0901.4816
openalex publication_date 2009/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Schrödinger 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.