Probability Bracket Notation, Wick-Matsubara Relation, Density Operators, and Microscopic Probability Modeling

TL;DR

This paper introduces a probability bracket notation (PBN) and explores its connection to Wick rotation and Matsubara relations, providing a new framework for microscopic probability modeling in quantum and thermodynamic systems.

Contribution

It develops the PBN framework, linking quantum operators and entropy in probability space, and extends Wick-Matsubara relations to microscopic probabilistic processes.

Findings

Energy expectation approaches the ground state over time.

Von Neumann entropy vanishes in the long-term limit.

Reproduces thermodynamic quantities like internal energy using PBN.

Abstract

Following the Dirac vector bracket notation (VBN), we proposed the probability bracket notation (PBN) in our previous paper. We mentioned that under the special Wick rotation (imaginary time), a stationary Schrodinger equation in the Hilbert space transforms into the master equation of a microscopic probabilistic process (MPP) in the probability space. In this article, we first study the MPP of the system of a single particle, we show that the energy expectation of the MPP eventually approaches the lowest energy level in its initial condition and its von Neumann entropy finally vanishes. Then we explore the MPP for the quantum system of identical particles in the Fock space, we recover the expected occupation number of particles and the grand partition function in quantum statistics by connecting time with temperature (the Wick-Matsubara relation). We also reproduce the internal energy of an ideal gas in thermodynamics by using the relation. To address the entropy issue and relate the PBN with research topics of statistics in the literature, we express the density operators and the von Neumann entropy in the probability space by using the PBN. The Wick-Matsubara relation plus the PBN might provide a new way of microscopic probability modeling.