A Proof for Poisson Bracket in Non-commutative Algebra of Quantum Mechanics

TL;DR

This paper demonstrates that the Poisson bracket in quantum mechanics can be derived from fundamental principles using Fourier transforms and Kramers-Kronig identities, eliminating the need for it to be postulated.

Contribution

It introduces a derivation of the Poisson bracket from basic concepts, replacing the traditional postulate with deeper mathematical foundations.

Findings

Poisson bracket derived from Fourier and Kramers-Kronig identities

Definition of Hermitian time-operator and its properties

Replaces postulate with fundamental mathematical concepts

Abstract

The widely accepted approach to the foundation of quantum mechanics is that the Poisson bracket, governing the non-commutative algebra of operators, is taken as a postulate with no underlying physics. In this manuscript, it is shown that this postulation is in fact unnecessary and may be replaced by a few deeper concepts, which ultimately lead to the derivation of Poisson bracket. One would only need to use Fourier transform pairs and Kramers-Kronig identities in the complex domain. We present a definition of Hermitian time-operator and discuss some of its basic properties.