General ordering theorem

TL;DR

The paper introduces the General Ordering Theorem (GOT), a systematic method to relate different operator orderings in quantum mechanics, unifying and simplifying key algebraic theorems like Magnus and Baker-Campbell-Hausdorff.

Contribution

It presents the GOT, a novel, general framework for relating various operator orderings, and demonstrates its power by deriving important algebraic theorems in a simplified form.

Findings

GOT unifies relations among operator orderings.

Recovered Magnus expansion and BCH formula as special cases.

Provides compact, simplified expressions for complex algebraic theorems.

Abstract

The problem of ordering operators has afflicted quantum mechanics since its foundation. Several orderings have been devised, but a systematic procedure to move from one ordering to another is still missing. The importance of establishing relations among different orderings is demonstrated by Wick's theorem (which relates time ordering to normal ordering), which played a crucial role in the development of quantum field theory. We prove the General Ordering Theorem (GOT), which establishes a relation among any pair of orderings, that act on operators satisfying generic (i.e. operatorial) commutation relations. We expose the working principles of the GOT by simple examples, and we demonstrate its potential by recovering two famous algebraic theorems as special instances: the Magnus expansion and the Baker-Campbell-Hausdorff formula. Remarkably, the GOT establishes a formal relation between these two theorems, and it provides compact expressions for them, unlike the notoriously complicated ones currently known.