Related papers: General ordering theorem

General ordering theorem

URL: http://arxiv.org/abs/2302.01264v2
Date: Wed, 24 May 2023 14:07:14 GMT
Title: General ordering theorem
Authors: Luca Ferialdi
Abstract summary: We prove the General Ordering Theorem (GOT), which establishes a relation among any pair of orderings. We show that it acts on operators satisfying generic (i.e. operatorial) commutation relations. Remarkably, it establishes a formal relation between these two theorems, and it provides compact expressions for them, unlike the notoriously complicated ones currently known.
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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.

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