Commutation relations of operator monomials

TL;DR

This paper derives a formula expressing the commutator of operator monomials with constant relations using Bernoulli numbers, highlighting their role in quantum identities and applications in ordering and Ehrenfest theorem.

Contribution

It introduces a new formula linking commutators of operator monomials to Bernoulli numbers, enhancing understanding of quantum operator identities.

Findings

Expressed commutators in terms of Bernoulli numbers and anticommutators.

Connected Bernoulli numbers to quantum identities like Baker-Campbell-Hausdorff.

Proposed applications to ordering problems and Ehrenfest theorem.

Abstract

In this short paper, the commutator of monomials of operators obeying constant commutation relations is expressed in terms of anticommutators. The formula involves Bernoulli numbers or Euler polynomials evaluated in zero. The role of Bernoulli numbers in quantum-mechanical identities such as the Baker-Campbell-Hausdorff formula is emphasized and applications connected to ordering problems as well as to Ehrenfest theorem are proposed.