A note on the Zassenhaus product formula

TL;DR

This paper introduces a straightforward computational method for calculating the terms in the Zassenhaus product formula for non-commuting operators, supported by a computer implementation and a conjecture on translating results into nested commutators.

Contribution

It presents a simple calculation method for Zassenhaus formula terms, includes a computer implementation, and proposes a conjecture relating these results to nested commutators.

Findings

Method successfully computes Zassenhaus terms up to high order

Computer implementation validates the calculation approach

Conjecture on nested commutators verified up to order 17

Abstract

We provide a simple method for the calculation of the terms c_n in the Zassenhaus product $e^{a+b}=e^a e^b \prod_{n=2}^{\infty} e^{c_n}$ for non-commuting a and b. This method has been implemented in a computer program. Furthermore, we formulate a conjecture on how to translate these results into nested commutators. This conjecture was checked up to order n=17 using a computer.