New BCH-like relations of the su(1, 1), su(2) and so(2, 1) Lie algebras

TL;DR

This paper introduces new BCH-like relations for su(1, 1), su(2), and so(2, 1) Lie algebras, simplifying the composition of group elements and aiding in quantum system evolution calculations.

Contribution

The work presents novel BCH-like formulas for specific Lie algebras, enabling straightforward composition and application to time-dependent quantum Hamiltonians.

Findings

Recovered the composition law of two squeezing operators.

Demonstrated the use of relations to compute time evolution operators.

Validated results through self-consistent checks.

Abstract

In this work we demonstrate new BCH-like relations involving the generators of the su(1, 1), su(2) and so(2, 1) Lie algebras. We use our results to obtain in a straightforward way the composition of an arbitrary number of elements of the corresponding Lie groups. In order to make a self-consistent check of our results, as a first application we recover the non-trivial composition law of two arbitrary squeezing operators. As a second application, we show how our results can be used to compute the time evolution operator of physical systems described by time-dependent hamiltonians given by linear combinations of the generators of the aforementioned Lie algebras.