The flow method for the Baker-Campbell-Hausdorff formula: exact results

TL;DR

This paper introduces the flow method, a novel approach leveraging geometric numerical integration techniques to compute exact expressions for the Baker-Campbell-Hausdorff formula by embedding Lie algebras into vector field algebras.

Contribution

The paper presents a new general method, called the flow method, for deriving exact BCH formula expressions through embedding Lie algebras into vector field algebras and analyzing flows.

Findings

Effective for cases with analytically computable flows

Demonstrated on benchmark examples

Discusses extensions for intractable cases

Abstract

Leveraging techniques from the literature on geometric numerical integration, we propose a new general method to compute exact expressions for the BCH formula. In its utmost generality, the method consists in embedding the Lie algebra of interest into a subalgebra of the algebra of vector fields on some manifold by means of an isomorphism, so that the BCH formula for two elements of the original algebra can be recovered from the composition of the flows of the corresponding vector fields. For this reason we call our method the flow method. Clearly, this method has great advantage in cases where the flows can be computed analytically. We illustrate its usefulness on some benchmark examples where it can be applied directly, and discuss some possible extensions for cases where an exact expression cannot be obtained.