A permutation-based power series representation of the Baker-Campbell-Hausdorff formula

TL;DR

This paper introduces a novel permutation-based power series representation of the Baker-Campbell-Hausdorff formula, simplifying coefficients and revealing new identities useful in physics.

Contribution

It presents a new form of power series coefficients using elementary permutations, improving upon previous hyperbolic function-based representations.

Findings

New permutation-based coefficient form derived

Simplified power series representation established

Proved new identities with physical applications

Abstract

The Baker-Campbell-Hausdorff formula was recently resummed exactly in one variable, and left as a power series in the other (Moodie and Long 2021 J. Phys. A: Math. Theor. 54 015208). The coefficients of the power series were provided as a sum of products of three hyperbolic functions that are analogous to the familiar commutator expansion. We find a new form of the power series coefficients that is a linear combination of just one of the hyperbolic functions. This linear combination can be understood through elementary permutations of the arguments of the hyperbolic function. We use generating functions and hyperbolic identities to relate the representations. The permutation representation is radically different to previously known structures in the Baker-Campbell-Hausdorff formula and naturally supersedes the previous power series representation for future use, in our opinion. It also allows us to prove a set of intriguing marching identities that are useful in physical applications and were discovered in the initial work.