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

TL;DR

This work replaces the original BCH coefficient structure, which was a sum of products of three hyperbolic functions, with a permutation-driven representation that relies on a single hyperbolic quantity $h_N$ and a systematic use of generating functions. The authors derive edge identities to reduce multifactor hyperbolic terms, construct a symmetric partition representation, and then obtain a permutation representation via the coth angle-addition identity, culminating in a compact formula for the BCH coefficients. They prove the equivalence of the new and original representations through a reversal-symmetry argument and establish marching identities that enable all-orders perturbation theory. The permutation framework offers a radically different and potentially more practical basis for applying the BCH expansion in physics, including diagonalizing perturbation problems and generating perturbative corrections with clear denominators and denominators-related hyperbolic structures.

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.