Disentangling q-exponentials: A general approach

TL;DR

This paper develops a general method to express the q-exponential of a sum of non-commuting operators as an infinite product of q-exponentials involving q-commutators, extending the q-Zassenhaus formula.

Contribution

It provides a systematic approach to disentangle q-exponentials into products with arbitrary bases, confirming the simplest case and extending previous fourth-order results.

Findings

Explicit calculation of operators up to sixth order

Confirmation of the simplest q-Zassenhaus formula for specific bases

General method allowing arbitrary bases in the infinite product

Abstract

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson $q$-exponential of the sum of two non-$q$-commuting operators as an (in general) infinite product of $q$-exponential operators involving repeated $q$-commutators of increasing order, $E_q(A+B) = E_{q^{\alpha_0}}(A) E_{q^{\alpha_1}}(B) \prod_{i=2}^{\infty} E_{q^{\alpha_i}}(C_i)$. By systematically transforming the $q$-exponentials into exponentials of series and using the conventional Baker-Campbell-Hausdorff formula, we prove that one can make any choice for the bases $q^{\alpha_i}$, $i=0$, 1, 2, ..., of the $q$-exponentials in the infinite product. An explicit calculation of the operators $C_i$ in the successive factors, carried out up to sixth order, also shows that the simplest $q$-Zassenhaus formula is obtained for $\alpha_0 = \alpha_1 = 1$, $\alpha_2 = 2$, and $\alpha_3 = 3$. This confirms and reinforces a result of Sridhar and Jagannathan, based on fourth-order calculations.