Generating q-Commutator Identities and the q-BCH Formula

Abstract

Motivated by the physical applications of q-calculus and of q-deformations, the aim of this paper is twofold. Firstly, we prove the q-deformed analogue of the celebrated theorem by Baker, Campbell, and Hausdorff for the product of two exponentials. We deal with the q-exponential function expq⁡(x)=∑n=0∞(xn/[n]q!), where [n]q=1+q+⋯+qn-1 denotes, as usual, the nth q-integer. We prove that if x and y are any noncommuting indeterminates, then expq⁡(x)expq⁡(y)=expq⁡(x+y+∑n=2∞Qn(x,y)), where Qn(x,y) is a sum of iterated q-commutators of x and y (on the right and on the left, possibly), where the q-commutator [y,x]q≔yx-qxy has always the innermost position. When [y,x]q=0, this expansion is consistent with the known result by Schützenberger-Cigler: expq⁡(x)expq⁡(y)=expq⁡(x+y). Our result improves and clarifies some existing results in the literature. Secondly, we provide an algorithmic procedure for obtaining identities between iterated q-commutators (of any length) of x and y. These results can be used to obtain simplified presentation for the summands of the q-deformed Baker-Campbell-Hausdorff Formula.