Generalizations of the Kermack-McCrea Identity

Abstract

An identity for exponentials of operators, which has been of use in quantum mechanics, was proved in 1931 by Kermack and McCrea. Let X be a real or complex Banach space; and let A, B ∈ L(X) be such that A and B commute with [A, B]=AB-BA. Then
exp(A+B)=exp(A/2)exp(B)exp(A/2).
The objective of this paper is to obtain similar formulas for exp (A+B) in case A and B do not commute with [A,B], but do commute with higher order commutators. Exp (A+B) will then be expressed as a palindromic product of exponentials of operators. For example, if A and B commute with [[A, B], A] and [[A, B], B], and if λ=(1-3√4)^-1≅-1.7024, then
exp(A+B)=exp(1-λ/4 A)exp(1-λ/2 B)exp(1+λ/4 A)exp(λB)exp(1+λ/4 A)exp(1-λ/2 B)exp(1-λ/4 A).