Generalizations of the Kermack-McCrea Identity

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Authors

  • Department of Mathematics, University of Arizona, Tuscon, Arizona-85721 ,US
  • Department of Mathematics, University of Arizona, Tuscon, Arizona-85721 ,US

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).

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Published

1991-12-01

How to Cite

Lomont, J. S., & Cheema, M. S. (1991). Generalizations of the Kermack-McCrea Identity. The Journal of the Indian Mathematical Society, 57(1-4), 79–93. Retrieved from https://www.informaticsjournals.com/index.php/jims/article/view/21905