Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box

Appendix D

The Baker–Hausdorff formula

The algebra of operators is noncommutative, so that all of the ordinary algebra, calculus, and analysis with ordinary numbers becomes of small utility for operators. Thus, for a single operator, α, ordinary functions of this operator, such as A = exp α, can be defined, for example, by power series. These functions obey the rules of ordinary analysis even though α is an operator. But if another operator β is introduced with which α does not commute, the question of functions of the two variables α, β is beset with commutation difficulties and the simplest theorems of analysis are lost. For example, if B = exp β, it is not true that BA, that is exp β exp α, is equal to exp(β + α).

—Richard Phillips Feynman [200]

This is a very useful tool to disentangle exponentials of certain operators often appear in quantum optics.¹ Suppose that  and are two operators such that , where commutes with both  and . Then the Baker–Hausdorff formula holds:

Notice ...