Lie product formula

In mathematics, the Lie product formula, named for Sophus Lie (1875), states that for arbitrary n × n real or complex matrices A and B,

${\displaystyle e^{A+B}=\lim _{N\rightarrow \infty }(e^{A/N}e^{B/N})^{N},}$

where eA denotes the matrix exponential of A. The Lie–Trotter product formula (Trotter 1959) and the Trotter–Kato theorem (Kato 1978) extend this to certain unbounded linear operators A and B.

This formula is an analogue of the classical exponential law

${\displaystyle e^{x+y}=e^{x}e^{y}\,}$

which holds for all real or complex numbers x and y. If x and y are replaced with matrices A and B, and the exponential replaced with a matrix exponential, it is usually necessary for A and B to commute for the law to still hold. However, the Lie product formula holds for all matrices A and B, even ones which do not commute.

It is a trivial corollary of the Baker–Campbell–Hausdorff formula.

The formula has applications, for example, in the path integral formulation of quantum mechanics. It allows one to separate the Schrödinger evolution operator into alternating increments of kinetic and potential operators. The same idea is used in the construction of splitting methods for the numerical solution of differential equations. Moreover, the Lie product theorem is sufficient to prove the Feynman–Kac formula.