Lie product formula

In mathematics, the Lie product formula states that for arbitrary matrices A and B,

$e^{A+B}=lim_{N\rightarrow \infty }(e^{A/N}e^{B/N})^{N}$.

This formula is an analogue of the classical exponential law

$e^{x+y}=e^{x}e^{y}$

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 don't commute. More generally, the Lie product identity holds when A and B are taken to be in certain classes of special linear operators.

The formula has applications, for example, in the path integral formulation of quantum mechanics. It allows one to separate the Schrodinger evolution operator into alternating increments of kinetic and potential operators.

References

Mathematical Theory of Feynman Path Integrals. Springer Berlin / Heidelberg, ISSN 0075-8434 (Print) 1617-9692 (Online), http://www.springerlink.com/content/w82057554x1w/?p=438a9838c41148099a8aa1e5a7dfb723&pi=0