mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Lie product formula, named for

Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. He also made substantial cont ...

(1875), but also widely called the Trotter product formula, named after Hale Trotter, states that for arbitrary ''m'' × ''m'' real or

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

''A'' and ''B'',

e^ = \lim_ (e^e^)^n,

where ''e''^''A'' denotes the

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exp ...

of ''A''. The Lie–Trotter product formula and the Trotter–Kato theorem extend this to certain unbounded linear operators ''A'' and ''B''. This formula is an analogue of the classical exponential law

e^ = 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

, 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. The Lie product formula is conceptually related to the

Baker–Campbell–Hausdorff formula In mathematics, the Baker–Campbell–Hausdorff formula gives the value of Z that solves the equation e^X e^Y = e^Z for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultima ...

, in that both are replacements, in the context of noncommuting operators, for the classical exponential law. The formula has applications, for example, in the

path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or ...

of quantum mechanics. It allows one to separate the Schrödinger evolution operator (''propagator'') into alternating increments of kinetic and potential operators (the Suzuki–Trotter decomposition, after Trotter and Masuo Suzuki). 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. The Trotter–Kato theorem can be used for approximation of linear C₀-semigroups.

Proof

By the

(e^e^)^n = e^ \to e^

Notes

References

* * * * * * * * * * * * * {{Citation , first=V.S. , last=Varadarajan , title=Lie Groups, Lie Algebras, and Their Representations , publisher=Springer-Verlag , year=1984 , isbn=978-0-387-90969-1, pp. 99. Matrix theory Lie groups

Proof

See also

Notes

References