mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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. Life and career Marius Sophu ...

(1875), but also widely called the Trotter product formula, named after

Hale Trotter Hale Freeman Trotter (30 May 1931 – 17 January 2022)biographical information from ''American Men and Women of Science'', Thomson Gale 2004 was a Canadian-American mathematician, known for the Lie–Trotter product formula, the Steinhaus–Johns ...

, states that for arbitrary ''m'' × ''m''

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

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 most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

''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 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 exponential gives ...

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 Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...

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 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 action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional in ...

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 Masuo (written: 益男, 斗雄 or 満寿夫) is a masculine Japanese given name. Notable people with the name include: * (born 1958), Japanese actor and voice actor * (born 1947), Japanese sumo wrestler * (1934–1997), Japanese artist, writer and ...

). The same idea is used in the construction of splitting methods for the numerical solution of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s. Moreover, the Lie product theorem is sufficient to prove the

Feynman–Kac formula The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947, when Kac and Feynman were both Cornell faculty, Kac attended a present ...

. The Trotter–Kato theorem can be used for approximation of linear C₀-semigroups.

References

* Sophus Lie and Friedrich Engel (1888, 1890, 1893). ''Theorie der Transformationsgruppen'' (1st edition, Leipzig; 2nd edition, AMS Chelsea Publishing, 1970) * . * * * * * * * , pp. 99. *{{cite journal, first=Masuo, last= Suzuki, title=Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, journal=Comm. Math. Phys., volume=51, issue= 2, year=1976, pages= 183–190, doi=10.1007/bf01609348, s2cid= 121900332 Matrix theory Lie groups

See also

References