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,

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

where e^A 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

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

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.

See also

• Time-evolving block decimation

References

• Sophus Lie and Friedrich Engel (1888, 1890, 1893). Theorie der Transformationsgruppen (1st edition, Leipzig; 2nd edition, AMS Chelsea Publishing, 1970) ISBN 0828402329
• Albeverio, Sergio A.; Høegh-Krohn, Raphael J. (1976), Mathematical Theory of Feynman Path Integrals: An Introduction, Lecture Notes in Mathematics, 423 (1st ed.), Berlin, New York: Springer-Verlag, doi:10.1007/BFb0079827, ISBN 978-3-540-07785-5.
• Hall, Brian C. (2003), Lie groups, Lie algebras, and representations: an elementary introduction, Springer, ISBN 978-0-387-40122-5, pp. 35.
• Hazewinkel, Michiel, ed. (2001), "Trotter product formula", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Kato, Tosio (1978), "Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups", Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), Adv. in Math. Suppl. Stud., 3, Boston, MA: Academic Press, pp. 185–195, MR 538020
• Trotter, H. F. (1959), "On the product of semi-groups of operators", Proceedings of the American Mathematical Society, 10 (4): 545–551, doi:10.2307/2033649, ISSN 0002-9939, JSTOR 2033649, MR 0108732
• Varadarajan, V.S. (1984), Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, ISBN 978-0-387-90969-1, pp. 99.
• Suzuki, Masuo (1976). "Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems". Comm. Math. Phys. 51: 183–190. doi:10.1007/bf01609348.