Lie product formula

In mathematics, the Lie product formula, named for Sophus Lie (1875), but also widely called the Trotter product formula,^[1] named after Hale Trotter, states that for arbitrary m × m real or complex matrices A and B,^[2] $${\displaystyle e^{A+B}=\lim _{n\rightarrow \infty }(e^{A/n}e^{B/n})^{n},}$$ where e^A denotes the matrix exponential of A. The Lie–Trotter product formula^[3] and the Trotter–Kato theorem^[4] extend this to certain unbounded linear operators A and B.^[5]

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.

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 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.^[6]

The Trotter–Kato theorem can be used for approximation of linear C₀-semigroups.^[7]

• Time-evolving block decimation

• Albeverio, Sergio A.; Høegh-Krohn, Raphael J. (1976), Mathematical Theory of Feynman Path Integrals: An Introduction, Lecture Notes in Mathematics, vol. 423 (1st ed.), Berlin, New York: Springer-Verlag, doi:10.1007/BFb0079827, hdl:10852/44049, ISBN 978-3-540-07785-5
• Appelbaum, David (2019). "The Feynman-Kac Formula via the Lie-Kato-Trotter Product Formula". Semigroups of Linear Operators : With Applications to Analysis, Probability and Physics. Cambridge University Press. pp. 123–125. ISBN 978-1-108-71637-6.
• Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, ISBN 978-1461471158
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-0-387-40122-5
• "Trotter product formula", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Cohen, Joel E.; Friedland, Shmuel; Kato, Tosio; Kelly, F. P. (1982). "Eigenvalue inequalities for products of matrix exponentials" (PDF). Linear Algebra and Its Applications. 45: 55–95. doi:10.1016/0024-3795(82)90211-7.
• Ito, Kazufumi; Kappel, Franz (1998). "The Trotter-Kato Theorem and Approximation of PDEs". Mathematics of Computation. 67 (221): 21–44. doi:10.1090/S0025-5718-98-00915-6. JSTOR 2584971.
• Joel E. Cohen; Shmuel Friedland; Tosio Kato; F. P. Kelly (1982), "Eigenvalue inequalities for products of matrix exponentials" (PDF), Linear Algebra and Its Applications, 45: 55–95, doi:10.1016/0024-3795(82)90211-7
• 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., vol. 3, Boston, MA: Academic Press, pp. 185–195, MR 0538020
• Lie, Sophus; Engel, Friedrich (1970). Theorie der Transformationsgruppen (in German). New York: American Mathematical Soc. ISBN 0-8284-0232-9.
• 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
• 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 (2): 183–190. doi:10.1007/bf01609348. S2CID 121900332.
• Varadarajan, V.S. (1984), Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, ISBN 978-0-387-90969-1, pp. 99.