Lie product formula: Difference between revisions
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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 |
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 |
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[[Category:Mathematics]] |
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{{Uncategorized|date=October 2008}} |
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[[Category:Lie theory]] |
Revision as of 19:53, 9 October 2008
In mathematics, the Lie product formula states that for arbitrary matrices A and B,
.
This formula is an analogue of the classical exponential law
which holds for all real or complex numbers and . If and are replaced with matrices and , and the exponential replaced with a matrix exponential, it is usually necessary for and to commute for the law to still hold. However, the Lie product formula holds for all matrices and , 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