[Home]History of Commutator

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Revision 4 . . November 21, 2001 1:53 am by AxelBoldt [commutators algebra elements]
Revision 3 . . November 21, 2001 1:51 am by AxelBoldt [Fleshed out commutators of algebras]
Revision 2 . . (edit) November 21, 2001 12:06 am by Gareth Owen
Revision 1 . . November 21, 2001 12:05 am by Goochelaar [New]
  

Difference (from prior major revision) (author diff)

Changed: 3c3
Commutators are also defined for other algebraic structures, especially rings and associative algebras. Here, the commutator [a,b] of two elements a and b is also called the Lie bracket and is defined by [a,b] = ab - ba. It is zero if and only if a and b commute. By using the Lie bracket, every associative algebra can be turned into a Lie algebra. The commutator of two operators defined on a Hilbert space is an important concept in quantum mechanics since it measures how well the two observables described by the operators can be measured simultaneously. The Uncertainty Principle is ultimately a theorem about these commutators.
Commutators are also defined for rings and associative algebras. Here, the commutator [a,b] of two elements a and b is also called the Lie bracket and is defined by [a,b] = ab - ba. It is zero if and only if a and b commute. By using the Lie bracket, every associative algebra can be turned into a Lie algebra. The commutator of two operators defined on a Hilbert space is an important concept in quantum mechanics since it measures how well the two observables described by the operators can be measured simultaneously. The Uncertainty Principle is ultimately a theorem about these commutators.

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