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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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