Commutator

In mathematics the commutator of two elements g and h of a group G is the element g^-1 h^-1 gh, often denoted by [g,h]. It is equal to 1 if and only if g and h commute, i.e., if and only if gh = hg. The subgroup generated by all commutators is called the derived group of G.

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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