A commutator is a mathematical operation used in various fields, including quantum mechanics and group theory. It measures the extent to which two elements, such as operators or matrices, fail to commute, meaning the order of their application matters. The commutator of two elements A and B is defined as [A, B] = AB - BA. In quantum mechanics, commutators play a crucial role in understanding the behavior of physical systems. For example, the position and momentum operators have a non-zero commutator, indicating that they cannot be precisely measured simultaneously, a principle known as the Heisenberg uncertainty principle.