Kommutator

The commutator, sometimes referred to as the Kommutator in German contexts, is a concept that measures how far two elements fail to commute in a given algebraic or physical setting. It appears in several disciplines, with definitions that are tailored to the structure in question, but the central idea remains consistent: a nonzero commutator signals noncommutativity.

In group theory, the commutator of two elements a and b in a group G is defined as [a,B] = a^{-1} b^{-1} a b. Equivalently, [a,b] = e if and only if ab = ba. The collection of all commutators generates the derived (or commutator) subgroup [G,G], the smallest normal subgroup for which the quotient G/[G,G] is abelian.

In Lie algebras and in matrix algebra, the commutator is a bilinear, antisymmetric operation [X,Y] that often plays the role of a Lie bracket. It satisfies the Jacobi identity in Lie algebras and, for matrices, is simply the matrix product difference AB − BA.

In quantum mechanics and related fields, the commutator of operators A and B is [A,B] = AB − BA. This quantity is fundamental to uncertainty principles; if [A,B] ≠ 0, the corresponding observables cannot be simultaneously sharp. A classic example is the canonical pair of position and momentum with [x,p] = iħ I.

Common properties include [A,B] = −[B,A] and tr([A,B]) = 0 for square matrices. The commutator thus encodes essential information about noncommutativity, symmetry, and the structure of the underlying mathematical or physical system.