Kommutation

Kommutation is the relation between two operators expressing that the order of their application does not affect the result. For linear operators A and B on a vector space, they commute if AB = BA. The commutator, denoted [A,B], is defined as AB − BA; it vanishes exactly when A and B commute.

In quantum mechanics and related theories, observables are represented by operators. Two observables can be measured precisely and simultaneously if their operators commute. If [A,B] ≠ 0, measuring one observable generally disturbs the other, and uncertainty relations impose limits on joint knowledge. A classic example is position x and momentum p, with [x,p] = iħ, meaning they do not commute. Another example is the components of angular momentum, [Lx,Ly] = iħLz, so only one component can be sharply defined at a time. Pauli matrices, used to describe spin-1/2, also fail to commute: [σx, σy] = 2iσz.

Mathematically, the commutator is a measure of non-commutativity in an algebra. In the language of Lie algebras, it defines the Lie bracket that governs the structure of the algebra of generators. If two linear operators commute, they can often be simultaneously analyzed or diagonalized in a suitable basis, simplifying spectral problems and dynamics. The concept is foundational in studying symmetries, conservation laws, and the algebraic structure of physical theories.