noncommuting

Noncommuting refers to a situation where the order of applying two operations affects the result. In mathematics, a binary operation is commutative if changing the order does not change the outcome. When that property fails, the structure is described as noncommutative. Noncommutativity is common in many algebraic systems, such as algebras of matrices and linear operators, where the product AB does not necessarily equal BA.

A simple concrete example uses 2x2 matrices. Let A be the matrix [[0, 1], [0, 0]] and B be [[0, 0], [1, 0]]. Then AB equals [[1, 0], [0, 0]] while BA equals [[0, 0], [0, 1]], so AB ≠ BA, illustrating noncommutativity.

In physics, noncommutativity has important implications. Observables are represented by operators on a Hilbert space, and if two observables do not commute, they cannot have simultaneously well-defined values. This is formalized by the commutator [A, B] = AB − BA. The canonical example is position and momentum, with [x, p] = iħ. Spin components also fail to commute, reflecting intrinsic quantum angular momentum properties.

Noncommutativity extends beyond quantum mechanics. It appears in noncommutative algebras and in the mathematical field of noncommutative geometry, which generalizes geometric notions to settings where coordinates do not commute. Not all systems are noncommutative; commutativity is a special property that characterizes abelian structures, such as abelian groups.