anticommute

Anticommute is a relation between two objects in an algebra in which their product changes sign when their order is swapped: ab = -ba. Equivalently, the anticommutator {a,b} = ab + ba equals zero. Opposite notion is commutativity, where ab = ba for all pairs. Anticommutation is not universal; some pairs may anticommute while others commute or have neither relation defined.

In linear algebra and matrix theory, several families are defined by anticommutation relations. The Pauli matrices σ1, σ2, σ3 satisfy σ_i σ_j = - σ_j σ_i for i ≠ j, and σ_i^2 = I. Their anticommutator for distinct indices vanishes: {σ_i, σ_j} = 0. Fermionic creation and annihilation operators satisfy {c_i, c_j} = 0 and {c_i†, c_j†} = 0, with {c_i, c_j†} = δ_ij, expressing antisymmetric statistics.

In exterior algebra and differential geometry, basis 1-forms e_i anticommute under the wedge product: e_i ∧ e_j = - e_j ∧ e_i for i ≠ j. In Clifford algebras, generators anticommute up to a scalar: e_i e_j + e_j e_i = 2 g_ij, where g is a metric, leading to both anticommutation and nonzero squares.

In graded algebra, parity determines signs: the product of two odd elements picks up a minus sign, while even elements commute with all elements. The concept of graded or supercommutators generalizes the idea of anticommutativity.

Its use pervades quantum mechanics, quantum field theory, and geometry, where anticommutativity encodes exclusion principles, antisymmetric wavefunctions, and the structure of fermionic variables.