Anticommutation

Anticommutation is a relation between two elements a and b of an algebra in which the order of multiplication reverses the sign: ab = −ba. The sum ab + ba is called the anticommutator, denoted {a, b}, and it vanishes when a and b anticommute. This concept is contrasted with commuting, where ab = ba and the commutator [a, b] = ab − ba equals zero.

In physics and mathematics, anticommutation appears in several standard contexts. Pauli matrices satisfy {σi, σj} = 2 δij I, which implies σi^2 = I and, for i ≠ j, σi σj = − σj σi. Fermionic creation and annihilation operators obey canonical anticommutation relations: {c_p, c_q} = 0 and {c_p, c_q†} = δpq. These relations encode the Pauli exclusion principle at the operator level. Grassmann variables, used to describe fermionic degrees of freedom in functional integrals, also anticommute: θi θj = − θj θi.

In algebra, Clifford algebras are generated by elements e_i obeying e_i e_j + e_j e_i = 2 δij, which yields e_i e_j = − e_j e_i for i ≠ j. Anticommutation thus captures the algebraic structure of fermionic systems and spinors.

Applications and implications include the spin-statistics connection in quantum theory, where fermions are described by anticommuting operators and bosons by commuting observables with different commutation relations. Anticommutation is also relevant in graded and superalgebras, where it generalizes the notion of symmetry between objects of different parity.