Antisymmetrisiert

Antisymmetrisiert (German: "antisymmetrized") denotes the result of applying an antisymmetrization operation to a multilinear object such as a tensor, a function of several variables, or a many-particle wavefunction. Antisymmetrization produces an alternating object that changes sign under odd permutations of its arguments and vanishes whenever two arguments coincide.

Formally, for an n-index tensor T, the antisymmetrized tensor A[T] is given by the average over permutations weighted by the sign of the permutation: A[T] = (1/n!) ∑_{σ∈S_n} sgn(σ) P_σ(T), where P_σ permutes the tensor indices according to σ. This normalized antisymmetrizer is a projection operator (A^2 = A) onto the subspace of alternating tensors. For two indices the operation yields A[T]_{ij} = (T_{ij} − T_{ji})/2. Unnormalized sums of signed permutations scale accordingly (the unnormalized operator squares to n! times itself).

Antisymmetrization is central in the construction of exterior algebras and differential forms via the wedge product, and in linear algebra through determinants as fully antisymmetric multilinear forms. In quantum mechanics antisymmetrization enforces fermionic exchange symmetry, producing Slater determinants that implement the Pauli exclusion principle. In representation theory and Young tableau methods, antisymmetrizers appear as components of Young symmetrizers that project onto irreducible representations of symmetric groups. The concept contrasts with symmetrization, which produces objects invariant under index permutations.