Anticommutativity

An antisymmetric function or skew-symmetric function in mathematics a function of several variables, in which the exchange of two variables reverses the sign of the function. Important special cases of antisymmetric functions are antikommutative links and alternating multilinear forms. In quantum mechanics, fermions are precisely those particles whose wavefunction is antisymmetric with respect to the exchange of particle positions. The counterpart to the anti- symmetric functions are symmetric functions.

Definition

Are and two vector spaces (often over the real or complex numbers ), it means a multivariate function is antisymmetric if for all permutations and all vectors

Holds, where the sign of the permutation.

Examples

Concrete examples

The subtraction

Is antisymmetric, because by interchanging the two operands and the sign of the result is reversed. Antisymmetric functions of three variables, for example,

Or

More general examples

• The cross product of two vectors is antisymmetric
• The Lie bracket of two vectors is also antisymmetric
• Antikommutative a binary operation is an antisymmetric function of the two operands
• The determinant of a matrix is an antisymmetric function of the column vectors of the matrix
• An alternate form is a multi- linear function in the anti-symmetric Skalarkörper which is linear in each case

Style

For the detection of anti- symmetry of a function does not need to be checked all the possible permutations of the symmetric group. After each permutation can be written as a consecutive execution of transpositions of the form, a function is already antisymmetric if and only if the function value is reversed by interchanging any two variables and, hence,

Is with. For other possible criteria used to establish the antisymmetry see Symmetric Function # Other criteria that must be applied in each case with change of sign.

Properties

The anti- symmetric functions form a subspace in the vector space of all functions from to ( with componentwise addition and scalar multiplication ), that is

• A scalar multiple of an antisymmetric function is again an antisymmetric function and
• The sum of two anti-symmetric functions is also again antisymmetric

The zero function is trivially antisymmetric.

Antisymmetrization

By antisymmetrization, that is a weighted summation over all possible permutations of the form

Can be any non- antisymmetric function assign an associated antisymmetric function. The Antisymmetrisierungsoperator leads by a projection on the subspace of antisymmetric functions. If a product of functions, each by only one variable depend ( in quantum chemistry is called such a function Hartree- product), can be written as Slater.