Antisymmetric function

This article covers anti-symmetric functions of several variables; for axis and point symmetry of real functions of a variable, see even and odd functions .

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

The counterpart to the antisymmetric functions are symmetric functions .

definition

If and are two vector spaces (mostly over the real or complex numbers), then a multivariate function is called antisymmetric if for all permutations and all vectors ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle f\colon V^{n}\to W}$ ${\displaystyle \sigma \in S_{n}}$ ${\displaystyle x_{1},\dotsc ,x_{n}\in V}$

${\displaystyle f(x_{1},\dotsc ,x_{n})=\operatorname {sgn} (\sigma )f(x_{\sigma (1)},\dotsc ,x_{\sigma (n)})}$

holds, where is the sign of the permutation. ${\displaystyle \operatorname {sgn} }$

Examples

Concrete examples

The subtraction

${\displaystyle f(x_{1},x_{2})=x_{1}-x_{2}=-(x_{2}-x_{1})=-f(x_{2},x_{1})}$

is antisymmetric, because swapping the two operands and reverses the sign of the result. For example, antisymmetric functions of three variables are ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$

${\displaystyle f(x_{1},x_{2},x_{3})=(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{1})}$

or

${\displaystyle f(x_{1},x_{2},x_{3})=x_{1}^{2}(x_{2}-x_{3})+x_{2}^{2}(x_{3}-x_{1})+x_{3}^{2}(x_{1}-x_{2})}$.

More general examples

• the cross product of two vectors is antisymmetric
• the Lie bracket of two vectors is also antisymmetric
• an anti- commutative two - digit combination is an anti-symmetric function of the two operands
• the determinant of a matrix is ​​an antisymmetric function of the column vectors of the matrix
• an alternating multilinear form is an antisymmetric function in the scalar body that is linear in every argument

further criteria

To prove the antisymmetry of a function, not all possible permutations of the symmetric group have to be checked. Since every permutation can be written as a sequential 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 , that is ${\displaystyle n!}$ ${\displaystyle S_{n}}$ ${\displaystyle (i~j)}$ ${\displaystyle x_{i}}$ ${\displaystyle x_{j}}$

${\displaystyle f(\dotsc ,x_{i},\dotsc ,x_{j},\dotsc )=-f(\dotsc ,x_{j},\dotsc ,x_{i},\dotsc )}$

for with is. For further possible criteria for verifying the antisymmetry, see Symmetrical functions that must be used with a change in sign. ${\displaystyle i,j\in \{1,\ldots ,n\}}$ ${\displaystyle i<j}$

properties

The antisymmetric functions form a subspace in the vector space of all functions from to (with the component-wise addition and scalar multiplication ), that is ${\displaystyle V^{n}}$ ${\displaystyle W}$

• a scalar multiple of an antisymmetric function is again an antisymmetric function and
• the sum of two antisymmetric functions is also antisymmetric again,

where the null function is trivially antisymmetric.

Antisymmetrization

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

${\displaystyle {\begin{aligned}Af(x_{1},\dotsc ,x_{n})&={\frac {1}{n!}}\sum _{\sigma \in S_{n}}\operatorname {sgn} (\sigma )f(x_{\sigma (1)},\dotsc ,x_{\sigma (n)})\\&={\frac {1}{n!}}\sum _{\sigma \in S_{n} \atop {\text{gerade}}}f(x_{\sigma (1)},\dotsc ,x_{\sigma (n)})-{\frac {1}{n!}}\sum _{\sigma \in S_{n} \atop {\text{ungerade}}}f(x_{\sigma (1)},\dotsc ,x_{\sigma (n)})\end{aligned}}}$

every non-antisymmetric function can be assigned an associated antisymmetric function . The antisymmetrization operator carries out a projection onto the subspace of the antisymmetric functions. If a product is a product of functions that only depend on a single variable (in quantum chemistry such a function is called a Hartree product), one can also write as a Slater determinant . ${\displaystyle f}$ ${\displaystyle Af}$ ${\displaystyle A}$ ${\displaystyle f(x_{1},\dotsc ,x_{n})}$ ${\displaystyle Af(x_{1},\dotsc ,x_{n})}$

See also

• Alternating group
• Graßmann algebra
• Levi Civita symbol

literature

• Ilka Agricola , Thomas Friedrich : Vector analysis: differential forms in analysis, geometry and physics . Springer, 2010, ISBN 3-8348-1016-9 .

Web links

• Robert Milson, Thomas Foregger, Joe Corneli: Antisymmetric Mapping . In: PlanetMath . (English)