Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged.[1][2] The index subset must generally either be all covariant or all contravariant.

For example,

${\displaystyle T_{ijk\dots }=-T_{jik\dots }=T_{jki\dots }=-T_{kji\dots }=T_{kij\dots }=-T_{ikj\dots }}$

holds when the tensor is antisymmetric with respect to its first three indices.

If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor of order p may be referred to as a p-form, and a completely antisymmetric contravariant tensor may be referred to as a p-vector.

Antisymmetric and symmetric tensors

A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0.

For a general tensor U with components ${\displaystyle U_{ijk\dots }}$ and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:

 ${\displaystyle U_{(ij)k\dots }={\frac {1}{2}}(U_{ijk\dots }+U_{jik\dots })}$ (symmetric part) ${\displaystyle U_{[ij]k\dots }={\frac {1}{2}}(U_{ijk\dots }-U_{jik\dots })}$ (antisymmetric part).

Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in

${\displaystyle U_{ijk\dots }=U_{(ij)k\dots }+U_{[ij]k\dots }.}$

Notation

A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor M,

${\displaystyle M_{[ab]}={\frac {1}{2!}}(M_{ab}-M_{ba}),}$

and for an order 3 covariant tensor T,

${\displaystyle T_{[abc]}={\frac {1}{3!}}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).}$

In any 2 and 3 dimensions, these can be written as

${\displaystyle M_{[ab]}={\frac {1}{2!}}\,\delta _{ab}^{cd}M_{cd},}$
${\displaystyle T_{[abc]}={\frac {1}{3!}}\,\delta _{abc}^{def}T_{def}.}$

where ${\displaystyle \delta _{ab\dots }^{cd\dots }}$ is the generalized Kronecker delta, and we use the Einstein notation to summation over like indices.

More generally, irrespective of the number of dimensions, antisymmetrization over p indices may be expressed as

${\displaystyle T_{[a_{1}\dots a_{p}]}={\frac {1}{p!}}\delta _{a_{1}\dots a_{p}}^{b_{1}\dots b_{p}}T_{b_{1}\dots b_{p}}.}$

In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as:

${\displaystyle T_{ij}={\frac {1}{2}}(T_{ij}+T_{ji})+{\frac {1}{2}}(T_{ij}-T_{ji})}$

This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.

Examples

Totally antisymmetric tensors include: