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. The index subset must generally either be all covariant or all contravariant.

For example,

$T_{ijk\dots }=-T_{jik\dots }=T_{jki\dots }=-T_{kji\dots }=T_{kij\dots }=-T_{ikj\dots }$ 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 $U_{ijk\dots }$  and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:

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

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

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,

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

and for an order 3 covariant tensor T,

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

In any number of dimensions, these are equivalent to

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

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

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

In the above,

$\delta _{ab\dots }^{cd\dots }$

is the generalized Kronecker delta of the appropriate order.

Examples

Totally antisymmetric tensors include: