Roothaan equations

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 on it first three indices.

If a tensor changes sign under exchange of any pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor 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.

Example

An important antisymmetric tensor in physics is the electromagnetic tensor F in electromagnetism.

See also

• Levi-Civita symbol
• Symmetric tensor
• Antisymmetric matrix
• Exterior algebra
• Ricci calculus

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• [1] - mathworld, wolfram