Antisymmetric tensor

An antisymmetric tensor changes sign under exchange of two indices in a specified subset of its indices. A totally antisymmetric tensor has this property for every pair of indices.

To vary an antisymmetric tensor without treating its entries as independent physical variables, temporarily allow an unrestricted array $M^{\mu \nu }$ and add ${\lambda }_{\mu \nu }(M^{\mu \nu }+M^{\nu \mu })/2$ to the Lagrangian density, with symmetric Lagrange multiplier $\lambda$. Variation of $\lambda$ enforces antisymmetry. The antisymmetric part of the unrestricted $M$ equation eliminates $\lambda$ and gives the physical Euler-Lagrange field equation. Equivalently, vary only the six independent entries in four spacetime dimensions. Quadratic contractions must be differentiated before applying antisymmetry, or their factors and signs can be lost.

For a real antisymmetric matrix, transpose gives $\mathrm{tr}(A^{2j+1})=\mathrm{tr}((-A{)}^{2j+1})=-\mathrm{tr}(A^{2j+1})$, so this trace vanishes. Every complete pairwise tensor contraction of copies of $A$, using the Euclidean Kronecker delta, splits into closed index cycles. Each cycle is a trace of a power of $A$, up to a sign from index order. An odd total number of copies forces an odd-length cycle, so the whole contraction vanishes. This assertion concerns pairwise index contraction with the metric; inserting additional tensors would be a different claim.

A tensor changes sign whenever two indices are interchanged. In three dimensions, a rank-two antisymmetric tensor is $A_{ij}=a_k{ϵ}_{kij}$, a rank-three one is $A_{ijk}=a{ϵ}_{ijk}$, and every totally antisymmetric tensor of rank greater than three vanishes.