[Physics] the difference between a skew-symmetric and an antisymmetric tensor

definitiontensor-calculus

What is the difference between a skew-symmetric and an anti-symmetric tensor?
If they represent the same tensor, then why use different labeling.

Best Answer

I) Many English words come in both a Greek and a Latin version. The prefix anti- is from Greek and the prefix skew- is from French.

Most authors would define an anti-symmetric and a skew-symmetric (possibly higher-order) tensor as precisely the same thing.

II) However, in the context of supernumber-valued tensors, some authors define a second-order anti-symmetric tensor/matrix as

$$\tag{A} A_{ab}=(-1)^{(|a|+1)(|b|+1)}A_{ba},$$

while a second-order skew-symmetric tensor/matrix obeys

$$\tag{S} S_{ab}=-(-1)^{|a||b|}S_{ba},$$

cf. Ref. 1. Here $|a|$ denotes the Grassmann-parity of the coordinate index $a$.

References:

  1. D. Leites, Seminar on supersymmetry. Vol. 1. Algebra and Calculus, 2006.