Square brackets in indices?

Question

What do these brackets within the indices mean in an equation like $$ \delta ^\mu _{[\alpha} \eta _{\beta ]\nu} ?$$

I can't find a text, which uses this notation, that explains it.

Answer 1

That's shorthand for a longer expression. The original expression is to be written, minus the same expression with the $\alpha$ and $\beta$ swapped. Then divide by two. This is the antisymmetric part of the tensor.

$$ T_{[\alpha\beta]} = {1 \over 2}(T_{\alpha\beta} - T_{\beta\alpha}) $$

The indices being antisymmetrized may be on the same tensor or belong to separate tensors as in your example.

If parentheses ( ) are used instead of [], add instead of subtract. This provides the symmetric part of the tensor expression. Adding the symmetric part and the antisymmetric parts should give the original tensor.

$$ T_{\alpha\beta} = T_{[\alpha\beta]} + T_{(\alpha\beta)} $$ There are more complex possibilities, such as three tensor indices indicated by square brackets, which are to understood as longer expressions involving sums and differences, and dividing by the count of permutations.

Any text on tensor algebra should cover this stuff. I got it (being a physicist) from Misner, Thorne & Wheeler's famous book, Gravitation. This kind of notation is decades old.

Answer 2

It means that you antisymmetrize. For example,

$$\Gamma_{[ab]} = \frac{1}{2} \left(\Gamma_{ab} - \Gamma_{ba} \right)$$

Similarly, parentheses are used for symmetrization:

$$\Gamma_{(ab)} = \frac{1}{2} \left(\Gamma_{ab} + \Gamma_{ba} \right)$$

This allows you to express concisely that every (rank 2) tensor is the sum of its symmetric and antisymmetric parts:

$$\Gamma_{ab} = \Gamma_{(ab)} + \Gamma_{[ab]}$$

In the case of three indices, the (anti)symmetrization takes place over all permutations of the three indices:

$$T_{[abc]} = \frac{1}{6} \left( T_{abc} + T_{bca} + T_{cab} - T_{bac} - T_{cba} - T_{acb}\right)$$

and in general, for $n$ indices,

$$T_{[a_1\dots a_n]} = \frac{1}{n!} \sum_{\sigma\in S_n} (-1)^{{\rm sign}(\sigma)}T_{a_{\sigma(1)} \dots a_{\sigma(n)}}$$

Edit: Here's a reference: Wikipedia