Apologies for such a basic question, but I cannot find an unambiguous answer.
Say I have 3 tensors;
- $A$ with components $A_{ij}$
- $B$ with components $B_{ij}$
- $C$ with components $C_{ij}$
Such that the commutator $[A,B]=C$. If I wanted to write this commutator in index notation with Einstein summation convention should I write;
$$ [A,B] \to A_{ij}B_{kl} – B_{kl}A_{ij}$$
But then I end up with 4 indices and the result should really have 2, or;
$$ [A,B] \to A_{ik}B_{kj} – B_{ik}A_{kj}$$
which, summing over the k's gives the number of indices I wanted?
Best Answer
The second option, namely
$$ A_{ik}B_{kj} - B_{ik}A_{kj} = C_{ij}. $$
The commutator $[A,B]$ is defined through matrix multiplication $[A,B] = AB - BA$ and to describe the $ij$ component of the matrix multiplication $AB$ you need to introduce a dummy index $k$ and sum over it to get $(AB)_{ij} = A_{ik} B_{kj}$ which is the multiplication of row elements of $A$ by column elements of $B$ that happens when you multiply the matrices A and B together.