[Physics] Pauli matrices and the Levi-Civita symbol

Question

This is just a quick question. I would figure this out myself if I wouldn't have an exam about this tomorrow.

I am working on the non-relativistic approximation of the Dirac equation for an electron in an EM field. On one point, I need the following relation:

$$
\epsilon^{klm} \sigma^{m} = \sigma^m \epsilon^{mkl}
$$
where $\sigma^m$ denotes the $m$th Pauli matrix and $\epsilon^{klm}$ denotes the Levi-Civita symbol and the Einstein summation convention is used.

The question is: does this relation hold in general for the Levi-Civita symbol or is this specific for the Pauli matrices?

TIA, Eric.

Answer 1

Unless I am missing something the relation is trivial since starting with

$$\epsilon^{klm}\sigma^m = \sigma^m \epsilon^{mkl}$$

and permuting the $m$ past the $l$ gives a factor of -1

$$(-1)\epsilon^{kml}\sigma^m = \sigma^m \epsilon^{mkl}$$

and permuting the $m$ past the $k$ gives a factor of -1

$$(-1)^2\epsilon^{mkl}\sigma^m = \sigma^m \epsilon^{mkl}$$

and since $(-1)^2=+1$

$$\epsilon^{mkl}\sigma^m = \sigma^m \epsilon^{mkl}$$

or

$$\sigma^m \epsilon^{mkl}= \sigma^m \epsilon^{mkl}.$$