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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.

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1 Answer

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

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

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