What does a left-right arrow in a tensor formula mean?

Question

I need help with some some notation I've not seen before. Is using the left-right arrow in this formula $$[P^μ,M^{ρσ}]=i\hbar(g^{\mu\sigma}P^\rho-(\rho\leftrightarrow\sigma))$$ equivalent to writing $$[P^μ,M^{ρσ}]=i\hbar(g^{\mu\sigma}P^\rho-g^{\mu\rho}P^\sigma)$$ if not, what does it mean?

Answer 1

Yes.

$$ f(A,B) \pm (A\leftrightarrow B) ~:=~ f(A,B)\pm f(B,A). $$

The notation is useful as a shorthand, or to convey a symmetry/antisymmetry that may often otherwise be less apparent.

See e.g. the last equation in my Phys.SE answer here for a nested example.

Answer 2

I've never seen it either. It looks like it means "a term just like the one I just wrote, but with the indicies $\sigma$ and $\rho$ exchanged."