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?

Where did you see the left-right arrow notation? Though I suspect that your interpretation is correct, it might be useful to see the original source. — Kyle Kanos, Apr 29 '15 at 17:29
You are correct, this notation is relatively common (but pretty lazy), and usually means the same expression as given previously but with the two symbols exchanged. — Mark Mitchison, Apr 29 '15 at 17:30

score 1 · Accepted Answer · edited Apr 13 '17 at 12:39

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.

edited Apr 13 '17 at 12:39

Community

1

answered Apr 29 '15 at 18:13

Qmechanic

201,751

score 0 · Answer 2 · answered Apr 29 '15 at 17:29

0

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."

answered Apr 29 '15 at 17:29

garyp

22,210

1

I've seen it regularly and I can confirm that's what it means. (I would post an answer but it'd say the same thing as yours) – David Z Apr 29 '15 at 17:30
@DavidZ Post it as an answer, and I'll delete mine since you have definitive information. – garyp Apr 29 '15 at 17:43
Ah, looks like Qmechanic already did so. – David Z Apr 30 '15 at 05:27

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

2 Answers2