Commutator notation in classical field theory

[paco_uk]

Homework Statement

Could someone please explain what is meant by the term:

[itex]\partial_{[ \mu}F_{\nu \rho ]} [/itex]

Homework Equations

I have come across this in the context of Maxwells equations where [tex]F^{\mu \nu}[/tex] is the field strength tensor and apparently:

[tex]\partial_{[ \mu}F_{\nu \rho ]} =0[/tex]

follows "trivially" from

[tex] F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}[/tex]

The Attempt at a Solution

I don't understand what the notation means and I haven't found it defined anywhere. Do all three indices get permuted somehow?

 

[I like Serena]

Would you please fix your latex tags?
They should be [ tex ]...[ /tex ] (that is: forward slash, but no spaces within the tag).

You'll see that your equation will look "strange" then.
Could you perhaps fix that as well?

 

[paco_uk]

Sorry about the mess. I've cleared up the latex tags. What's there now does indeed look strange to me but is what is written in the lecture notes. I don't know what the commutator bracket round the subscripts means.

 

[stevenb]

I don't understand what the notation means and I haven't found it defined anywhere. Do all three indices get permuted somehow?

Yes. This is the antisymmetrizing operation. A good explanation is given in Prof. Maloney's notes and lecture recordings. It's also found in many good reference books on GR and differential geometry.

http://www.physics.mcgill.ca/~maloney/514/

Here is a PDF of the relevant lecture notes, but it's helpful to listen to the lectures if you have time.

 

[dextercioby]

The notation makes sense, if you learn about differential forms on the space-time manifold. The F is then a 2-form, and dF =0 is a consequence of the 2-nd degree nilpotency of the exterior differential.

 

[paco_uk]

Thanks very much. I've never done a course on General Relativity so I'm pretty shaky on all this. I look forward to listening to the lectures.