Could someone please explain this 4-vector/tensor notation?

If you can answer any of the questions below, your help will be greatly appreciated.

There's an equation from E&M (I believe the definition of the antisymmetric field tensor) that reads:

[tex]
F_{\mu \nu} = \frac{\partial A_\nu}{\partial x^\mu} - \frac{\partial A_\mu}{\partial x^\nu},
[/tex]

where (of course) [itex]A[/itex] is the 4-vector potential. I have no idea how to parse this equation...what's going on with taking the partial of [itex]A_\nu[/itex] with respect to [itex]x^\mu[/itex]? What's this notation describing, or telling me to do? I just don't see it.

Also (and this pertains to relativistic quantum mechanics), what's meant by something like

[tex]
\partial_\mu [\partial_\nu \Psi(x^\mu)],
[/tex]

where (of course) [itex]\Psi(x^\mu)[/itex] is the wave function? I.e., what does it mean to operate on [itex]\partial_\nu[/itex] of something with [itex]\partial_\mu[/itex]? Particularly when that something I'm operating on is a function of [itex]x^\mu[/itex]? (By the way: In case the notation is unfamiliar to you, [itex]\partial^\mu = \partial / \partial x_\mu[/itex].) Does that turn into something? Can I compactify that? Or do I just have to leave it as [itex]\partial_\mu [\partial_\nu \Psi(x^\mu)][/itex]? Also, would I get an equivalent expression if I interchanged the order of [itex]\partial_\mu[/itex] and [itex]\partial_\nu[/itex]; i.e., if I tried to evaluate [itex]\partial_\nu [\partial_\mu \Psi(x^\mu)][/itex]? Or would I get something quite different?

Does all of this boil down to "we need two distinct dummy indices, [itex]\mu[/itex] and [itex]\nu[/itex], to indicate that there shouldn't be any summation going on?"

(P.S. - I would like to thank David J. Griffiths for getting me into the habit of italicizing words I want to emphasize.)
 
305
2
[tex]
F_{\mu \nu} = \frac{\partial A_\nu}{\partial x^\mu} - \frac{\partial A_\mu}{\partial x^\nu},
[/tex]

where (of course) [itex]A[/itex] is the 4-vector potential. I have no idea how to parse this equation...what's going on with taking the partial of [itex]A_\nu[/itex] with respect to [itex]x^\mu[/itex]?
Given two four vectors, one for the field, and one for the spacetime
position

[tex]
\vec{A} = (A_0, -A_1, -A_2, -A_3)
[/tex]
[tex]
\vec{x} = (x^0, x^1, x^2, x^3) = (ct, x, y, z)
[/tex]

This says that there's 16 scalar quantities that can be calculated by
taking derivatives. Two examples are:

[tex]
F_{23} =
\frac{\partial A_3}{\partial x^2} - \frac{\partial A_2}{\partial x^3} =
\frac{\partial A_3}{\partial y} - \frac{\partial A_2}{\partial z}
[/tex]

[tex]
F_{33} =
\frac{\partial A_3}{\partial x^3} - \frac{\partial A_3}{\partial x^3} = 0
[/tex]

For your other question, I'd assume summation over [itex]\mu[/itex] is implied.
 
Thanks a lot for your help!

What if I confront something like

[tex]
\frac{\partial F^{\mu \nu}}{\partial x^\nu} = \partial_\nu F^{\mu \nu}?
[/tex]

Does that imply a sum over [itex]\nu[/itex]?
 

nicksauce

Science Advisor
Homework Helper
1,272
5
Yes, if an index is repeated as an upper index and a lower index, it implies a sum over that index.
 

Want to reply to this thread?

"Could someone please explain this 4-vector/tensor notation?" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top