Sigma Sub/Sup i,j: Differences & Help

[lour]

What's the difference between sigma sub i,j and sigma sup i,j??thanks.

 

[Avodyne]

One is a Greek letter with two Latin subscripts, and the other is a Greek letter with two Latin superscripts.

Seriously, you have to say what the symbols mean before a question like this can be answered. Common meanings of sigma in physics include a Pauli matrix, a cross section, a conductivity, etc, etc.

 

[lour]

It's $$(\gamma^{\mu}(p_{\mu}-\frac{e}{c}A_{\mu})+\frac{Keh}{4mc^{2}}\sigma_{\mu\nu}F^{\mu\nu}-mc)\Psi=0$$.I don't know what $$\sigma_{\mu\nu}F^{\mu\nu}$$ means.$$F^{\mu\nu}=\frac{\partial_{A^{\mu}}}{\partial_{x_{\nu}}}-\frac{\partial_{A^{\nu}}}{\partial_{x_{\mu}}}$$.Can someone tell me?Help appreciated

 

[jtbell]

I can't help you because I don't know that equation. However, I'm curious... what is that equation supposed to be about?

 

[Manilzin]

In your equation,

$$\sigma_{\mu\nu}$$

means the mu'th-nu'th component of the tensor (or matrix) sigma. When you have an expression like

$$\sigma_{\mu\nu}F^{\mu\nu}$$,

Einstein's summing convention is implied - that is, you should sum over repeated indices, in this case mu and nu, from zero to three. It is a kind of "dot product" between the matrices sigma and F. Typically, you will need to know $$\sigma_{\mu\nu}$$ for all mu and nu to actually calculate this. The difference between upper and lower indices is that (depending on convention), for a four-vector,

$$f^{\mu} = g^{\mu\nu}f_{\nu}$$

where g is the 4x4 matrix that has zero in all positions when you're not on the diagonal, and it has 1 in its first diagonal position and -1 in the last three positions. Thus, $$f^0 = f_0$$, and $$f^i = -f_i$$ for i = 1, 2 or 3. For a matrix, we would then write

$$\sigma^{\mu\nu} = g^{\alpha\mu}g^{\beta\nu}\sigma_{\alpha\beta}$$

It's not very simple, but this is standard notation in relativity, so if you get the hang of this, a lot of stuff becomes easier..

 

[lour]

Thanks a lot for all your help.The equation is from one of my homework problems,it is kind of Dirac equation,"introduce an anomalous magnetic monent"-my homework states,.If you are interested,I can send you the whole problem(I'm working on it,I bet you won't like it)

 

[Avodyne]

[tex]\sigma^{\mu\nu}=\frac{i}{4}(\gamma^\mu\gamma^\nu-\gamma^\nu\gamma^\mu)[/itex][/tex]