Solving Indices Problem in QFT

[anony]

Hi,

I always seem to have a problem with my indices...

$$\Lambda^{\mu}_{\;\;\nu}\Lambda_{\bar{\nu}}^{\;\;\mu}= \frac{\partial x'^{\nu}}{\partial x^{\mu}} \frac{\partial x_{\bar{\nu}}'}{\partial x_{\mu}}$$

Now, the first term I'm relatively sure is right (first lambad corresponds to first of the derivatives on RHS. That second one seems completely wrong. Where do I get it from?

$$\frac{\partial}{\partial x_{\bar{\nu}} } = \frac{\partial x_{\mu}'}{\partial x_{\bar{\nu}}} \frac{\partial }{\partial x_{\mu}'} = \Lambda_{\mu}^{\;\;{\bar{\nu}}} \frac{\partial}{\partial x_{\mu}' }$$

What am I doing wrong?!

 

[dextercioby]

That the LHS is wrong you can see because the <mu> is not sitting once 'downstairs' and other time 'upstairs'. So the LHS should have its first term with the <mu> 'downstairs' (SE) and the <nu> 'upstairs' (NW).

 

[jfy4]

Yes, with the partial derivatives, when the index is low in the denominator, that corresponds to a raised index, and visa versa.

$$\Lambda^{\mu}_{\;\;\nu}\Lambda_{\bar{\nu}}^{\;\;\mu}= \frac{\partial x'^{\nu}}{\partial x^{\mu}} \frac{\partial x_{\bar{\nu}}'}{\partial x_{\mu}}$$

This would be the appropriate way to do this, and this is standard convention

$$\Lambda_{\mu}^{\;\;\nu}\Lambda_{\alpha}^{\;\;\beta}=\frac{\partial x'^\nu}{\partial x^{\mu}}\frac{\partial x'^{\beta}}{\partial x^{\alpha}}$$

If you want them summed, then you have to make the $\beta$ into a $\mu$.