I am still stuck in my homework, but this time at another point, so this is a continuation of my last question to that subject.

It is given a classical field theory for a 4-vector field $A_{\mu}$ and assume that the terms in the lagrangian density have two powers of the field and two powers of derivatives. Further, $\mathscr{L}$ must be a lorentz skalar.

We have a given Lagrangian: $$\mathscr{L} = C_{1} (\partial_{\nu} A_{\mu}) (\partial^{\nu} A^{\mu}) + C_2 (\partial_{\nu} A_{\mu}) (\partial^{\mu} A^{\nu}) + C_3 A_{\mu} A^\mu.$$ From that we should compute the canonical momentum: $$\pi_\nu = \frac{\partial \mathscr{L}}{\partial \partial_0A^\nu}.$$

So the first term with $C_1$ I calculated by writing it into a vector notation. But now I got stuck at the $C_2$ term because now, the derivatives indices are not the same anymore.

In the following you see my Attempt:

$\frac{\partial}{\partial (\partial_0 A^\nu)} (\partial_{\nu}A_\mu \partial^\mu A^\nu) = \left( \begin{array}{c}\frac{\partial}{\partial_0 A^\nu} [(\partial_{\nu}A_0 \partial^\mu A^0)] \\ \frac{\partial}{\partial_0 A^\nu} [(\partial_{\nu}A_1 \partial^\mu A^1)] \\ \frac{\partial}{\partial_0 A^\nu} [(\partial_{\nu}A_2 \partial^\mu A^2)] \\ \frac{\partial}{\partial_0 A^\nu} [(\partial_{\nu}A_3 \partial^\mu A^3)]\end{array} \right)$

So if the two partials would have the same indices, I could put a sum into the brackets and than I have the result for $C_1$ but now I have 2 different indices, so I can not sum, so can someone please say me how I can go on in that vector notation, because than it is in detail and I can understand well what's going on.

Best regards

  • 2
    $\begingroup$ Hint: don't use the same symbol for dummy indices and free indices (i.e., $\nu$). $\endgroup$ Apr 20, 2018 at 16:04