Make sure the Lagrangian symmetry？

[centry57]

Is the Lagrangian of the neutral Proca field
$$\mathcal{L}=-\frac{1}{16\pi}\left(F^{\mu\nu}F_{\mu\nu}-2m^2 A_{\mu} A^{\mu}\right)$$
symmetric?
And How to make sure whether it's symmetric.

 

[Altabeh]

Is the Lagrangian of the neutral Proca field
$$\mathcal{L}=-\frac{1}{16\pi}\left(F^{\mu\nu}F_{\mu\nu}-2m^2 A_{\mu} A^{\mu}\right)$$
symmetric?
And How to make sure whether it's symmetric.

Symmetric in what?

AB

 

[centry57]

I mean if the Stress-energy momentum has the form $$T_{\mu\nu}=T_{\nu\mu}$$

 

[haushofer]

Well, take the functional derivative with respect to the metric of this Lagrangian. What do you get?

If the EM-tensor is not symmetric, you can construct the so-called Belinfante tensor; see for instance DiFrancesco's "Conformal Field Theory", 2.5.1 :)