# Make sure the Lagrangian symmetry？

1. Jun 26, 2010

### 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.

2. Jun 26, 2010

### Altabeh

Symmetric in what?

AB

3. Jun 27, 2010

### centry57

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

4. Jun 30, 2010

### 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 :)