Lagrangian for electromagnetic field

[eoghan]

Hi!
In some texts (Sakurai - advanced qm and others) I found this expression for the lagrangian of an em field:
$$L=F_{\mu \nu}F_{\mu \nu}$$
but I'm a bit confused... L must be a Lorentz invariant, so I would write instead:
$$L=F_{\mu \nu}F^{\mu \nu} \;\;$$
Which form is the correct one? Or are they both correct?

 

[Bill_K]

The second one. Some texts are overly casual about their use of the summation convention. Generally they also use an imaginary fourth component, so there's no need for an explicit Lorentz metric or a minus sign in the summation.

 

[eoghan]

Ok, so Sakurai uses an imaginary component with an euclidean metric and so there is no difference between covariant and contravariant vectors.
Another question, the full density of Lagrangian is
$$\mathfrak{L}=-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}-\frac{1}{c}J_{\mu}A^{\mu}$$
But now, how can I incorporate the mechanical term of the particles? I mean, how can I add to the density of Lagrangian the mechanical term
$$L=-\frac{mc^2}{\gamma}$$
The problem is that the latter is a Lagrangian, while the former is a density of Lagrangian.
My goal is to get the full Hamiltonian of a point charge interacting with an em field:
$$H=\left[ \int d^3 x \frac{1}{2}(E^2+B^2)\right]+c\sqrt{m^2c^2+(\vec p - q\vec A)^2}+q\phi$$