Different weights for time and spatial derivative in Lagrangian Density I'm new to QFT and trying to understand the form of the Lagrangian densitys used. As a simple model you often see a Lagrangian density of the form
$${\mathcal L} = \frac{1}{2} \partial_j \phi_n \partial^j \phi_n + \frac{1}{2}m^2 \phi_n \phi_n.\tag{1}$$
The two terms corresponding to the time and spatial derivatives are often refered to as the kinetic and gradient energy. I see a strong analogy to the Lagrangian density of a vibrating string:
$$\mathcal{L} = \frac{\mu}{2} \left(\frac{\partial \phi}{\partial t}\right)^2 - \frac{E}{2} \left(\frac{\partial \phi}{\partial x} \right)^2.\tag{2}$$
Now the question that arraises to me is why there are different factors before the time and spatial derivatives in the second Lagrangian density but not in the first one. I guess the answer is that the first Lagrangian density has to be relativisticaly covariant while the second one isn't. But then my question is how can you treat fields with Lagrangian densitys like the second one in a covariant manner?
 A: 
I guess the answer is that the first Lagrangian density has to be relativisticaly covariant while the second one isn't.

Yep, that's why.

But then my question is how can you treat fields with Lagrangian densitys like the second one in a covariant manner?

By introducing a nontrivial coupling tensor:
$$
{\mathcal L} = \frac{1}{2} M_{\alpha\beta} \partial_\alpha \phi \partial^\beta \phi + \frac{1}{2}m^2 \phi \phi,
\tag{1'}
$$
with $M_{\alpha\beta}=M_{\beta\alpha}$ a covariant tensor. Generically, this tensor will be diagonal in some frames (like the original frame in which the formulation $\frac{\mu}{2} \left({\partial \phi}/{\partial t}\right)^2 - \frac{E}{2} \left({\partial \phi}/{\partial x} \right)^2$ is formulated), but it will acquire cross-terms if you boost away from those frames unless $M_{\alpha\beta} = K g_{\alpha\beta}$ is proportional to the metric $g_{\alpha\beta}$ that's used to define those boosts.
A: FWIW, if $\mu$ & $E$ in eq. (2) are two positive constants, they lead to a characteristic speed $c=\sqrt{E/\mu}$. They can be scaled away in (2) via appropriate scaling of the $t$ & $x$ variables to reach (1).
