Why in field theory Lagrangian is an integral of space-time

[kof9595995]

I remember when I learned some basic continuum mechanics, Lagrangian is just a integral of lagrangian density over space, which is quite easy to accept because it's just a continuous version of L=T-U. Now I'm trying to start a bit QFT and notice that Lagrangian is an integral over space-time, and I just can't quite accept the integral over time part.
I suppose in classical relativistic field theory one must include the integral over time to get the correct equation of motion, but I can't find any introductory article about this on the net, so could you guys provide some guidance? I just want to convince myself that we need the integral over time.

 

[Thaakisfox]

Here is simple intuitive response. The whole point of relativity theory is that we deal with space and time as one entity i.e. space-time. The transformations act on these variables together. The operator doesn't know about "time" or "space" it only knows about "space-time".
We want the Lagrangian to transform as a scalar under Lorentz transformations (otherwise there would be no point), hence we need a Lorentz invariant measure which is we integrate over space-time. (In simple minkowski spacetime the invariant measure is just the integration over spacetime, but when the metric is more complicated then we need some extra stuff to get an invariant measure.)

 

[Petr Mugver]

It's a matter of what are the dynamical variables of the system, and on what parameters they depend. In particle theory, the dynamical variables are the positions and velocities of the particles, that depend on just one parameter, time. In field theory, the dynamical variables are...the fields, and they depend on four coordinates. Space is no longer a dynamical variable, but a parameter.

In mathematical terms, particle lagrangian theory is just a special case of field theory. In general, you have a lagrangian that depends on a certain number of fields (and their first partial derivatives), which in turn depend on a certain number (n) of parameters. Relativistic field theory has n=4, lagrangian particle theory (relativistic or not) has n=1. Another example is a lagrangian formulation of a drum (n=2).

 

[kof9595995]

Thank you both for the reply, I think I simply confused the action S and Lagrangian. S=int L dt
and in field theory L=int (lagrangian density) d^3x, so S is an integral of space time, not L.