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.

 

[PJC01]

The concept of a Lagrangian in field theory is based on the principle of least action, which states that a physical system will follow the path of least action in space-time. The action is defined as the integral of the Lagrangian over space and time.

In classical mechanics, the Lagrangian is simply the difference between the kinetic and potential energy of a system. However, in field theory, the Lagrangian is a function of the fields themselves, and not just their values at a particular point in space. This is why the Lagrangian is integrated over all of space and time, as it takes into account the behavior of the fields at every point in the system.

In order to fully describe the dynamics of a field, we need to consider its behavior over all of space and time. This is why the integral over time is necessary in field theory, as it allows us to take into account the changes in the field over a period of time.

Furthermore, in classical relativistic field theory, the concept of space and time are intertwined, and cannot be separated. This is why the Lagrangian is integrated over space-time, as it reflects the fundamental nature of space and time in relativity.

In conclusion, the integral over space-time in the Lagrangian is necessary in field theory to fully describe the dynamics of a field and to take into account the fundamental nature of space and time in relativity. While it may seem confusing at first, it is a crucial component in understanding the behavior of fields in the physical world.