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In calculating the action in classical field theory, why do we integrate over all of spacetime, thus over all of time, while we don't have to do that in ordinary particle action?

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  • $\begingroup$ Why do you think particle actions aren’t integrated over all of time? $\endgroup$ – G. Smith Oct 14 at 17:17
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    $\begingroup$ No that's not what I meant. I mean ordinarly we can integrate the Lagrangian between t1 and t2, but in field theory, in Minkowski spacetime for example, we always do that over the whole time axis. So, why is that? $\endgroup$ – Arthur Oct 14 at 17:21
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    $\begingroup$ That's not true in general. Who are we? $\endgroup$ – Qmechanic Oct 14 at 17:30
  • $\begingroup$ "We" means the people doing calculations. I know that it's very useful that way, because when we do a Lorentz transformation to some field, it suffices for the Lagrangian density to be covariant, since we don't pick up a Jacobi factor, and we don't need to worry about the shape of the boundary of the region we are integrating over in spacetime. But if we didn't do that, then we have to be careful about the boundary, when proving that the action is invariant, or am I mistaken somewhere? I would appreciate the help. But if I am not mistaken, then is there a stronger reason for that? . $\endgroup$ – Arthur Oct 14 at 17:36

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