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  1. Consider a correlation function between two points $A(x_1,t_1)$ and $B(x_2,t_2)$, we need to integrate over paths which could be infinite long. But the time length $(t_1-t_2)$ is finite, so if $A$ and $B$ are the coordinates of one single particle, then all of the paths from $A$ to $B$ should be time-like curves, the maximum length should be $c\times(t_1-t_2)$, which is not infinitely long. It seems special relativity could be violated here.

  2. Consider any loop integration of higher order correction to a Feynman diagram calculation, we have infinitely many off-shell processes and "internal virtual particles", which could be created and annihilated without taking any time. Does this violate special relativity?

Is there any better reason than simply saying these are "internal virtual processes"? They do affect our final real physical observations!

for the second case, one might argue that no particles here but only fluctuating fields? But when, where, and how are those fields created? This picture is not that physically clear to me.

Any better insight?

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Related: physics.stackexchange.com/q/18835/2451 –  Qmechanic Feb 11 '12 at 10:00
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1 Answer 1

Concerning the path integral, there is no problem with integration over too long paths. Integration is not a particle motion, but is a sum like $A=a+b+c+d+...$. You may plug in the right-hand side anything resulting in zero, for example, $z-z$, but it does not give you the right to consider your $z$ as physical motion. The path integral is a sum of many things that are not related to the physical motion. The fact of using the Lagrangian does not makes each particular term (path) physical. Only sum (integral) makes sense.

Off-shell "processes" are just Fourier-presentations of functions of time and space, each Fourier variable $p_x$, $p_y$, $p_z$, and $\omega$ are dumb independent variables. They are not restricted with any relationship like $p^2-\omega ^2 =m^2$ and their denotions as physical variables are just deceptive.

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