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I'n new to QFT, and recently lerned about the propagator of a free scalar field theory in Minkowski-space, which according to our lecture notes looks like $$G(p, q) = \frac{1}{q^\mu q_\mu + M^2} \delta(p-q).$$ It also says there that the two poles of the propagator correspond to particle and antiparticle. Now for me this gives rise to a few related questions:

  1. In Euclidean-space according to the lecture the propagators are the transition amplitudes of particles, e.g. in position space $G(x, y)$ would be the propability $\langle 0|\phi(x)\phi^\dagger(y)|0\rangle$ that a particle emerges from the vacuum at $y$, travels to $x$ and vanishes there. Can I interpret $G(p, p)$ in Minkowski-Space the same way as a particle with four-momentum $p$ that emerges from vacuum and then vanishes after some time?
  2. If that would be the case, is $G(p, q)$ then somehow related to virtual particles and vacuum fluctuations? This seems to be the case for me as I kind of understand $G(p, p)$ as propability for vacuum flucutations of a particle with given four-momentum $p$ that doens't even vanish off-shell.
  3. I'm given to understand that after fourier-transforming $G(p, q)$ to time-momentum-space ($E = \omega \to t$) the propagator (depending on how one shifts the poles for integration) looks something like $G((t, \mathbf{p}), (\tau, \mathbf{q})) = \delta(\mathbf{p}-\mathbf{q})\left[\Theta(t-\tau)e^{iE\tau}+\Theta(\tau-t)e^{-iE\tau}\right]$, which reflects causality (or "anti-causality" for antiparticles in this case). If one removes the Heaviside-functions and fourier-transforms back to frequency-momentum-space one would get something like $G'(p, q) = \delta(q^\mu q_\mu - M^2) \delta(p - q)$ (I think) where the propagator is non-zero only on-shell. Given I had the right intuition in points 1. and 2. would this mean that off-shell fluctuations are actually a consequence of causality?
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  • $\begingroup$ if you do not get a satisfying answer here try at physicsoverflow.org $\endgroup$ – anna v Dec 27 '18 at 17:24
  • $\begingroup$ @annav, what's the difference with "overflow" and "SE physics"? Why there are two similar "physics" things here? Can I copy-paste the same questions on both places? $\endgroup$ – Cham Dec 27 '18 at 17:45
  • $\begingroup$ the overflow is mainly for theoretical physics, graduate level. it is not a part of .SE, more mathematical.. Sure you can copy paste the same question $\endgroup$ – anna v Dec 27 '18 at 19:16

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