# Are vacuum-fluctuations a consequence of causality?

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?
• if you do not get a satisfying answer here try at physicsoverflow.org – anna v Dec 27 '18 at 17:24
• @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? – Cham Dec 27 '18 at 17:45
• 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 – anna v Dec 27 '18 at 19:16