Editing Propagator (section)

===Faster than light?===
{{More citations needed section|date=November 2022}}
The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator is ''nonzero'' outside of the [[light cone]], though it falls off rapidly for spacelike intervals. Interpreted as an amplitude for particle motion, this translates to the virtual particle travelling faster than light. It is not immediately obvious how this can be reconciled with causality: can we use faster-than-light virtual particles to send faster-than-light messages?

The answer is no: while in [[classical mechanics]] the intervals along which particles and causal effects can travel are the same, this is no longer true in quantum field theory, where it is [[commutator]]s that determine which operators can affect one another.

So what ''does'' the spacelike part of the propagator represent? In QFT the [[vacuum]] is an active participant, and [[particle number]]s and field values are related by an [[uncertainty principle]]; field values are uncertain even for particle number ''zero''. There is a nonzero [[probability amplitude]] to find a significant fluctuation in the vacuum value of the field {{math|Φ(''x'')}} if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some extent. The nonzero time-ordered product for spacelike-separated fields then just measures the amplitude for a nonlocal correlation in these vacuum fluctuations, analogous to an [[EPR paradox|EPR correlation]].  Indeed, the propagator is often called a ''two-point correlation function'' for the [[free field]].

Since, by the postulates of quantum field theory, all [[observable]] operators commute with each other at spacelike separation, messages can no more be sent through these correlations than they can through any other EPR correlations; the correlations are in random variables.

Regarding virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle-[[antiparticle]] pair that eventually disappears into the vacuum, or for detecting a virtual pair emerging from the vacuum.  In [[Richard Feynman|Feynman]]'s language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone.  However, no signaling back in time is allowed.

====Explanation using limits====
This can be made clearer by writing the propagator in the following form for a massless particle:
<math display="block">G^\varepsilon_F(x, y) = \frac{\varepsilon}{(x - y)^2 + i \varepsilon^2}.</math>

This is the usual definition but normalised by a factor of <math>\varepsilon</math>. Then the rule is that one only takes the limit <math>\varepsilon \to 0</math> at the end of a calculation.

One sees that 
<math display="block">G^\varepsilon_F(x, y) = \frac{1}{\varepsilon} \quad\text{if}~~~ (x - y)^2 = 0,</math>
and
<math display="block">\lim_{\varepsilon \to 0} G^\varepsilon_F(x, y) = 0 \quad\text{if}~~~ (x - y)^2 \neq 0.</math>
Hence this means that a single massless particle will always stay on the light cone. It is also shown that the total probability for a photon at any time must be normalised by the reciprocal of the following factor:
<math display="block">
\lim_{\varepsilon \to 0} \int |G^\varepsilon_F(0, x)|^2 \, dx^3 
 = \lim_{\varepsilon \to 0} \int \frac{\varepsilon^2}{(\mathbf{x}^2 - t^2)^2 + \varepsilon^4} \, dx^3
 = 2 \pi^2 |t|.
</math>
We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.