# Correlators for space-like photon bouncing

Consider the diagram below of two detectors-emitters exchanging photons: Thin lines are null-paths between detectors. Detectors are also synchronized in such a way that measurements outside the blue or red windows are ignored. So first detector can only react to photons received in the red window, and the 2nd detector can only react to photons on the blue window.

First detector has some amplitude $\psi_A$ to emit photon from some point $A$ inside the apparatus red window, and second detector has some amplitude $\phi_C$ to emit photon from inside blue window. The key assumption here is that that second apparatus is made such that amplitude $\phi_C$ depends on a photon having being received at some point $B$ inside the measurement blue window. So $\phi_C(B)$ actually depends on past local events in the apparatus.

Now, while the observables and correlators for photons separated by space-like paths like $A-B$ or $C-D$ must commute, Feynmann propagators only vanish exponentially inside the space-like separations, and presumably a measurement of photons at some point $D$ inside the red measurement window of detector 1 are not restricted by space-like commutator rules, presumably the correlations between observations $A$ and $D$ do not need to be zero

Question: Can we expect the correlators between $A$ and $D$ events be zero, or just exponentially small in proportion to the length of the red and blue reaction windows (and their separation from the lightcones)?

I have 2 questions plus an optional third, as I’m not sure I’ve fully understood the problem:

1) what does it mean that the probability of second apparatus to shoot a photon from C depends on the detection of a photon in B? Do you mean that there the emission from the second device is not causal and the second photon “sees” the future of the detection in B and is emitted accordingly?

Or just that the probability of detection in B has some fixed value and probability of emission in C is consequently fixed and will emit accordingly, like a bet on a soccer match?

Is vertical line a time scale such that horizontal lines are same time?

2) why do you say that A-B and C-D are spacelike? If they’re paths of two photons I would have said that they’re time-like paths; Am I missing some crucial point?

[ 3) are you assuming flat Minkowski space? ]

Thanks for clarifying

Francesco

• 1) the photon absorption at B happens before C is emitted, and in those case one assumes that the amplitude of emitting C is a function of whether a photon was received or not before – lurscher Jul 15 '18 at 14:45
• 2) Feynmann propagators do not vanish in the space-like regions but are exponentially attenuated. For more information check the links in this question: physics.stackexchange.com/q/314821/955 – lurscher Jul 15 '18 at 14:47
• 3) yeah space can be assumed flat everywhere – lurscher Jul 15 '18 at 14:48
• For a sensible theory, we don’t need to require Feynman propagators to vanish for spacetime separations; we only need any observable built through them to vanish in that case, for example the commutator of two (free) fields vanishes for spacelike separations; said this, which is the actual quantity that you would like to evaluate between A and D to see if it’s nonzero? – Francesco Bernardini Jul 15 '18 at 15:16
• we want to estimate the correlator between photon emission at A and photon capture at D, I think we do not need to assume any time ordering for A and D, as long as they happen within the red measurement window – lurscher Jul 15 '18 at 15:38