Path integral and causally disconnected parts of universe....

[asimov42]

One more question: it appears that portions of the universe are expanding away from us faster than the speed of light. Given this, particles in two 'parts' of the universe that are no longer causally connected should not be able to influence each other (due to speed of light constraint).

So, say I have an electron sitting happily in one part of the universe and a proton in that moves into a portion of space receding from the electron at a speed faster than light.

Now, the quantum wave function for each particle should occupy all of spacetime, and so when I compute the path integral, in theory, I should include terms for the electron and the proton interacting - however, any measurable interaction would violate relativity (by transmitting information). So, is it the case that the relevant terms in the path integral 'cancel' in some way, such that relativity is respected?

Thanks!

 

[mfb]

The current recession speed is not the relevant point, you also have to take into account the past and/or future recession speed. Things at the edge of the causally connected part of the universe (as seen by us) currently have a recession speed slightly above c, the speed was significantly above c in the past.

Now, the quantum wave function for each particle should occupy all of spacetime

Decoherence prevents that. If you imagine a universe where no decoherence happens, then electron and proton are not localized - it does not make sense to talk about their distance as single value.

Interactions, at least in QFT, are always local.

 

[asimov42]

Thanks mfb - sorry, still not quite clear - so because of decoherence, the two particle wave functions would effectively not interact? Is this guaranteed (in a probabilistic sense)? I'm not overly familiar with decoherence, but I'm assuming this is the 'solution' to the having to compute the path integral that involves their (the particles') interaction?

 

[asimov42]

Just to be clear though - the full path integral would include terms for the interactions of the particles, even if they're causally disconnected? It seems like decoherence sort of 'sweeps under the rug' some of the these issues. Is decoherence in QFT really the solution to the problem of influence between causally disconnected regions of spacetime? Prior to rise of the decoherence idea, how was this dealt with?

At the very least, to make it crystal clear in my mind, particles in regions of space moving apart at > c cannot influence one another's behaviour? (because, again, influence would amount to the flow of information between the particles).

Thanks, and sorry for all the questions.

 

[asimov42]

Actually, mfb, perhaps the easiest thing would be just to expand on what you mean by all interactions in QFT being local? (and sorry, above I'm calling the electron and proton 'particles', but I realize they really should be field excitations).

 

[rubi]

The path integral must of course contain information about all of spacetime, since you can calculate correlations of spacelike separated observables. However, if you are only interested in observables with support in one region of spacetime, you will get results that are independent of the rest of spacetime. It's like taking a partial trace in the Hamiltonian formulation.

 

[mfb]

so because of decoherence, the two particle wave functions would effectively not interact?

No, the wave functions as you imagine them do not exist in the first place.

At the very least, to make it crystal clear in my mind, particles in regions of space moving apart at > c cannot influence one another's behaviour?

As I said above, the apparent recession velocity is not the relevant parameter. The CMB light we see today (=interaction) was emitted by things where the distance to us increased faster than the speed of light at any point in time. There are regions in space that cannot influence each other, correct.

Actually, mfb, perhaps the easiest thing would be just to expand on what you mean by all interactions in QFT being local? (and sorry, above I'm calling the electron and proton 'particles', but I realize they really should be field excitations).

What is unclear about locality? An event A cannot influence an event B if B is not in the future light cone of A. If you consider a single moment in time, then nothing has a "range" - because everything can only influence the fields at the same place.