I’ve made a sketch to illustrate the simplest version of the problem.
Horizontal direction is spacelike, vertical direction is timelike. On the left we have a classically relativistic theory. Everything reduces to properties localized at individual space-time points (the blue dots), so there’s no significance to a change of slicing (black vs pink), you’re just grouping and re-grouping the dots differently.
On the right we have a quantum theory. There’s a quantum state on each slice. A red circle around two dots indicates entanglement. How can we apply relativity here? Well, we can represent the same slicing differently in two coordinate systems, changing the space-time “tilt” of the slices (this is what I’ve illustrated). But if you adopt a truly different slicing, one that cuts across the original slicing (as pink cuts across black on the far left), you don’t have a recipe for specifying what the quantum states on the new slices should be; because the quantum states on the original slicing do not decompose into properties located solely at single space-time points. The entanglement is made up of properties depending on two or more locations at once.
In practical QFT, the situation isn’t usually this straightforward. E.g. perturbation theory is often introduced by talking about pure momentum states which notionally fill the whole of space-time, and not just a single slice. The Feynman diagrams which add up to give S-matrix elements then appear to represent space-time networks of point interactions between completely delocalized objects. I think it’s just incredibly un-obvious how to turn that into a clear wavefunction-is-real ontology. What’s the master wavefunction containing all the diagrammatic processes? What space is it defined over? Does this master wavefunction have a representation in terms of an instantaneous wavefunction on slicings that evolves over time? If it does, how do you change slicings? If it doesn’t, what’s the relation between the whole (master wavefunction) and the parts (history-superpositions as summed up in a Feynman diagram)?
And even that is all still rather elementary compared to the full complexity of what people do in QFT. So how to define relativistic MWI is a major challenge. I hope that the “simplest version of the problem”, that I started with, conveys some of why this is so.
Entanglement is over mutually-timelike regions, not merely simultaneous moments, so your diagrams are misleading. Try redrawing your ellipses of entanglement so they’re legal spacetime entities. If you redraw ALL of the entanglement this way, then it will transform just fine.
Entanglement is over mutually-timelike regions, not merely simultaneous moments
Entangled regions should be spacelike-separated from each other; do you mean that each individual region will have some internal timelike extension? Is this about the smearing of field operators to create normalizable states?
Maybe we should have this discussion privately. I’m quite keen to discuss technicalities but I don’t want to spam the site with it.
I misused a phrasing there—mutually timelike regions can only be 1 dimensional or less, just as mutually spacelike regions can only be 3 or fewer dimensional.
Entanglement is between points that are spacelike separated, but the boundaries of this entanglement—the processes that create or destroy it—are purely causal and local.
We can continue in PM. I just wanted to clear that up, since I ended on something that was flat-out wrong.
I’ve made a sketch to illustrate the simplest version of the problem.
Horizontal direction is spacelike, vertical direction is timelike. On the left we have a classically relativistic theory. Everything reduces to properties localized at individual space-time points (the blue dots), so there’s no significance to a change of slicing (black vs pink), you’re just grouping and re-grouping the dots differently.
On the right we have a quantum theory. There’s a quantum state on each slice. A red circle around two dots indicates entanglement. How can we apply relativity here? Well, we can represent the same slicing differently in two coordinate systems, changing the space-time “tilt” of the slices (this is what I’ve illustrated). But if you adopt a truly different slicing, one that cuts across the original slicing (as pink cuts across black on the far left), you don’t have a recipe for specifying what the quantum states on the new slices should be; because the quantum states on the original slicing do not decompose into properties located solely at single space-time points. The entanglement is made up of properties depending on two or more locations at once.
In practical QFT, the situation isn’t usually this straightforward. E.g. perturbation theory is often introduced by talking about pure momentum states which notionally fill the whole of space-time, and not just a single slice. The Feynman diagrams which add up to give S-matrix elements then appear to represent space-time networks of point interactions between completely delocalized objects. I think it’s just incredibly un-obvious how to turn that into a clear wavefunction-is-real ontology. What’s the master wavefunction containing all the diagrammatic processes? What space is it defined over? Does this master wavefunction have a representation in terms of an instantaneous wavefunction on slicings that evolves over time? If it does, how do you change slicings? If it doesn’t, what’s the relation between the whole (master wavefunction) and the parts (history-superpositions as summed up in a Feynman diagram)?
And even that is all still rather elementary compared to the full complexity of what people do in QFT. So how to define relativistic MWI is a major challenge. I hope that the “simplest version of the problem”, that I started with, conveys some of why this is so.
Entanglement is over mutually-timelike regions, not merely simultaneous moments, so your diagrams are misleading. Try redrawing your ellipses of entanglement so they’re legal spacetime entities. If you redraw ALL of the entanglement this way, then it will transform just fine.
Entangled regions should be spacelike-separated from each other; do you mean that each individual region will have some internal timelike extension? Is this about the smearing of field operators to create normalizable states?
Maybe we should have this discussion privately. I’m quite keen to discuss technicalities but I don’t want to spam the site with it.
I misused a phrasing there—mutually timelike regions can only be 1 dimensional or less, just as mutually spacelike regions can only be 3 or fewer dimensional.
Entanglement is between points that are spacelike separated, but the boundaries of this entanglement—the processes that create or destroy it—are purely causal and local.
We can continue in PM. I just wanted to clear that up, since I ended on something that was flat-out wrong.