What do you mean by the technical challenge posed by relativity? As far as I can tell relativistic quantum mechanics is unusually satisfactory and uncontroversial.
In relativistic QFT, you start with something that is formally Lorentz-invariant (a Lagrangian density) and you end up with predictions that are invariant (or covariant), but to get from the Lagrangian to the predictions, you utilize objects and procedures which are frame-dependent and usually gauge-dependent too. Many basic properties of QFTs are still unproven (inconsistency of QED at Landau scale, existence of mass gap in QCD), and the theories have a somewhat heuristic existence, as an open-ended set of approximations and calculational procedures.
From an instrumentalist perspective, such procedures are relativistic if they start relativistic and end relativistic, with any frame-fixing and gauge-fixing that you do along the way “cancelling out” by the time you get to the end. But wavefunction realism is about reifying some of the objects which you use in the middle of that process, and they will usually bear the imprint of a preferred frame and a preferred gauge. Some of those objects are invariant or covariant (Heisenberg-picture operators, the S-matrix, asymptotic field states, some Feynman diagrams), but I sure don’t see how to turn them into the basis of an ontologically relativistic wavefunction realism, which is what relativistic MWI would have to be.
If all the rules governing the wavefunction are relativistic, then the result is relativistic. You can look at it however you like, including non-relativistic ways. I don’t see a problem here.
A lot of the time, the rules aren’t relativistic, just the input and output. MWI is a theory about what’s inside the black box of reality: wavefunctions! Wavefunctions are what’s real, it says. But the wavefunctions that actually get used in QFT are defined on a preferred time-slicing (closed time-path formalism), or they only exist asymptotically, in the infinite past or future (S-matrix)… The mathematical manipulations in QFT can be quite baroque, and I am very far from seeing how you could make them all relativistic at every stage.
The only way I can see to do it, is to break the theory down to the level of individual space-time points, allow continua of duplicates of points, define the analogue of Lorentz transformations in the resulting infinite-dimensional space, and then define a way to build the configurations (that enter into a wavefunctional in the position basis) out of these points, while also associating amplitudes or proto-amplitudes with the way that the points are glued together, so that configurations can have global amplitudes attached to them as required. It would be a sort of bottom-up approach to the “spacetime state realism” described by Wallace and Timpson, and it might not even be a well-defined approach for a QFT that isn’t UV-complete, like QED.
That was all a mouthful to say, and I regret introducing such complexity into the discussion, but that is how I think when I ask myself “how could you describe a quantum multiverse that is genuinely relativistic?” To ontologically ground QFT, you have to specify what it is that exists in the ontology, and you have to explain what the QFT calculational procedures mean ontologically—why they work, what real things they refer to. And if you’re going to ground it all in wavefunctions, and you don’t want to be stuck in a preferred frame, then you have to do something drastic.
the wavefunctions that actually get used in QFT are defined on a preferred time-slicing (closed time-path formalism), or they only exist asymptotically, in the infinite past or future (S-matrix)
When you describe a state, you need to choose a method of describing it, yes. But you can choose to describe it in any frame you like, and you can transform from one such description to another in a different frame. This is an artifact of the descriptions, not the thing in itself.
Like, you have a covariant quantity. You can do all sorts of symbolic math with it and it’s totally relativistic. But then you want to do a calculation. You’re going to have to pick a frame so you can work with actual numbers. These numbers are not vectors or tensors—they’re scalars. They themselves do not obey the transformation laws of the entities they represent.
BUT that doesn’t mean that using your description involves invoking a preferred frame. You know how to turn that description into a description in any other frame you like, and if you do, the results come out the same.
So, the time-slicing method is perfectly legit. In principle, you could use any mutually-time-like-separated slice, but it’s usually inconvenient to do so.
(edit: I meant, ‘you could use any arbitrary mutually-SPACE-like-separated curve, but it’s usually inconvenient to pick anything more complicated than strictly necessary’)
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.
What do you mean by the technical challenge posed by relativity? As far as I can tell relativistic quantum mechanics is unusually satisfactory and uncontroversial.
In relativistic QFT, you start with something that is formally Lorentz-invariant (a Lagrangian density) and you end up with predictions that are invariant (or covariant), but to get from the Lagrangian to the predictions, you utilize objects and procedures which are frame-dependent and usually gauge-dependent too. Many basic properties of QFTs are still unproven (inconsistency of QED at Landau scale, existence of mass gap in QCD), and the theories have a somewhat heuristic existence, as an open-ended set of approximations and calculational procedures.
From an instrumentalist perspective, such procedures are relativistic if they start relativistic and end relativistic, with any frame-fixing and gauge-fixing that you do along the way “cancelling out” by the time you get to the end. But wavefunction realism is about reifying some of the objects which you use in the middle of that process, and they will usually bear the imprint of a preferred frame and a preferred gauge. Some of those objects are invariant or covariant (Heisenberg-picture operators, the S-matrix, asymptotic field states, some Feynman diagrams), but I sure don’t see how to turn them into the basis of an ontologically relativistic wavefunction realism, which is what relativistic MWI would have to be.
If all the rules governing the wavefunction are relativistic, then the result is relativistic. You can look at it however you like, including non-relativistic ways. I don’t see a problem here.
A lot of the time, the rules aren’t relativistic, just the input and output. MWI is a theory about what’s inside the black box of reality: wavefunctions! Wavefunctions are what’s real, it says. But the wavefunctions that actually get used in QFT are defined on a preferred time-slicing (closed time-path formalism), or they only exist asymptotically, in the infinite past or future (S-matrix)… The mathematical manipulations in QFT can be quite baroque, and I am very far from seeing how you could make them all relativistic at every stage.
The only way I can see to do it, is to break the theory down to the level of individual space-time points, allow continua of duplicates of points, define the analogue of Lorentz transformations in the resulting infinite-dimensional space, and then define a way to build the configurations (that enter into a wavefunctional in the position basis) out of these points, while also associating amplitudes or proto-amplitudes with the way that the points are glued together, so that configurations can have global amplitudes attached to them as required. It would be a sort of bottom-up approach to the “spacetime state realism” described by Wallace and Timpson, and it might not even be a well-defined approach for a QFT that isn’t UV-complete, like QED.
That was all a mouthful to say, and I regret introducing such complexity into the discussion, but that is how I think when I ask myself “how could you describe a quantum multiverse that is genuinely relativistic?” To ontologically ground QFT, you have to specify what it is that exists in the ontology, and you have to explain what the QFT calculational procedures mean ontologically—why they work, what real things they refer to. And if you’re going to ground it all in wavefunctions, and you don’t want to be stuck in a preferred frame, then you have to do something drastic.
When you describe a state, you need to choose a method of describing it, yes. But you can choose to describe it in any frame you like, and you can transform from one such description to another in a different frame. This is an artifact of the descriptions, not the thing in itself.
Like, you have a covariant quantity. You can do all sorts of symbolic math with it and it’s totally relativistic. But then you want to do a calculation. You’re going to have to pick a frame so you can work with actual numbers. These numbers are not vectors or tensors—they’re scalars. They themselves do not obey the transformation laws of the entities they represent.
BUT that doesn’t mean that using your description involves invoking a preferred frame. You know how to turn that description into a description in any other frame you like, and if you do, the results come out the same.
So, the time-slicing method is perfectly legit. In principle, you could use any mutually-time-like-separated slice, but it’s usually inconvenient to do so. (edit: I meant, ‘you could use any arbitrary mutually-SPACE-like-separated curve, but it’s usually inconvenient to pick anything more complicated than strictly necessary’)
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.