Quantum field theory is very flexible and can take many forms. In particle physics one mostly cares about quantum fields in flat space—the effects of spatial curvature are nonexistent e.g. in particle collider physics—and this is really the paradigmatic form of QFT as far as a physicist is concerned. There is a lot that can be done with QFT in curved space, but ultimately that takes us towards the fathomless complexities of quantum gravity. I expect that the final answer to the meaning of quantum mechanics lies there, so it is not a topic one can avoid in the long run. But I do not think that adding gravity to the mix simplifies MWI’s problem with relativity, unless you take Julian Barbour’s option and decide to prefer the position basis on the grounds that there is no time evolution in quantum gravity. That is an eccentric combination of views and I think it is a side road to nowhere, on the long journey to the truth. Meanwhile, in the short term, considering the nature of quantum field theory in Minkowski space has the value, that it shows up a common deficiency in Many Worlds thinking.
Non-parallel slices… In general, we are talking about time evolution here. For example, we may consider a wavefunction on an initial hypersurface, and another wavefunction on a final hypersurface, and ask what is the amplitude to go from one to the other. In a Schrodinger picture, you might obtain this amplitude by evolving the initial wavefunction forward in time to the final hypersurface, and then taking the inner product with the final wavefunction, which tells you “how much” (as orthonormal might put it) of the time-evolved wavefunction consists of the desired final wavefunction; what the overlap is.
Non-parallel spacelike hypersurfaces will intersect somewhere, but you could still try to perform a similar calculation. The first difficulty is, how do you extrapolate from the ‘initial’ to the ‘final’ hypersurface? Ordinary time evolution won’t do, because the causal order (which hypersurface comes first) will be different on different sides of the plane of intersection. If I was trying to do this I would resort to path integrals: develop a Green’s function or other propagator-like expression which provides an amplitude for a transition from one exact field configuration on one hypersurface, to a different exact field configuration on the other hypersurface, then express the initial and final wavefunctions in the configuration basis, and integrate the configuration-to-configuration transition amplitudes accordingly. One thing you might notice is that the amplitude for configuration-to-configuration transition, when we talk about configurations on intersecting hypersurfaces, ought to be zero unless the configurations exactly match on the plane of intersection.
It’s sort of an interesting problem mathematically, but it doesn’t seem too relevant to physics. What might be relevant is if you were dealing with finite (bounded) hypersurfaces—so there was no intersection, as would be inevitable in flat space if they were continued to infinity. Instead, you’re just dealing with finite patches of space-time, which have a different spacelike ‘tilt’. Again, the path integral formalism has to be the right way to do it from first principles. It’s really more general than anything involving wavefunctions.
It’s sort of an interesting problem mathematically, but it doesn’t seem too relevant to physics
I disagree, certainly this is where all of the fun stuff happens in classical relativity.
Anyway, I guess I buy your explanation that time evolution identifies the state spaces of two parallel hypersurfaces, but my quarter-educated hunch is that you’ll find it’s not so in general.
Quantum field theory is very flexible and can take many forms. In particle physics one mostly cares about quantum fields in flat space—the effects of spatial curvature are nonexistent e.g. in particle collider physics—and this is really the paradigmatic form of QFT as far as a physicist is concerned. There is a lot that can be done with QFT in curved space, but ultimately that takes us towards the fathomless complexities of quantum gravity. I expect that the final answer to the meaning of quantum mechanics lies there, so it is not a topic one can avoid in the long run. But I do not think that adding gravity to the mix simplifies MWI’s problem with relativity, unless you take Julian Barbour’s option and decide to prefer the position basis on the grounds that there is no time evolution in quantum gravity. That is an eccentric combination of views and I think it is a side road to nowhere, on the long journey to the truth. Meanwhile, in the short term, considering the nature of quantum field theory in Minkowski space has the value, that it shows up a common deficiency in Many Worlds thinking.
Non-parallel slices… In general, we are talking about time evolution here. For example, we may consider a wavefunction on an initial hypersurface, and another wavefunction on a final hypersurface, and ask what is the amplitude to go from one to the other. In a Schrodinger picture, you might obtain this amplitude by evolving the initial wavefunction forward in time to the final hypersurface, and then taking the inner product with the final wavefunction, which tells you “how much” (as orthonormal might put it) of the time-evolved wavefunction consists of the desired final wavefunction; what the overlap is.
Non-parallel spacelike hypersurfaces will intersect somewhere, but you could still try to perform a similar calculation. The first difficulty is, how do you extrapolate from the ‘initial’ to the ‘final’ hypersurface? Ordinary time evolution won’t do, because the causal order (which hypersurface comes first) will be different on different sides of the plane of intersection. If I was trying to do this I would resort to path integrals: develop a Green’s function or other propagator-like expression which provides an amplitude for a transition from one exact field configuration on one hypersurface, to a different exact field configuration on the other hypersurface, then express the initial and final wavefunctions in the configuration basis, and integrate the configuration-to-configuration transition amplitudes accordingly. One thing you might notice is that the amplitude for configuration-to-configuration transition, when we talk about configurations on intersecting hypersurfaces, ought to be zero unless the configurations exactly match on the plane of intersection.
It’s sort of an interesting problem mathematically, but it doesn’t seem too relevant to physics. What might be relevant is if you were dealing with finite (bounded) hypersurfaces—so there was no intersection, as would be inevitable in flat space if they were continued to infinity. Instead, you’re just dealing with finite patches of space-time, which have a different spacelike ‘tilt’. Again, the path integral formalism has to be the right way to do it from first principles. It’s really more general than anything involving wavefunctions.
I disagree, certainly this is where all of the fun stuff happens in classical relativity.
Anyway, I guess I buy your explanation that time evolution identifies the state spaces of two parallel hypersurfaces, but my quarter-educated hunch is that you’ll find it’s not so in general.