A wavefunction may evolve over time, but configuration space does not.
This is probably not true. To really get off the ground with quantum field theory, you have to attach an a priori different Hilbert space of states to each space-like slice of spacetime, and make sense of what equations of motion could mean in this setting—at least this is my limited understanding. I haven’t been following your discussion and I don’t know how it affects the MWI.
you have to attach an a priori different Hilbert space of states to each space-like slice of spacetime
That is a valid formalism but then all the Hilbert spaces are copies of the same Hilbert space, and in the configuration basis, the state vectors are still wavefunctionals over an identical configuration space. The only difference is that the configurations are defined on a different hypersurface; but the field configurations are otherwise the same.
ETA: This comment currently has one downvote and no follow-up, which doesn’t tell me what I got “wrong”. But I will use the occasion to add some more detail.
In perturbative quantum field theory, one of the basic textbook approaches to calculation is the “interaction picture”, a combination of the Schrodinger picture in which the state vector evolves with time and the operators do not, and the Heisenberg picture, in which the state vector is static and the operators evolve with time. In Veltman’s excellent book Diagrammatica, I seem to recall a discussion of the interaction picture, in which it was motivated in terms of different Hilbert spaces. But I could be wrong, he may just have been talking about the S-matrix in general, and I no longer have the book.
The Hilbert spaces of quantum field theory usually only have a nominal existence anyway, because of renormalization. The divergences mean that they are ill-defined; what is well-defined is the renormalization procedure, which is really a calculus of infinite formal series. It is presumed that truly fundamental, “ultraviolet-complete” theories should have properly defined Hilbert spaces, and that the renormalizable field theories are well-defined truncations of unspecified UV-complete theories. But the result is that practical QFT sees a lot of abuse of formalism when judged by mathematical standards.
So it’s impossible to guess whether Sewing-Machine is talking about a way that Hilbert spaces are used in a particular justification of a practical QFT formalism; or perhaps it is the way things are done in one of the attempts to define a mathematically rigorous approach to QFT, such as “algebraic QFT”. These rigorous approaches usually have little to say about the field theories actually used in particle physics, because the latter are renormalizable gauge theories and only exist at that “procedural” level of calculation.
But it should in any case be obvious that the configuration space of field theory in flat space is the same on parallel hypersurfaces. It’s the same fields, the same geometry, the same superposition principle. Anyone who objects is invited to provide a counterargument.
Mathematicians work with toy models of quantum field theories (eg topological QFTs) whose purpose would be entirely defeated if all slices had the same fields on them. For instance, the topology of a slice can change, and mathematicians get pretty excited by theories that can measure such changes. Talking about flat spacetime I suppose such topology changes are irrelevant, and you’re saying moreover that there are no subtler geometric changes that are relevant at all? What if I choose two non-parallel slices?
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.
This is probably not true. To really get off the ground with quantum field theory, you have to attach an a priori different Hilbert space of states to each space-like slice of spacetime, and make sense of what equations of motion could mean in this setting—at least this is my limited understanding. I haven’t been following your discussion and I don’t know how it affects the MWI.
That is a valid formalism but then all the Hilbert spaces are copies of the same Hilbert space, and in the configuration basis, the state vectors are still wavefunctionals over an identical configuration space. The only difference is that the configurations are defined on a different hypersurface; but the field configurations are otherwise the same.
ETA: This comment currently has one downvote and no follow-up, which doesn’t tell me what I got “wrong”. But I will use the occasion to add some more detail.
In perturbative quantum field theory, one of the basic textbook approaches to calculation is the “interaction picture”, a combination of the Schrodinger picture in which the state vector evolves with time and the operators do not, and the Heisenberg picture, in which the state vector is static and the operators evolve with time. In Veltman’s excellent book Diagrammatica, I seem to recall a discussion of the interaction picture, in which it was motivated in terms of different Hilbert spaces. But I could be wrong, he may just have been talking about the S-matrix in general, and I no longer have the book.
The Hilbert spaces of quantum field theory usually only have a nominal existence anyway, because of renormalization. The divergences mean that they are ill-defined; what is well-defined is the renormalization procedure, which is really a calculus of infinite formal series. It is presumed that truly fundamental, “ultraviolet-complete” theories should have properly defined Hilbert spaces, and that the renormalizable field theories are well-defined truncations of unspecified UV-complete theories. But the result is that practical QFT sees a lot of abuse of formalism when judged by mathematical standards.
So it’s impossible to guess whether Sewing-Machine is talking about a way that Hilbert spaces are used in a particular justification of a practical QFT formalism; or perhaps it is the way things are done in one of the attempts to define a mathematically rigorous approach to QFT, such as “algebraic QFT”. These rigorous approaches usually have little to say about the field theories actually used in particle physics, because the latter are renormalizable gauge theories and only exist at that “procedural” level of calculation.
But it should in any case be obvious that the configuration space of field theory in flat space is the same on parallel hypersurfaces. It’s the same fields, the same geometry, the same superposition principle. Anyone who objects is invited to provide a counterargument.
Mathematicians work with toy models of quantum field theories (eg topological QFTs) whose purpose would be entirely defeated if all slices had the same fields on them. For instance, the topology of a slice can change, and mathematicians get pretty excited by theories that can measure such changes. Talking about flat spacetime I suppose such topology changes are irrelevant, and you’re saying moreover that there are no subtler geometric changes that are relevant at all? What if I choose two non-parallel slices?
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.