This article (PDF) gives a nice (and fairly accessible) summary of some of the issues involved in extending MWI to QFT.
Thanks for that; it’s quite an interesting article, and I’m still trying to absorb it. However, one thing that seems pretty clear to me is that for EY’s intended philosophical purposes, there really is no important distinction between “wavefunction realism” (in the context of NRQM) and “spacetime state realism” (in the context of QFT). Especially since I consider this post to be mostly wrong: locality in configuration space is what matters, and configuration space is a vector space (specifically a Hilbert space) -- there is no preferred (orthonormal) basis.
I assume you’re referring to the infinities that arise in QFT when we integrate over arbitrarily short length scales. I don’t think this shows a lack of rigor in QFT
If the “problem” is merely that certain integrals are divergent, then I agree. No one says that the fact that int_{0}{1}frac{1}{x},dx
diverges shows a lack of rigor in real analysis!
What concerns me is whether any actual mathematical lies are being told—such as integrals being assumed to converge when they haven’t yet been proved to do so. Or something like the early history of the Dirac delta, when physicists unashamedly spoke of a “function” with properties that a function cannot, in fact, have.
If QFT is merely a physical lie—i.e., “not a completely accurate description of the universe”—and not a mathematical one, then that’s a different matter, and I wouldn’t call it an issue of “rigor”.
However, one thing that seems pretty clear to me is that for EY’s intended philosophical purposes, there really is no important distinction between “wavefunction realism” (in the context of NRQM) and “spacetime state realism” (in the context of QFT).
I’m a little unclear about what EY’s intended philosophical purposes are in this context, so this might well be true. One possible problem worth pointing out is that spacetime state realism involves an abandonment of a particular form of reductionism. Whether or not EY is committed to this form of reductionism somebody more familiar with the sequences than I would have to judge.
According to spacetime state realism, the physical state of a spacetime region is not supervenient on the physical states of its subregions, i.e. the physical state of a spacetime region could be different without any of its subregions being in different states. This is because subregions can be entangled with one another in different ways without altering their local states. This is not true of wavefunction realism set in configuration space. There, the only way a region of configuration space could have different physical properties is if some of its subregions had different properties.
Also, I think it’s possible that the fact that the different “worlds” in spacetime state realism are spatially overlapping (as opposed to wavefunction realism, where they are separated in configuration space) might lead to interesting conceptual differences between the two interpretations. I haven’t thought about this enough to give specific reasons for this suspicion, though.
Especially since I consider this post to be mostly wrong: locality in configuration space is what matters, and configuration space is a vector space (specifically a Hilbert space) -- there is no preferred (orthonormal) basis.
I’m not sure exactly what you’re saying here, but if you’re rejecting the claim that MWI privileges a particular basis, I think you’re wrong. Of course, you could treat configuration space itself as if it had no preferred basis, but this would still amount to privileging position over momentum. You can’t go from position space to momentum space by a change of coordinates in configuration space. Configuration space is always a space of possible particle position configurations, no matter how you transform the coordinates.
I think you might be conflating configuration space with the Hilbert space of wavefunctions on configuration space. In this latter space, you can transform from a basis of position eigenstates to a basis of momentum eigenstates with a coordinate transformation. But this is not configuration space itself, it is the space of square integrable functions on configuration space. [I’m lying a little for simplicity: Position and momentum eigenstates aren’t actually square integrable functions on configuration space, but there are various mathematical tricks to get around this complication.]
What concerns me is whether any actual mathematical lies are being told—such as integrals being assumed to converge when they haven’t yet been proved to do so. Or something like the early history of the Dirac delta, when physicists unashamedly spoke of a “function” with properties that a function cannot, in fact, have.
If this is your standard for lack of rigor, then perhaps QFT hasn’t been rigorously formulated yet, but the same would hold of pretty much any physical theory. I think you can find places in pretty much every theory where some such “mathematical lie” is relied upon. There’s an example of a standard mathematical lie told in NRQM earlier in my post.
In many of these cases, mathematicians have formulated more rigorous versions of the relevant proofs, but I think most physicists tend to be blithely ignorant of these mathematical results. Maybe QFT isn’t rigorously formulated according to the mathematician’s standards of rigor, but it meets the physicist’s lower standards of rigor. There’s a reason most physicists working on QFT are uninterested in things like Algebraic Quantum Field Theory.
I’m a little unclear about what EY’s intended philosophical purposes are in this context
As I read him, he mainly wants to make the point that “simplicity” is not the same as “intuitiveness”, and the former trumps the latter. It may seem more “humanly natural” for there to be some magical process causing wavefunction collapse than for there to be a proliferation of “worlds”, but because the latter doesn’t require any additions to the equations, it is strictly simpler and thus favored by Occam’s Razor.
I think you might be conflating configuration space with the Hilbert space of wavefunctions on configuration space.
Yes, sorry. What I actually meant by “configuration space” was “the Hilbert space that wavefunctions are elements of”. That space, whatever you call it (“state space”?), is the one that matters in the context of “wavefunction realism”.
(This explains an otherwise puzzling passage in the article you linked, which contrasts the “configuration space” and “Hilbert space” formalisms; but on the other hand, it reduces my credence that EY knows what he’s talking about in the QM sequence, since he doesn’t seem to talk about the space-that-wavefunctions-are-elements-of much at all.)
If this is your standard for lack of rigor, then perhaps QFT hasn’t been rigorously formulated yet, but the same would hold of pretty much any physical theory
This is contrary to my understanding. I was under the impression that classical mechanics, general relativity, and NRQM had all by now been given rigorous mathematical formulations (in terms of symplectic geometry, Lorentzian geometry, and the theory of operators on Hilbert space respectively).
Maybe QFT isn’t rigorously formulated according to the mathematician’s standards of rigor, but it meets the physicist’s lower standards of rigor. There’s a reason most physicists working on QFT are uninterested in things like Algebraic Quantum Field Theory.
The mathematician’s standards are what interests me, and are what I mean by “rigor”. I don’t consider it a virtue on the part of physicists that they are unaware of or uninterested in the mathematical foundations of physics, even if they are able to get away with being so uninterested. There is a reason mathematicians have the standards of rigor they do. (And it should of course be said that some physicists are interested in rigorous mathematics.)
Thanks for that; it’s quite an interesting article, and I’m still trying to absorb it. However, one thing that seems pretty clear to me is that for EY’s intended philosophical purposes, there really is no important distinction between “wavefunction realism” (in the context of NRQM) and “spacetime state realism” (in the context of QFT). Especially since I consider this post to be mostly wrong: locality in configuration space is what matters, and configuration space is a vector space (specifically a Hilbert space) -- there is no preferred (orthonormal) basis.
If the “problem” is merely that certain integrals are divergent, then I agree. No one says that the fact that int_{0}{1}frac{1}{x},dx diverges shows a lack of rigor in real analysis!
What concerns me is whether any actual mathematical lies are being told—such as integrals being assumed to converge when they haven’t yet been proved to do so. Or something like the early history of the Dirac delta, when physicists unashamedly spoke of a “function” with properties that a function cannot, in fact, have.
If QFT is merely a physical lie—i.e., “not a completely accurate description of the universe”—and not a mathematical one, then that’s a different matter, and I wouldn’t call it an issue of “rigor”.
I’m a little unclear about what EY’s intended philosophical purposes are in this context, so this might well be true. One possible problem worth pointing out is that spacetime state realism involves an abandonment of a particular form of reductionism. Whether or not EY is committed to this form of reductionism somebody more familiar with the sequences than I would have to judge.
According to spacetime state realism, the physical state of a spacetime region is not supervenient on the physical states of its subregions, i.e. the physical state of a spacetime region could be different without any of its subregions being in different states. This is because subregions can be entangled with one another in different ways without altering their local states. This is not true of wavefunction realism set in configuration space. There, the only way a region of configuration space could have different physical properties is if some of its subregions had different properties.
Also, I think it’s possible that the fact that the different “worlds” in spacetime state realism are spatially overlapping (as opposed to wavefunction realism, where they are separated in configuration space) might lead to interesting conceptual differences between the two interpretations. I haven’t thought about this enough to give specific reasons for this suspicion, though.
I’m not sure exactly what you’re saying here, but if you’re rejecting the claim that MWI privileges a particular basis, I think you’re wrong. Of course, you could treat configuration space itself as if it had no preferred basis, but this would still amount to privileging position over momentum. You can’t go from position space to momentum space by a change of coordinates in configuration space. Configuration space is always a space of possible particle position configurations, no matter how you transform the coordinates.
I think you might be conflating configuration space with the Hilbert space of wavefunctions on configuration space. In this latter space, you can transform from a basis of position eigenstates to a basis of momentum eigenstates with a coordinate transformation. But this is not configuration space itself, it is the space of square integrable functions on configuration space. [I’m lying a little for simplicity: Position and momentum eigenstates aren’t actually square integrable functions on configuration space, but there are various mathematical tricks to get around this complication.]
If this is your standard for lack of rigor, then perhaps QFT hasn’t been rigorously formulated yet, but the same would hold of pretty much any physical theory. I think you can find places in pretty much every theory where some such “mathematical lie” is relied upon. There’s an example of a standard mathematical lie told in NRQM earlier in my post.
In many of these cases, mathematicians have formulated more rigorous versions of the relevant proofs, but I think most physicists tend to be blithely ignorant of these mathematical results. Maybe QFT isn’t rigorously formulated according to the mathematician’s standards of rigor, but it meets the physicist’s lower standards of rigor. There’s a reason most physicists working on QFT are uninterested in things like Algebraic Quantum Field Theory.
As I read him, he mainly wants to make the point that “simplicity” is not the same as “intuitiveness”, and the former trumps the latter. It may seem more “humanly natural” for there to be some magical process causing wavefunction collapse than for there to be a proliferation of “worlds”, but because the latter doesn’t require any additions to the equations, it is strictly simpler and thus favored by Occam’s Razor.
Yes, sorry. What I actually meant by “configuration space” was “the Hilbert space that wavefunctions are elements of”. That space, whatever you call it (“state space”?), is the one that matters in the context of “wavefunction realism”.
(This explains an otherwise puzzling passage in the article you linked, which contrasts the “configuration space” and “Hilbert space” formalisms; but on the other hand, it reduces my credence that EY knows what he’s talking about in the QM sequence, since he doesn’t seem to talk about the space-that-wavefunctions-are-elements-of much at all.)
This is contrary to my understanding. I was under the impression that classical mechanics, general relativity, and NRQM had all by now been given rigorous mathematical formulations (in terms of symplectic geometry, Lorentzian geometry, and the theory of operators on Hilbert space respectively).
The mathematician’s standards are what interests me, and are what I mean by “rigor”. I don’t consider it a virtue on the part of physicists that they are unaware of or uninterested in the mathematical foundations of physics, even if they are able to get away with being so uninterested. There is a reason mathematicians have the standards of rigor they do. (And it should of course be said that some physicists are interested in rigorous mathematics.)