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komponisto comments on Stupid Questions Open Thread Round 3 - Less Wrong
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This article (PDF) gives a nice (and fairly accessible) summary of some of the issues involved in extending MWI to QFT. See sections 4 and 8 in particular. Their focus in the paper is wavefunction realism, but given that MWI (at least the version advocated in the Sequences) is committed to wavefunction realism, their arguments apply. They offer a suggestion of the kind of theory that they think can replace MWI in the relativistic context, but the view is insufficiently developed (at least in that paper) for me to fully evaluate it.
A quick summary of the issues raised in the paper:
In NRQM, the wave function lives in configuration space, but there is no well-defined particle configuration space in QFT since particle number is not conserved and particles are emergent entities without precisely defined physical properties.
A move to field configuration space is unsatisfactory because quantum field theories admit of equivalent description using many different choices of field observable. Unlike NRQM, where there are solid dynamical reasons for choosing the position basis as fundamental, there seems to be no natural or dynamically preferred choice in QFT, so a choice of a particular field configuration space description would amount to ad hoc privileging.
MWI in NRQM treats physical space as non-fundamental. This is hard to justify in QFT, because physical space-time is bound up with the fundamentals of the theory to a much greater degree. The dynamical variables in QFT are operators that are explicitly associated with space-time regions.
This objection is particularly clever and interesting, I think. In MWI, the history of the universe is fully specified by giving the universal wavefunction at each time in some reference frame. In a relativistic context, one would expect that all one needs to do in order to describe how the universe looks in some other inertial reference frame is to perform a Lorentz transformation on this history. If the history really tells us everything about the physical state of the universe, then it gives us all the information required to determine how the universe looks under a Lorentz transformation. But in relativistic quantum mechanics, this is not true. Fully specifying the wavefunction (defined on an arbitrarily chosen field configuration space, say) at all times is not sufficient to determine what the universe will look like under a Lorentz transformation. See the example on p. 21 in the paper, or read David Albert's paper on narratability. This suggests that giving the wavefunction at all times is not a full specification of the physical properties of the universe.
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. Thanks to the development of renormalization group theory in the 70s, we know how to do functional integrals in QFT with an imposed cutoff at some finite short length scale. QFT with a cutoff doesn't suffer from problems involving infinities. Of course, the necessity of the cutoff is an indication that QFT is not a completely accurate description of the universe. But we already know that we're going to need a theory of quantum gravity at the Planck scale. In the domain where it works, QFT is reasonably rigorously defined, I'd say.
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
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.)