It would be nice if the universe were finite, but you can’t demand that a priori; it’s enough that the infinite mathematical object obeys simple rules.
I’m saying that if we lived in another universe, and someone came along and described to us the wavefunction for the Schrodinger equation, and asked how we should regard the size of some part of the configuration space compared to some other part, the L^2 norm is the blindingly obvious mathematical answer because of the properties of the wavefunction. And so if we (outside the system) were looking for a “typical” instance of a configuration corresponding to a mind, we would weight the configurations by the L^2 norm of the wavefunction.
Because (as it turns out) the wavefunction has a distinguished exceptionally-low-entropy state corresponding to the Big Bang, the configurations where the wavefunction is relatively large encode in various ways the details of (practically unique) intermediate stages between the Big Bang state and the one under consideration: that is, they encode unique histories (1). So a “typical” instance of a configuration containing a mind turns out to be one that places it within a context of a unique and lawful history satisfying the Born probabilities, because the L^2 norm of the wavefunction over the set where they hold to within epsilon is much, much larger than the L^2 norm of the rest. So to the extent that I’m a typical instance of mind-configurations similar to me, I should expect to remember and see evidence of histories satisfying the Born probabilities.
...seriously, I don’t see why people get worked up over this. OK, Eliezer has his infinite-set atheism, and you have your insistence on a naive theory of qualia, but what about everyone else?
(1) This is not a conjecture, it is not controversial, it is something you can prove mathematically about the Schrodinger equation in various contexts.
how we should regard the size of some part of the configuration space compared to some other part, the L^2 norm is the blindingly obvious mathematical answer because of the properties of the wavefunction.
Does “part of the configuration space” refer to a single state vector, or a whole region that a state vector might belong to? My impression is that measuring the latter sort of thing is problematic from a rigorous mathematical standpoint. Is this correct, and does it have consequences for your discussion?
I say the former; people scared of continuous densities might prefer the latter, at which point they have the traditional sorites paradox of how large an epsilon-neighborhood to draw; but in practical terms, this isn’t so bad because (if we start with low entropy) decoherence rapidly separates the wavefunction into thin wisps with almost-zero values taken between them.
Okay, I have tried to understand what sort of ontology could answer to your description. A key consideration: you say we should judge “the size of some part of the configuration space compared to some other part” according to “the L^2 norm of the wavefunction”. You also talk about “mind-configurations similar to me”.
A wavefunction may evolve over time, but configuration space does not. Configuration space is a static arena, and the amplitudes associated with configurations change (unless we’re talking about a timeless wavefunction of the universe; I’ll come to that later). In general, I infer from your discussion that configurations are real—they are the worlds or branches—and the wavefunction determines a measure on configuration space. The measure can’t be identified with the wavefunction—the phase information is lost—so, if we are to treat the wavefunction as also real, we seem to have a dualism remotely similar to Bohmian mechanics: The wavefunction is real, and evolves over time, and there is also a population of configurations—the worlds—whose relative multiplicity changes according to the changing measure.
I want to note one of the peculiarities of this perspective. Since configuration space does not change, and since the different configurations are the worlds, then at every moment in the history of the universe, every possible configuration exists (presumably except for those isolated configurations which have an individual measure of exactly zero). What distinguishes one moment from the next is that there is “more” or “less” of each individual configuration. If we take the use of the mathematical continuum seriously, then it seems that there must be an uncountable number of copies of each configuration at each moment, and the measure is telling us the relative sizes of these uncountable sets.
This scenario might be simplified a little if you had a timeless wavefunction of the universe, if the basic configurations were combinatorial (discrete degrees of freedom rather than continuous), and if amplitudes / probabilities were rational numbers. This would allow your multiverse to consist of a countable number of configurations, each duplicated only finitely often, and without the peculiar phenomenon of all configurations having duplicates at every moment in the history of the universe. This would then land us in a version of Julian Barbour’s Platonia.
There are three features of this analysis that I would emphasize. First, relativity in any space-time sense has disappeared. The worlds are strictly spatial configurations. Second, configurations must be duplicated (whether only finitely often, or uncountably infinitely often), in order for the Born frequencies to be realized. Otherwise, it’s like the parable of the car dealer. Just associating a number with a configuration does not by itself make the events in that configuration occur more frequently. Third, the configurations are distinct from the wavefunction. The wavefunction contains information not contained in the measure, namely the phase relations. So we have a Bohm-like dualism, except, instead of histories guided by a pilot wave, we have disconnected universe-moments whose multiplicities are determined by the Born rule.
There are various ways you could adjust the details of this ontology—which, I emphasize, is an attempt to spell out the ontological commitments implied by what you said. For example, your argument hinged on typicality—being a typical mind-configuration. So maybe, instead of saying that configurations are duplicated, you could simply say that configurations only get to exist if their amplitude is above some nonzero threshold, and then you could argue that Born frequencies are realized inside the individual universe-configuration. This would be a version of Everett’s original idea, I believe. I thought it had largely been abandoned by modern Many Worlds advocates—for example, Robin Hanson dismisses it on the way to introducing his idea of mangled worlds—but I would need to refresh my knowledge of the counterarguments to personally dismiss it.
In any case, you may wish to comment on (1) my assertion that this approach requires dualism of wavefunction and worlds (because the wavefunction can’t be identified with the ensemble of worlds, on account of containing phase information), (2) my assertion that this approach requires world duplication (in order to get the frequencies right), and (3) the way that configuration has supplied a definitely preferred basis in my account. Most Many Worlds people like to avoid a preferred basis, but I don’t see how you can identify the world we actually experience with a wavefunction-part unless you explicitly say that yes, that wavefunction-part has a special status compared to other possible local basis-decompositions. Alternatively, you could assert that several or even all possible basis-decompositions define a “valid” set of worlds, but validity here has to mean existing—so along with the ensemble of spatial configurations, distinct from the wavefunction, you will end up with other ensembles of worlds, corresponding to the basis wavefunctions in other choices of basis, which will also have to be duplicated, etc., in order to produce the right frequencies.
To sum up, my position is that if you do try to deliver on the claims regarding how Many Worlds works, you have to throw out relativity as anything more than a phenomenological fact; you have to have duplication of worlds in order to get the Born frequencies; and the resulting set of worlds can’t be identified with the wavefunction itself, so you end up with a Bohm-like dualism.
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.
It would be nice if the universe were finite, but you can’t demand that a priori; it’s enough that the infinite mathematical object obeys simple rules.
I’m saying that if we lived in another universe, and someone came along and described to us the wavefunction for the Schrodinger equation, and asked how we should regard the size of some part of the configuration space compared to some other part, the L^2 norm is the blindingly obvious mathematical answer because of the properties of the wavefunction. And so if we (outside the system) were looking for a “typical” instance of a configuration corresponding to a mind, we would weight the configurations by the L^2 norm of the wavefunction.
Because (as it turns out) the wavefunction has a distinguished exceptionally-low-entropy state corresponding to the Big Bang, the configurations where the wavefunction is relatively large encode in various ways the details of (practically unique) intermediate stages between the Big Bang state and the one under consideration: that is, they encode unique histories (1). So a “typical” instance of a configuration containing a mind turns out to be one that places it within a context of a unique and lawful history satisfying the Born probabilities, because the L^2 norm of the wavefunction over the set where they hold to within epsilon is much, much larger than the L^2 norm of the rest. So to the extent that I’m a typical instance of mind-configurations similar to me, I should expect to remember and see evidence of histories satisfying the Born probabilities.
...seriously, I don’t see why people get worked up over this. OK, Eliezer has his infinite-set atheism, and you have your insistence on a naive theory of qualia, but what about everyone else?
(1) This is not a conjecture, it is not controversial, it is something you can prove mathematically about the Schrodinger equation in various contexts.
Does “part of the configuration space” refer to a single state vector, or a whole region that a state vector might belong to? My impression is that measuring the latter sort of thing is problematic from a rigorous mathematical standpoint. Is this correct, and does it have consequences for your discussion?
I say the former; people scared of continuous densities might prefer the latter, at which point they have the traditional sorites paradox of how large an epsilon-neighborhood to draw; but in practical terms, this isn’t so bad because (if we start with low entropy) decoherence rapidly separates the wavefunction into thin wisps with almost-zero values taken between them.
Okay, I have tried to understand what sort of ontology could answer to your description. A key consideration: you say we should judge “the size of some part of the configuration space compared to some other part” according to “the L^2 norm of the wavefunction”. You also talk about “mind-configurations similar to me”.
A wavefunction may evolve over time, but configuration space does not. Configuration space is a static arena, and the amplitudes associated with configurations change (unless we’re talking about a timeless wavefunction of the universe; I’ll come to that later). In general, I infer from your discussion that configurations are real—they are the worlds or branches—and the wavefunction determines a measure on configuration space. The measure can’t be identified with the wavefunction—the phase information is lost—so, if we are to treat the wavefunction as also real, we seem to have a dualism remotely similar to Bohmian mechanics: The wavefunction is real, and evolves over time, and there is also a population of configurations—the worlds—whose relative multiplicity changes according to the changing measure.
I want to note one of the peculiarities of this perspective. Since configuration space does not change, and since the different configurations are the worlds, then at every moment in the history of the universe, every possible configuration exists (presumably except for those isolated configurations which have an individual measure of exactly zero). What distinguishes one moment from the next is that there is “more” or “less” of each individual configuration. If we take the use of the mathematical continuum seriously, then it seems that there must be an uncountable number of copies of each configuration at each moment, and the measure is telling us the relative sizes of these uncountable sets.
This scenario might be simplified a little if you had a timeless wavefunction of the universe, if the basic configurations were combinatorial (discrete degrees of freedom rather than continuous), and if amplitudes / probabilities were rational numbers. This would allow your multiverse to consist of a countable number of configurations, each duplicated only finitely often, and without the peculiar phenomenon of all configurations having duplicates at every moment in the history of the universe. This would then land us in a version of Julian Barbour’s Platonia.
There are three features of this analysis that I would emphasize. First, relativity in any space-time sense has disappeared. The worlds are strictly spatial configurations. Second, configurations must be duplicated (whether only finitely often, or uncountably infinitely often), in order for the Born frequencies to be realized. Otherwise, it’s like the parable of the car dealer. Just associating a number with a configuration does not by itself make the events in that configuration occur more frequently. Third, the configurations are distinct from the wavefunction. The wavefunction contains information not contained in the measure, namely the phase relations. So we have a Bohm-like dualism, except, instead of histories guided by a pilot wave, we have disconnected universe-moments whose multiplicities are determined by the Born rule.
There are various ways you could adjust the details of this ontology—which, I emphasize, is an attempt to spell out the ontological commitments implied by what you said. For example, your argument hinged on typicality—being a typical mind-configuration. So maybe, instead of saying that configurations are duplicated, you could simply say that configurations only get to exist if their amplitude is above some nonzero threshold, and then you could argue that Born frequencies are realized inside the individual universe-configuration. This would be a version of Everett’s original idea, I believe. I thought it had largely been abandoned by modern Many Worlds advocates—for example, Robin Hanson dismisses it on the way to introducing his idea of mangled worlds—but I would need to refresh my knowledge of the counterarguments to personally dismiss it.
In any case, you may wish to comment on (1) my assertion that this approach requires dualism of wavefunction and worlds (because the wavefunction can’t be identified with the ensemble of worlds, on account of containing phase information), (2) my assertion that this approach requires world duplication (in order to get the frequencies right), and (3) the way that configuration has supplied a definitely preferred basis in my account. Most Many Worlds people like to avoid a preferred basis, but I don’t see how you can identify the world we actually experience with a wavefunction-part unless you explicitly say that yes, that wavefunction-part has a special status compared to other possible local basis-decompositions. Alternatively, you could assert that several or even all possible basis-decompositions define a “valid” set of worlds, but validity here has to mean existing—so along with the ensemble of spatial configurations, distinct from the wavefunction, you will end up with other ensembles of worlds, corresponding to the basis wavefunctions in other choices of basis, which will also have to be duplicated, etc., in order to produce the right frequencies.
To sum up, my position is that if you do try to deliver on the claims regarding how Many Worlds works, you have to throw out relativity as anything more than a phenomenological fact; you have to have duplication of worlds in order to get the Born frequencies; and the resulting set of worlds can’t be identified with the wavefunction itself, so you end up with a Bohm-like dualism.
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.