Which basis do you use in obtaining multiple worlds from a single wavefunction?
evhub replied
Any diagonal basis—the whole point of decoherence is that the wavefunction evolves into a diagonalizable form over time.
Then my next question would be, exactly when in this evolution does one world become many?
I also asked
How do you deal with relativity?
evhub replied
Just use the Relativistic Schrodinger equation.
In relativity, wavefunctions will only be defined with respect to a particular reference frame. You have to say which spacelike surfaces you are treating as surfaces of simultaneity; only then are you equipped to talk about e.g. EPR states. (The technical exception to this is asymptotic states at spacelike infinity.)
In relativistic quantum field theory, the wave equations have new meanings, they are now operator equations. You’re no longer talking about waves with definite values at space-time points (x,t), and a differential equation describing how those values vary. Instead you are talking about “observables” at space-time points (x,t), and operators which formally represent those observables, and the wave equation describes the algebraic relations among those operators; something which empirically translates into relationships among the observables, such as the uncertainty principle.
I don’t know how clear that explanation is, but the significant thing is that the field operators are consistent with relativity because they are anchored at individual space-time points, whereas wavefunctions are defined only with respect to a particular reference frame. The point being that this is a problem for an ontological interpretation which starts by saying that wavefunctions are what’s real.
exactly when in this evolution does one world become many?
See also discussion here; I’ll copy it for convenience:
Sometimes you find that the wavefunction |ψ⟩ is the sum of a discrete number of components |ψ⟩=|ψ1⟩+|ψ2⟩+⋯ , with the property that for any relevant observable A, ⟨ψi|A|ψj⟩≈0 for i≠j. (Here, “≈0” also includes things like “has a value that varies quasi-randomly and super-rapidly as a function of time and space, such that it averages to 0 for all intents and purposes”, and “relevant observable” likewise means “observable that might come up in practice, as opposed to artificial observables with quasi-random super-rapidly-varying spatial and time-dependence, etc.”).
When that situation comes up, if it comes up, you can start ignoring cross-terms, and calculate the time-evolution and other properties of the different |ψi⟩ as if they had nothing to do with each other, and that’s where you can use the term “branch” to talk about them.
There isn’t a sharp line for when the cross-terms are negligible enough to properly use the word “branch”, but there are exponential effects such that it’s very clearly appropriate in the real-world cases of interest.
Well, it seems like the most important part of your answer comes in a subsequent comment
“how many worlds are there” is not a question with a well-defined answer in Everett’s theory
As far as I am concerned, that renders the theory unviable. We-here (as opposed to our copies in slightly divergent branches) inhabit a particular world. We definitely exist, therefore the object in the theory corresponding to our existence must also definitely exist; therefore if its existence is only a matter of degree or definition, then the theory is wrong.
But at least you have clarified the kind of MWI that you are talking about—worlds are defined only vaguely or exactly, and cannot be counted. This is not the case in all forms of MWI, e.g. see “many interacting worlds”.
Do you have anything to say about the criticism from relativity? That in relativistic quantum field theory, wavefunctions only exist in the context of a particular frame, and so can’t be ontologically fundamental?
in relativistic quantum field theory, wavefunctions only exist in the context of a particular frame, and so can’t be ontologically fundamental?
I guess I don’t really understand what you’re getting at. For example, displacement and 4-velocity and electromagnetic 4-potential are all 4-vectors, such that their components are different in different frames. Whereas, say, the rest mass or electric charge of a particle is a Lorentz scalar, the same in every frame. Is your position that Lorentz scalars have a special status that Lorentz 4-vectors, 4-tensors, etc. don’t have, that allows them to be “ontologically fundamental”? If so, why? I haven’t ever thought of Lorentz scalars having a special status, and I don’t find that notion intuitive. Or sorry if I’m misunderstanding.
A wavefunction is spatially extended. Your description of MWI involves tracking how the properties of a wavefunction change over time. In relativity, that’s going to require choosing a reference frame, a particular division of space-time into space and time.
In a Copenhagen approach to, say, particle physics, that doesn’t matter, because everything that is frame-dependent vanishes by the end of the calculation (as does everything that is gauge-dependent). But I don’t see how you can reify wavefunctions without also having a preferred reference frame.
In quantum field theory the wave function is an operator at each point in spacetime, and it works out that everything is consistent with experiments across reference frame changes and nothing travels faster than the speed of light, etc. That’s all experimentally established. Can you say again what’s the problem?
everything that is frame-dependent vanishes by the end of the calculation
I mean, velocity is frame-dependent, right? You can measure velocity, it doesn’t vanish at the end of the calculation… It’s different in different reference frames, of course, and that’s fine, because its reference-frame-dependence is consistent with everything else and with experiments. So what do you mean? Sorry if I’m just not understanding you here, you can try again...
Hmm, I guess you could make it clearer by focusing on gauge dependence. “The wave function is gauge dependent, so how can you say it’s “real”?” Is that similar to your argument? If so, I guess I’m sympathetic to that argument, and I would say that the “real” thing is the equivalence class of wave functions up to gauge transformations, or something like that...
The point seems so simple to me, I am having trouble expressing it… A wavefunction is the instantaneous state of a quantum system. It is extended spatially. In relativistic space-time, to talk about the instantaneous state of an extended object, you have to define simultaneity. This means choosing a particular decomposition of space-time into spacelike hypersurfaces that are treated as surfaces of simultaneity. In a relativistic universe, you cannot talk about finite time evolution of spatially extended wavefunctions without first breaking space-time into space and time.
In particle physics a la Copenhagen, there is no ontological commitment to wavefunctions as things that exist. They are just part of a calculation. But we are told that in MWI, the universal wavefunction is real and it is a superposition of worlds. As I have just argued, you can’t do what you want to do—study how this wavefunction evolves over time—without first breaking space-time into space and time, so that you have the hypersurfaces of simultaneity on which the wavefunction is defined. So it seems that belief in the wavefunction as something real, requires belief in an ontologically preferred frame, with respect to which that wavefunction’s time evolution is defined.
Hmm. Again, “the universal wavefunction is real” is part of the theory but “it is a superposition of worlds” is not, the latter is just a way to talk loosely about particular situations that sometimes come up. I don’t think that people in different inertial reference frames have to agree about how many worlds there are, indeed I don’t even think people in the same inertial reference frame have to agree about how many worlds there are. It’s not part of the theory. The only other thing that is part of the theory is some kind of indexical axiom, like I think one version is “if the complex amplitude for me having a certain brain state approaches zero, then the probability that I will find myself experiencing having that brain state also approaches zero”, or things like that, I think.
In my experience when physicists challenge a proposal as being inconsistent with relativity, they try to come up with an example where two people in different reference frames would make different predictions about the same concrete experimental (or thought-experimental) result. Can you think of anything like that? It seems like you have a different demand, which is “people in different reference frames cannot disagree about the value of ontologically primitive things” even if the disagreement doesn’t shake out as a concrete prediction incompatibility. If so (and sorry if I’m misunderstanding), I guess I just don’t see why that’s important. Why can’t something be both ontologically primitive and reference frame dependent? Like velocity, to take an everyday example. I don’t know if it’s ontologically primitive, (partly because I’m not sure what ontologically primitive means), but anyway, I don’t see why reference frame dependence should count against it.
I don’t think that people in different inertial reference frames have to agree about how many worlds there are, indeed I don’t even think people in the same inertial reference frame have to agree about how many worlds there are.
At this point I have nothing to say, because there’s no coherent concept of ‘world’ left to debate.
I think one version is “if the complex amplitude for me having a certain brain state approaches zero, then the probability that I will find myself experiencing having that brain state also approaches zero”
This could become a version of ‘many-minds interpretation’. But now you need to make ‘mind’ a rigorous concept. There has to be something exact in the ontology that corresponds to the specificity of what we see! - whether it’s a whole ‘world’, or just an ‘observer experience’. If everything other than the universal wavefunction is fuzzy and vague and a matter of convention, you no longer have a theory corresponding to observed reality.
Why can’t something be both ontologically primitive and reference frame dependent? Like velocity, to take an everyday example
The 4-velocity (considered as an invariant geometric object, rather than in terms of covariant components) is the fundamental entity.
there’s no coherent concept of ‘world’ left to debate.
Good! Maybe we’re on the same page there. “World” is not part of the theory and is not a well-defined concept, in my opinion.
now you need to make ‘mind’ a rigorous concept
Hmm, I guess I would propose something like “the complete history of exactly which neurons in a brain fire at which times, to 1μs accuracy, is a mind, for present purposes”. Then I would argue that different “minds” don’t exhibit measurable quantum interference with each other, or we can say “different minds are in different worlds / branches” as a casual shorthand for that, if we want. And there is a well-defined (albeit complicated) way to project the universal wavefunction into the subspace of one “mind”, in order to calculate its quantum amplitude, and then you can apply the Born rule for the indexical calculation of how likely you are to find yourself in that mind. Something like that, I guess. I haven’t thought it through very carefully, I just think something vaguely like that could work, with a bit more effort to iron out the details. I’m not sure what’s in the literature, maybe there’s a better approach...
There isn’t a sharp line for when the cross-terms are negligible enough to properly use the word “branch”, but there are exponential effects such that it’s very clearly appropriate in the real-world cases of interest.
I agree that it isn’t a problem for practical purposes but if we are talking about a fundamental theory about reality shouldn’t questions like “How many worlds are there?” have unambiguous answers?
No … “how many worlds are there” is not a question with a well-defined answer in Everett’s theory. It’s like “How many grains of sand make up a heap?” … just a meaningless question. The notion that there is a specific, well-defined number of worlds is sometimes implied by the language used in simplifications / popularizations of the theory, but it’s not part of the actual theory, and really it can’t possibly be, I don’t think.
I agree that the question “how many worlds are there” doesn’t have a well-defined answer in the MWI. I disagree that it is a meaningless question.
From the bird’s-eye view, the ontology of the MWI seems pretty clear: the universal wavefunction is happily evolving (or is it?). From the frog’s-eye view, the ontology is less clear. The usual account of an experiment goes like this:
The system and the observer come together and interact
This leads to entanglement and decoherence in a certain basis
In the final state, we have a branch for each measurement outcome. i.e. there are now multiple versions of the observer
This seems to suggest a nice ontology: first there’s one observer, then the universe splits and afterwards we have a certain number of versions of the observer. I think questions like “When does the split happen?” and “How many versions?” are important because they would have well-defined answers if the nice ontology was tenable.
Unfortunately it isn’t, so the ontology is muddled. We have to use terms like “approximately zero” and “for all practical purposes” which takes us most of the way back to give the person who determines which approximations are appropriate and what is practical—aka the observer—an important part in the whole affair.
The ontology doesn’t feel muddled to me, although it does feel… not very quantum? Like a thing that seems to be happening with collapse postulates is that it takes seriously the “everything should be quantized” approach, and so insists on ending up with one world (or discrete numbers of worlds). MWI instead seems to think that wavefunctions, while having quantized bases, are themselves complex-valued objects, and so there doesn’t need to be a discrete and transitive sense of whether two things are ‘in the same branch’, and instead it seems fine to have a continuous level of coherence between things (which, at the macro-scale, ends up looking like being in a ‘definite branch’).
[I don’t think I’ve ever seen collapse described as “motivated by everything being quantum” instead of “motivated by thinking that only what you can see exists”, and so quite plausibly this will fall apart or I’ll end up thinking it’s silly or it’s already been dismissed for whatever reason. But somehow this does seem like a lens where collapse is doing the right sort of extrapolating principles where MWI is just blindly doing what made sense elsewhere. On net, I still think wavefunctions are continuous, and so it makes sense for worlds to be continuous too.]
Like, I think it makes more sense to think of MWI as “first many, then even more many,” at which point questions of “when does the split happen?” feel less interesting, because the original state is no longer as special. When I think of the MWI story of radioactive decay, for example, at every timestep you get two worlds, one where the particle decayed at that moment and one where it held together, and as far as we can tell if time is quantized, it must have very short steps, and so this is very quickly a very large number of worlds. If time isn’t quantized, then this has to be spread across continuous space, and so thinking of there being a countable number of worlds is right out.
I think it makes more sense to think of MWI as “first many, then even more many,” at which point questions of “when does the split happen?” feel less interesting, because the original state is no longer as special. [...] If time isn’t quantized, then this has to be spread across continuous space, and so thinking of there being a countable number of worlds is right out.
What I called the “nice ontology” isn’t so much about the number of worlds or even countability but about whether the worlds are well-defined. The MWI gives up a unique reality for things. The desirable feature of the “nice ontology” is that the theory tells us what a “version” of a thing is. As we all seem to agree, the MWI doesn’t do this.
If it doesn’t do this, what’s the justification for speaking of different versions in the first place? I think pure MWI makes only sense as “first one, then one”. After all, there’s just the universal wave function evolving and pure MWI doesn’t give us any reason to take a part of this wavefunction and say there are many versions of this.
Any diagonal basis—the whole point of decoherence is that the wavefunction evolves into a diagonalizable form over time.
Just use the Relativistic Schrodinger equation.
I asked
evhub replied
Then my next question would be, exactly when in this evolution does one world become many?
I also asked
evhub replied
In relativity, wavefunctions will only be defined with respect to a particular reference frame. You have to say which spacelike surfaces you are treating as surfaces of simultaneity; only then are you equipped to talk about e.g. EPR states. (The technical exception to this is asymptotic states at spacelike infinity.)
In relativistic quantum field theory, the wave equations have new meanings, they are now operator equations. You’re no longer talking about waves with definite values at space-time points (x,t), and a differential equation describing how those values vary. Instead you are talking about “observables” at space-time points (x,t), and operators which formally represent those observables, and the wave equation describes the algebraic relations among those operators; something which empirically translates into relationships among the observables, such as the uncertainty principle.
I don’t know how clear that explanation is, but the significant thing is that the field operators are consistent with relativity because they are anchored at individual space-time points, whereas wavefunctions are defined only with respect to a particular reference frame. The point being that this is a problem for an ontological interpretation which starts by saying that wavefunctions are what’s real.
See also discussion here; I’ll copy it for convenience:
Sometimes you find that the wavefunction |ψ⟩ is the sum of a discrete number of components |ψ⟩=|ψ1⟩+|ψ2⟩+⋯ , with the property that for any relevant observable A, ⟨ψi|A|ψj⟩≈0 for i≠j. (Here, “≈0” also includes things like “has a value that varies quasi-randomly and super-rapidly as a function of time and space, such that it averages to 0 for all intents and purposes”, and “relevant observable” likewise means “observable that might come up in practice, as opposed to artificial observables with quasi-random super-rapidly-varying spatial and time-dependence, etc.”).
When that situation comes up, if it comes up, you can start ignoring cross-terms, and calculate the time-evolution and other properties of the different |ψi⟩ as if they had nothing to do with each other, and that’s where you can use the term “branch” to talk about them.
There isn’t a sharp line for when the cross-terms are negligible enough to properly use the word “branch”, but there are exponential effects such that it’s very clearly appropriate in the real-world cases of interest.
Well, it seems like the most important part of your answer comes in a subsequent comment
As far as I am concerned, that renders the theory unviable. We-here (as opposed to our copies in slightly divergent branches) inhabit a particular world. We definitely exist, therefore the object in the theory corresponding to our existence must also definitely exist; therefore if its existence is only a matter of degree or definition, then the theory is wrong.
But at least you have clarified the kind of MWI that you are talking about—worlds are defined only vaguely or exactly, and cannot be counted. This is not the case in all forms of MWI, e.g. see “many interacting worlds”.
Do you have anything to say about the criticism from relativity? That in relativistic quantum field theory, wavefunctions only exist in the context of a particular frame, and so can’t be ontologically fundamental?
I guess I don’t really understand what you’re getting at. For example, displacement and 4-velocity and electromagnetic 4-potential are all 4-vectors, such that their components are different in different frames. Whereas, say, the rest mass or electric charge of a particle is a Lorentz scalar, the same in every frame. Is your position that Lorentz scalars have a special status that Lorentz 4-vectors, 4-tensors, etc. don’t have, that allows them to be “ontologically fundamental”? If so, why? I haven’t ever thought of Lorentz scalars having a special status, and I don’t find that notion intuitive. Or sorry if I’m misunderstanding.
A wavefunction is spatially extended. Your description of MWI involves tracking how the properties of a wavefunction change over time. In relativity, that’s going to require choosing a reference frame, a particular division of space-time into space and time.
In a Copenhagen approach to, say, particle physics, that doesn’t matter, because everything that is frame-dependent vanishes by the end of the calculation (as does everything that is gauge-dependent). But I don’t see how you can reify wavefunctions without also having a preferred reference frame.
In quantum field theory the wave function is an operator at each point in spacetime, and it works out that everything is consistent with experiments across reference frame changes and nothing travels faster than the speed of light, etc. That’s all experimentally established. Can you say again what’s the problem?
I mean, velocity is frame-dependent, right? You can measure velocity, it doesn’t vanish at the end of the calculation… It’s different in different reference frames, of course, and that’s fine, because its reference-frame-dependence is consistent with everything else and with experiments. So what do you mean? Sorry if I’m just not understanding you here, you can try again...
Hmm, I guess you could make it clearer by focusing on gauge dependence. “The wave function is gauge dependent, so how can you say it’s “real”?” Is that similar to your argument? If so, I guess I’m sympathetic to that argument, and I would say that the “real” thing is the equivalence class of wave functions up to gauge transformations, or something like that...
The point seems so simple to me, I am having trouble expressing it… A wavefunction is the instantaneous state of a quantum system. It is extended spatially. In relativistic space-time, to talk about the instantaneous state of an extended object, you have to define simultaneity. This means choosing a particular decomposition of space-time into spacelike hypersurfaces that are treated as surfaces of simultaneity. In a relativistic universe, you cannot talk about finite time evolution of spatially extended wavefunctions without first breaking space-time into space and time.
In particle physics a la Copenhagen, there is no ontological commitment to wavefunctions as things that exist. They are just part of a calculation. But we are told that in MWI, the universal wavefunction is real and it is a superposition of worlds. As I have just argued, you can’t do what you want to do—study how this wavefunction evolves over time—without first breaking space-time into space and time, so that you have the hypersurfaces of simultaneity on which the wavefunction is defined. So it seems that belief in the wavefunction as something real, requires belief in an ontologically preferred frame, with respect to which that wavefunction’s time evolution is defined.
Is that any clearer?
Hmm. Again, “the universal wavefunction is real” is part of the theory but “it is a superposition of worlds” is not, the latter is just a way to talk loosely about particular situations that sometimes come up. I don’t think that people in different inertial reference frames have to agree about how many worlds there are, indeed I don’t even think people in the same inertial reference frame have to agree about how many worlds there are. It’s not part of the theory. The only other thing that is part of the theory is some kind of indexical axiom, like I think one version is “if the complex amplitude for me having a certain brain state approaches zero, then the probability that I will find myself experiencing having that brain state also approaches zero”, or things like that, I think.
In my experience when physicists challenge a proposal as being inconsistent with relativity, they try to come up with an example where two people in different reference frames would make different predictions about the same concrete experimental (or thought-experimental) result. Can you think of anything like that? It seems like you have a different demand, which is “people in different reference frames cannot disagree about the value of ontologically primitive things” even if the disagreement doesn’t shake out as a concrete prediction incompatibility. If so (and sorry if I’m misunderstanding), I guess I just don’t see why that’s important. Why can’t something be both ontologically primitive and reference frame dependent? Like velocity, to take an everyday example. I don’t know if it’s ontologically primitive, (partly because I’m not sure what ontologically primitive means), but anyway, I don’t see why reference frame dependence should count against it.
At this point I have nothing to say, because there’s no coherent concept of ‘world’ left to debate.
This could become a version of ‘many-minds interpretation’. But now you need to make ‘mind’ a rigorous concept. There has to be something exact in the ontology that corresponds to the specificity of what we see! - whether it’s a whole ‘world’, or just an ‘observer experience’. If everything other than the universal wavefunction is fuzzy and vague and a matter of convention, you no longer have a theory corresponding to observed reality.
The 4-velocity (considered as an invariant geometric object, rather than in terms of covariant components) is the fundamental entity.
Good! Maybe we’re on the same page there. “World” is not part of the theory and is not a well-defined concept, in my opinion.
Hmm, I guess I would propose something like “the complete history of exactly which neurons in a brain fire at which times, to 1μs accuracy, is a mind, for present purposes”. Then I would argue that different “minds” don’t exhibit measurable quantum interference with each other, or we can say “different minds are in different worlds / branches” as a casual shorthand for that, if we want. And there is a well-defined (albeit complicated) way to project the universal wavefunction into the subspace of one “mind”, in order to calculate its quantum amplitude, and then you can apply the Born rule for the indexical calculation of how likely you are to find yourself in that mind. Something like that, I guess. I haven’t thought it through very carefully, I just think something vaguely like that could work, with a bit more effort to iron out the details. I’m not sure what’s in the literature, maybe there’s a better approach...
I agree that it isn’t a problem for practical purposes but if we are talking about a fundamental theory about reality shouldn’t questions like “How many worlds are there?” have unambiguous answers?
No … “how many worlds are there” is not a question with a well-defined answer in Everett’s theory. It’s like “How many grains of sand make up a heap?” … just a meaningless question. The notion that there is a specific, well-defined number of worlds is sometimes implied by the language used in simplifications / popularizations of the theory, but it’s not part of the actual theory, and really it can’t possibly be, I don’t think.
I agree that the question “how many worlds are there” doesn’t have a well-defined answer in the MWI. I disagree that it is a meaningless question.
From the bird’s-eye view, the ontology of the MWI seems pretty clear: the universal wavefunction is happily evolving (or is it?). From the frog’s-eye view, the ontology is less clear. The usual account of an experiment goes like this:
The system and the observer come together and interact
This leads to entanglement and decoherence in a certain basis
In the final state, we have a branch for each measurement outcome. i.e. there are now multiple versions of the observer
This seems to suggest a nice ontology: first there’s one observer, then the universe splits and afterwards we have a certain number of versions of the observer. I think questions like “When does the split happen?” and “How many versions?” are important because they would have well-defined answers if the nice ontology was tenable.
Unfortunately it isn’t, so the ontology is muddled. We have to use terms like “approximately zero” and “for all practical purposes” which takes us most of the way back to give the person who determines which approximations are appropriate and what is practical—aka the observer—an important part in the whole affair.
The ontology doesn’t feel muddled to me, although it does feel… not very quantum? Like a thing that seems to be happening with collapse postulates is that it takes seriously the “everything should be quantized” approach, and so insists on ending up with one world (or discrete numbers of worlds). MWI instead seems to think that wavefunctions, while having quantized bases, are themselves complex-valued objects, and so there doesn’t need to be a discrete and transitive sense of whether two things are ‘in the same branch’, and instead it seems fine to have a continuous level of coherence between things (which, at the macro-scale, ends up looking like being in a ‘definite branch’).
[I don’t think I’ve ever seen collapse described as “motivated by everything being quantum” instead of “motivated by thinking that only what you can see exists”, and so quite plausibly this will fall apart or I’ll end up thinking it’s silly or it’s already been dismissed for whatever reason. But somehow this does seem like a lens where collapse is doing the right sort of extrapolating principles where MWI is just blindly doing what made sense elsewhere. On net, I still think wavefunctions are continuous, and so it makes sense for worlds to be continuous too.]
Like, I think it makes more sense to think of MWI as “first many, then even more many,” at which point questions of “when does the split happen?” feel less interesting, because the original state is no longer as special. When I think of the MWI story of radioactive decay, for example, at every timestep you get two worlds, one where the particle decayed at that moment and one where it held together, and as far as we can tell if time is quantized, it must have very short steps, and so this is very quickly a very large number of worlds. If time isn’t quantized, then this has to be spread across continuous space, and so thinking of there being a countable number of worlds is right out.
What I called the “nice ontology” isn’t so much about the number of worlds or even countability but about whether the worlds are well-defined. The MWI gives up a unique reality for things. The desirable feature of the “nice ontology” is that the theory tells us what a “version” of a thing is. As we all seem to agree, the MWI doesn’t do this.
If it doesn’t do this, what’s the justification for speaking of different versions in the first place? I think pure MWI makes only sense as “first one, then one”. After all, there’s just the universal wave function evolving and pure MWI doesn’t give us any reason to take a part of this wavefunction and say there are many versions of this.