Does the MWI require "creation" of multiple worlds?

[Derek P]

2. MWI may be technically sound (eg. Aharonov and Rohrlich), but some MWI proponents (Wallace and Carroll) remain unsure if there is a correct justification of how probabilities enter the theory.

[
Last time I wrestled with this issue it seemed to me to come down to explaining

1. why a naive observer might mistake frequencies in a history for actual probabilities
2. why the statistics follow the Born rule
3. what is meant by a "typical" history

They all seemed rather trivial to me but someone, I think it was Ruth Kastner (apologies @rkastner if I'm wrong), was adamant that #3 cannot be resolved without invoking actual probability. I see the point. Even if the frequency in a real experiment converges to the ensemble expectation value, that only occurs in "most" cases. We could still be in a rogue world. But as that would apply to classical probability, indeed to all of science (because induction and probability are inseparable), it would seem to be a philosophical matter. We can eliminate physical probability but we need to be able to justify the assumption that the world I am in is probably fairly typical. The trouble is "the world I am in" does not have a probability measure. Or does it? A symmetrical finite-way split should result in equal probabilities, so the problem becomes easier to think about. And then the map from amplitude to frequentist probability necessarily follows Born's Rule because of the way orthogonal vectors add. But is the assumpion of equal weighting justifiable? My brain hurts.

 

[stevendaryl]

1. Bohmian mechanics and other hidden variable theories lack explicit constructions that will reproduce the standard model.
2. MWI may be technically sound (eg. Aharonov and Rohrlich), but some MWI proponents (Wallace and Carroll) remain unsure if there is a correct justification of how probabilities enter the theory.
3. Decoherent histories is likely to be technically correct, but it is not a realist solution to the measurement problem (traditionally, the measurement problem only admits realist solutions; if necessary - just add the qualifier "realist" to my remarks at the right places) because of the lack of a single fine-grained history (eg. Gell-Mann and Hartle). Here may I borrow bhobba's characterization: defining your difficulties away :)

My feeling is that the measurement problem is a red herring. For something to count as a measurement, a microscopic quantity must be amplified (typically using a metastable system) so that it has a macroscopic effect. So for empirical adequacy, a theory only needs to predict probabilities for macroscopic quantities. It doesn't need to have any special rules for measurement. However, for intellectual coherence, it's unsatisfying that macroscopic quantities should be treated as special. Decoherence explains why superpositions of states with different macroscopic properties will be problematic, but I don't think that actually solves the question of what's special about macroscopic quantities. It says: If you have to choose just one basis for describing things, it doesn't make sense to choose one other than one where macroscopic quantities have definite values. But it doesn't say why a basis needs to be chosen at all.

 

[Derek P]

Please point to a reference where the above 7 points are made precise. They are directly relevant for the question posed in #1.

That's just prevarication. The thread has moved on from simply finding a reference for the OP. (I'd have given it if I had one.)

 

[PeterDonis]

Shut up and calculate is what is meant by Copenhagen.

I don't think there is an agreed meaning for "Copenhagen interpretation" to begin with, but I certainly don't think most of the people using that term mean it this way.

Also, if it is meant this way, then "Copenhagen observer" would just mean "shut up and calculate observer", which obviously any interpretation of QM contains. So I don't understand your claim that MWI doesn't contain "Copenhagen observers".

 

[PeterDonis]

That's verging on being insulting. Please stick to analysing the physical theory before implying that its
proponents are delusional !

Please do not overreact. "In the head of the believer" is not implying any kind of delusion. It's just referring to a person's beliefs as opposed to what those beliefs are about. Even if "what those beliefs are about" does not exist, unless you are going to claim that normal humans never have false beliefs, it's hard to see such a statement as an insult; it's just a description of the human condition.

 

[PeterDonis]

What precisely constitutes a world in MWI?

The obvious answer to this is that a "world" is a term in the superposition I wrote down in the OP; i.e., each of the terms $|1>|U>$ and $|2>|D>$ is a "world". So "worlds" are picked out by the interaction between the measured system and the measuring apparatus and how the two become entangled.

Are these worlds just ''points of view'' (independent of reality), or are they dynamical objects in time?

With the answer I gave above, obviously it's the latter.

What precisely constitutes a split of one of these worlds? What triggers a split?

Again, with the answer I gave above, this is obvious: the entanglement interaction between the measured system and the measuring apparatus.

When precisely do these splits happen? Do they happen at all? Is it observer-dependent?

When the entanglement interactions happen. Of course they do. Of course not.

For an observer as a quantum object in the MWI for the whole universe, how is its perceived world characterized among all possible worlds?

If we want to talk about observers, then the kets $|U>$ and $|D>$ would include the brain states of the observers. So each "world" has its own "copy" of the observer, who observes the appropriate state of the measured system. (We could, if we wanted, split out these kets to include kets for the apparatus, the environment, the observer's eyes, the observer's brain, etc., and separately model the interactions that entangle all of these things, but that wouldn't change the substance of the description in the OP. It would just complicate it.)

Do different observers perceive different worlds? If yes, why?

Different "copies" of an observer perceive different worlds, because of the way the entanglement interactions work. See above.

If we want to include multiple observers, then all of their states would be included in the kets $|U>$ and $|D>$ (or we could split out all those entanglement interactions, as above, which would not change the substance, it would just complicate it), so corresponding copies of all observers in the same world would have consistent observations (all of them would observe the "up" or "down" result for the same measurement, in the case described in the OP).

What object inside a quantum universe described by MWI qualifies as an observer? What as a measurement? What constitutes a measurement result?

Like any QM treatment of macroscopic objects, including observers, nobody actually tries to describe them in detail; everybody just writes down kets like $|U>$ and $|D>$ and says those kets represent states of the observer (or observers) that correspond to particular measurement results being observed.

A measurement is an entanglement interaction, as above. A measurement result is one term in an entangled output state of such an interaction. (Note that in the OP I have assumed that the interaction is localized in spacetime, i.e., that it happens over a small region of space and a small interval of time, similar to the standard way that scattering is treated.)

All of this seems to me to be straightforward "MWI 101". I'm not trying to argue that it's "right" (or "wrong"); I'm just trying to be clear about what it says.

 

[PeterDonis]

See my critique of Everett here.

So if I'm reading this right, you're basically saying that the interaction I described in the OP can't be realized by a unitary operator if there is more than one possible result?

 

[A. Neumaier]

I'm not trying to argue that it's "right" (or "wrong"); I'm just trying to be clear about what it says.

Thanks, this is good enough for a discussion. I'll prepare a sensible answer.

 

[Derek P]

Please do not overreact. "In the head of the believer" is not implying any kind of delusion. It's just referring to a person's beliefs as opposed to what those beliefs are about. Even if "what those beliefs are about" does not exist, unless you are going to claim that normal humans never have false beliefs, it's hard to see such a statement as an insult; it's just a description of the human condition.

Okay. But I did say "verging on". I doubt whether Dr Neumaier actually intended a personal insult.

 

[stevendaryl]

So if I'm reading this right, you're basically saying that the interaction I described in the OP can't be realized by a unitary operator if there is more than one possible result?

I had not read @A. Neumaier's article on Everett, but I think I've known the conclusion for a long time. When people talk about measurements (this is not just Everett, I've seen it several places that weren't specifically about Many Worlds), they often say something like:

• Let $|u\rangle$ be the electron state corresponding to spin-up in the z-direction
  
• Let $|d\rangle$ be the state corresponding to spin-down in the z-direction.
• Let $|ready\rangle$ be the state of the measurement device before it measures the particle's spin.
• Let $|U\rangle$ be the state of the device after it measures spin-up.
• Let $|D\rangle$ be the state of the device after it measures spin-down.
• Then, we assume that the composite state satisfies: $|u\rangle |ready\rangle \Rightarrow |u\rangle |U\rangle$ and $|d\rangle |ready\rangle \Rightarrow |d\rangle |D\rangle$, where $\Rightarrow$ means "evolves into".

The difficulty with this is the $\Rightarrow$ can't possibly mean unitary evolution, because what's described isn't unitary. With unitary evolution, the reverse of any possible transition is also a possible transition, with the same transition probability. To get one-way evolution, which is what you need for measurement devices, you need irreversibility.

At that point, having realized this, I basically give up. I understand how irreversibility arises classically from reversible microscopic interactions, but I don't see immediately how to do the analogous thing quantum mechanically. Classical irreversibility arises from state counting. If macroscopic state $A$ has orders of magnitude fewer microscopic states than macroscopic state $B$, then you're going to see transitions from $A$ to $B$ but very rarely the reverse. So if I want to do irreversibility in QM, I assume that I want to say that there is not just a single state $|U\rangle$ corresponding to the device measuring spin-up, but many, many states that all indicate spin-up. So that makes the analysis not something that you can really do in a few lines of a PF post.

I'm not at all in agreement that this says anything about MWI versus other interpretations. It seems that the hard problem of analyzing measurement, quantum mechanically is interpretation-independent. Well, as long as you don't invoke the mystical powers of consciousness, which I guess some interpretations do, although I have never met anyone who takes that really seriously.

 

[PeterDonis]

The difficulty with this is the $\Rightarrow$ can't possibly mean unitary evolution, because what's described isn't unitary.

Yes, I see, and since that is what I was describing in the OP (just with the two terms summed), the answer to the question I posed that you quoted would be "yes".

However, now I'm wondering about the standard analysis of, say, the Stern-Gerlach experiment, where we have the $|u>$ and $|d>$ states corresponding to the "up" and "down" spin eigenstates of a spin-1/2 particle, the $|ready>$, $|U>$, and $|D>$ states corresponding to three different momentum eigenstates of the same particle (for definiteness, suppose that $|ready>$ corresponds to +x momentum, $|U>$ to +z momentum, and $|D>$ to -z momentum), and the unitary operator that induces the transition is just $\exp(i H t)$, where $H$ is the Hamiltonian including the $\mu B \cdot S$ magnetic moment coupling term that entangles the z-momentum and z-spin of the particle. If you and @A. Neumaier are correct, then there is something wrong with this standard analysis of a standard experiment, but I can't see what it would be.

 

[stevendaryl]

Yes, I see, and since that is what I was describing in the OP (just with the two terms summed), the answer to the question I posed that you quoted would be "yes".

However, now I'm wondering about the standard analysis of, say, the Stern-Gerlach experiment, where we have the $|u>$ and $|d>$ states corresponding to the "up" and "down" spin eigenstates of a spin-1/2 particle, the $|ready>$, $|U>$, and $|D>$ states corresponding to three different momentum eigenstates of the same particle (for definiteness, suppose that $|ready>$ corresponds to +x momentum, $|U>$ to +z momentum, and $|D>$ to -z momentum), and the unitary operator that induces the transition is just $\exp(i H t)$, where $H$ is the Hamiltonian including the $\mu B \cdot S$ magnetic moment coupling term that entangles the z-momentum and z-spin of the particle. If you and @A. Neumaier are correct, then there is something wrong with this standard analysis of a standard experiment, but I can't see what it would be.

You're using $|U\rangle$ and $|D\rangle$ differently than in my post. You're not meaning them to be macroscopic states (pointer states). So my point doesn't immediately apply.

 

[PeterDonis]

You're not meaning them to be macroscopic states (pointer states). So my point doesn't immediately apply.

I don't see why not. The argument that the evolution you described can't be unitary doesn't depend on those states being macroscopic. It only depends on the evolution not being reversible.

 

[stevendaryl]

I don't see why not. The argument that the evolution you described can't be unitary doesn't depend on those states being macroscopic. It only depends on the evolution not being reversible.

I found a paper that works out the Stern Gerlach case in detail. I'm not sure how it reconciles this problem.

https://arxiv.org/pdf/quant-ph/0511276.pdf

 

[PeterDonis]

I found a paper that works out the Stern Gerlach case in detail.

In their formalism, the specific case I described in the OP would correspond to $\theta_0 = \pi / 2$, $\phi_0 = 0$, i.e., magnetic moment pointed along the $x$ axis (so equal amplitudes for +z and -z spin; they have a factor of $i$ in the "down" spinor component, but that doesn't affect anything we're discussing here).

The state in their (1) corresponds to my initial product of kets; the exponential factor is the "ready" state of the "apparatus" (which in their formalism is z position rather than z momentum as I was stating previously), and the spinor factor is the linear combination of the "up" and "down" spin eigenstates (the upper and lower components of the spinor, in their notation).

The Hamiltonian in their (2) is manifestly Hermitian, so the time evolution induced by it will be unitary and hence reversible.

Their (3) shows the entanglement of the spin eigenstates and position in the $z$ direction, which corresponds to the final state I wrote down in the OP. So it seems like there is a unitary transformation $U = \exp(i H t)$, where $H$ is the Hamiltonian in (2) and $t$ is the time during which the coupling is "turned on" (during which the silver atom is inside the magnetic field, in the paper's terminology) that induces the evolution from (1) to (3).

 

[rootone]

Elaborate, please. I don't see the connection.

It rules out the possibility of there being two identical objects present at the same spatial location.
(In conventional space time)

 

[atyy]

I don't think there is an agreed meaning for "Copenhagen interpretation" to begin with, but I certainly don't think most of the people using that term mean it this way.

Also, if it is meant this way, then "Copenhagen observer" would just mean "shut up and calculate observer", which obviously any interpretation of QM contains. So I don't understand your claim that MWI doesn't contain "Copenhagen observers".

Well, what would you call the standard interpretation? Shut up and calculate refers to Copenhagen, and Copenhagen is the more proper term.

Landau and Lifshitz don't use the term Copenhagen, but explicitly mention Bohr for interpretation. Messiah and Weinberg explicitly state Copenhagen as their default interpretation.

 

[PeterDonis]

It rules out the possibility of there being two identical objects present at the same spatial location.

The MWI does not claim that this happens, so I still don't understand how this is relevant.

I think you have a misunderstanding as to what the MWI actually says. A quantum particle being in a superposition of different spin eigenstates, for example, at a given spatial position does not mean there are two particles at that position. Quantum states don't work like your classical intuitions are telling you they do.

 

[Derek P]

I had not read @A. Neumaier's article on Everett, but I think I've known the conclusion for a long time. When people talk about measurements (this is not just Everett, I've seen it several places that weren't specifically about Many Worlds), they often say something like:

• Let $|u\rangle$ be the electron state corresponding to spin-up in the z-direction
  
• Let $|d\rangle$ be the state corresponding to spin-down in the z-direction.
• Let $|ready\rangle$ be the state of the measurement device before it measures the particle's spin.
• Let $|U\rangle$ be the state of the device after it measures spin-up.
• Let $|D\rangle$ be the state of the device after it measures spin-down.
• Then, we assume that the composite state satisfies: $|u\rangle |ready\rangle \Rightarrow |u\rangle |U\rangle$ and $|d\rangle |ready\rangle \Rightarrow |d\rangle |D\rangle$, where $\Rightarrow$ means "evolves into".

The difficulty with this is the $\Rightarrow$ can't possibly mean unitary evolution, because what's described isn't unitary. With unitary evolution, the reverse of any possible transition is also a possible transition, with the same transition probability. To get one-way evolution, which is what you need for measurement devices, you need irreversibility.

At that point, having realized this, I basically give up. I understand how irreversibility arises classically from reversible microscopic interactions, but I don't see immediately how to do the analogous thing quantum mechanically. Classical irreversibility arises from state counting. If macroscopic state $A$ has orders of magnitude fewer microscopic states than macroscopic state $B$, then you're going to see transitions from $A$ to $B$ but very rarely the reverse. So if I want to do irreversibility in QM, I assume that I want to say that there is not just a single state $|U\rangle$ corresponding to the device measuring spin-up, but many, many states that all indicate spin-up. So that makes the analysis not something that you can really do in a few lines of a PF post.

I'm not at all in agreement that this says anything about MWI versus other interpretations. It seems that the hard problem of analyzing measurement, quantum mechanically is interpretation-independent. Well, as long as you don't invoke the mystical powers of consciousness, which I guess some interpretations do, although I have never met anyone who takes that really seriously.

I'm probably missing the point here but the final step appears to be a proper mixture, two mutually exclusive possible outcomes. Which is collapse of the wavefunction and decidedly non-unitary. But in the context of MWI, the final step is to a superposition of detector states which leads to two observation-worlds. It is unitary and, in principle, though not in practice, reversible.

 

[stevendaryl]

I'm probably missing the point here but the final step appears to be a proper mixture, two mutually exclusive possible outcomes. Which is collapse of the wavefunction and decidedly non-unitary. But in the context of MWI, the final step is to a superposition of detector states which leads to two observation-worlds. It is unitary and, in principle, though not in practice, reversible.

The issue isn't really about collapse of the wave function, it's about how to model classical irreversibility in quantum mechanics. A measurement process involves a microscopic cause triggering a macroscopic effect. For example, an electron hits a photographic plate and produces a black dot. That is an irreversible process, whether or not you bring in collapse. Collapse is about the question of what happens if the electron is in a superposition of two states, but I'm talking about a process that happens even when the electron is initially in a pure state (of the relevant observable---spin in the case of Stern-Gerlach). Such amplification processes cannot be described by unitary evolution, but not because of collapse, but because of classical irreversibility.

As I said, classical irreversibility is an entropy effect. There are many ways that a glass bottle can be broken, but only one way that it can be whole. So a transition involving the "brokenness" state of the bottle is practically speaking only one-way: Bottles break, but they don't heal themselves. That kind of entropy effect is involved whenever you magnify a microscopic signal so that it is strong enough to directly observe.

So what's really going on with a measurement of a microscopic quantity might better be described like this:

• There are many, many possible states of the device + environment that are macroscopically indistinguishable from the "ready" state. Let $N_{ready}$ be this number.
• There are many, many possible states of the device + environment that are macroscopically indistinguishable from the "having measured spin-up" state. Let $N_{U}$ be this number.

For a measurement to be possible, $N_{U} \gg N_{ready}$.

So you can't really accurately describe a measurement process using only a single "ready" state $|ready\rangle$ and a single "having measured spin-up" state $|U\rangle$.

That's actually why my description in terms of projection operators is more accurate.

 

[A. Neumaier]

Does the MWI require "creation" of multiple worlds?

Unitary evolution, postulated in MWI, implies that the wave function is that of the universe, since the universe is the only isolated system that exists - the effective (observable) dynamics of nonisolated systems is always dissipative. I try to understand you post #1 in the light of this and your explanations in #78.

a "world" is a term in the superposition I wrote down in the OP; i.e., each of the terms |$1>|U>$ and $|2>|D>$ is a "world". So "worlds" are picked out by the interaction between the measured system and the measuring apparatus and how the two become entangled.

This means that your worlds are not features of the (objective) universal wave function alone, but mathematical artifacts defined for the special purpose of analyzing a particular experiment. The worlds are created when the experiment is set up (the first moment where one can distinguish measured system and detector), and they are destroyed once the experiment is finished and something else is measured. [But your universe in #1 is exceedingly simple, having only 2*3 dimensions, with a fixed tensor product basis, from which one infers that there is no ''something else''. (I guess this is what you mean by ''highly schematic''.) ]

During the time where the experiment can faithfully be simplified to your setting, the number of worlds remains constant (2 in your case), and only the states in these worlds (R,U,D) evolve. Thus the resulting worlds are dynamical in time.

Note that nothing splits during the experiment, the splitting happens when the experiment is set up, where one world for each possible measurement result is created. The measurement result is therefore determined in advance by the world the detector is in, independent of the dynamics of the state, and independent of the interaction. In particular, the measurement says nothing about the state of the measured system, only something about the world in which the measurement happens. Moreover, the dynamics in each world is open since the future of world 1 depends also on the present state of world 2, and vice versa. In a sense, the other worlds serve as a reservoir of hidden variables of some kind for the dynamics of a given world.

Nothing about my surprising conclusions significantly changes in a more complex world, where R,U,D are replaced by highly complex states encoding the detector, any observers, and the environment.

My analysis differs significantly from the answers you gave in #78, so please correct me where I made an assumption not intended by you, or a logical error.

Let me also note that in quantum mechanics applied to the real world, there should be something objective about what is measured by what. Since everything objective is in MWI encoded into the state of the universe, the tensor product basis should be determined intrinsically by the state of the universe. Apparently it isn't - but this criticism is unrelated to your question.

 

[A. Neumaier]

So if I'm reading this right, you're basically saying that the interaction I described in the OP can't be realized by a unitary operator if there is more than one possible result?

Well, my answer (given in #93) turned out to say something quite different. If I understood your setting correctly, it has nothing to do with the notion of ''good observation'' I criticized in Everett.

 

[romsofia]

A lot of questions can be answered by reading this paper by the great Bryce DeWitt "Quantum mechanics and reality":
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.729.5437&rep=rep1&type=pdf

Honestly, his papers are the ones to read in regards to this issue.

 

[PeterDonis]

Unitary evolution, postulated in MWI, implies that the wave function is that of the universe, since the universe is the only isolated system that exists

Yes, agreed. And one issue with any actual description of a quantum experiment in terms of the MWI is that nobody actually writes down the wave function for the entire universe; they write down "highly schematic" abbreviated wave functions that only describe the very, very tiny portion of the universe that they are picking out as relevant to the experiment. For purposes of this discussion, I'm not raising that issue; I'm assuming that, for purposes of discussion, we can usefully talk about highly schematic wave functions like the one I wrote down in the OP as wave functions of a "system" whose evolution can be usefully modeled as unitary. But I agree that some people might think that eliminates too much.

During the time where the experiment can faithfully be simplified to your setting, the number of worlds remains constant (2 in your case), and only the states in these worlds (R,U,D) evolve. Thus the resulting worlds are dynamical in time.

Hm, yes, I see what you mean; there are two terms in the superposition both before and after the interaction. Which means that the definition of "worlds" I gave before--worlds are terms in the superposition--can't be the one that MWI advocates normally use, since MWI advocates describe the process I described in the OP as one world splitting into two.

For the process I described in the OP, I think an MWI advocate would say that "worlds" are defined by the "apparatus" kets; there is one world before the interaction because the state only contains $|R>$, one apparatus ket, and there are two worlds after the interaction because the state has two apparatus kets, $|U>$ and $|D>$. Or perhaps a better way of saying it is that the state before the interaction is separable, and the part describing the apparatus is in a single eigenstate; but after the interaction, the state is entangled and the apparatus by itself does not have a definite state; its state is entangled with that of the measured system, and that entanglement interaction is the "splitting" of one world into two. But because that interaction is unitary, nothing is actually being "created"; it's just a unitary process that entangles two subsystems.

So with this alternate definition of "worlds", what determines how many worlds there are is the apparatus: whatever basis is picked out by the physical configuration of the apparatus is the one that is used to determine how many "worlds" there are. I take it that this would be an MWI advocate's answer to the "preferred basis" problem (what picks out the basis in which the "worlds" are counted).

My analysis differs significantly from the answers you gave in #78, so please correct me where I made an assumption not intended by you, or a logical error.

I think the error was mine, in giving an incorrect definition (from the MWI advocate's perspective) of "worlds". I don't know if the alternate definition I gave above changes anything substantive about your criticism, though. It is still true that the "worlds" are determined by the apparatus (although I don't think it is true that the state of the apparatus in each "world", under my new definition, gives no information about the state of the measured system).

the dynamics in each world is open since the future of world 1 depends also on the present state of world 2, and vice versa.

I don't think this is true (even on my original wrong definition of "worlds"). Each term in the superposition evolves independently of the others. That's why we can use "effective collapse" in the math of QM to ignore all the other terms once we've observed a particular measurement result.

To be clear, I am not considering cases where the two terms will be made to interfere later; in the highly schematic picture I've been using, any later interference would be part of the "measurement interaction" (the $\rightarrow$ in the OP). The final state would be the one after all such interactions have been completed (and everything has been allowed to decohere).

 

[Derek P]

To be clear, I am not considering cases where the two terms will be made to interfere later; in the highly schematic picture I've been using, any later interference would be part of the "measurement interaction" (the $\rightarrow$ in the OP). The final state would be the one after all such interactions have been completed (and everything has been allowed to decohere).

Which is why some people say the idea of worlds should only be applied after decoherence. I guess that ties in with @stevendaryl's point. But in the end the process is clear enough - if you can call decoherence theory "clear"... which I suppose you can in an A level thread. But where in the process you say "here be worlds" is somewhat arbitrary. Rather like trying to say when "life begins" from a biological PoV. There was a time when I definitely didn't exist but I definitely do exist now. I think.

So with this alternate definition of "worlds", what determines how many worlds there are is the apparatus: whatever basis is picked out by the physical configuration of the apparatus is the one that is used to determine how many "worlds" there are. I take it that this would be an MWI advocate's answer to the "preferred basis" problem (what picks out the basis in which the "worlds" are counted).

I think the "preferred basis" is coarse-grained though I'm open to correction about that. Otherwise every particle could be an apparatus. @bhobba will know :)

 

[bhobba]

We seem to be getting caught up in measurements here. Decoherent histories only has them indirectly - specifically in that interpretation QM is a stochastic theory of histories:
https://arxiv.org/abs/quant-ph/0504155

Gell-Mann's point is that one can consider instead of one particular history existing then all can in separate worlds with one overall wave-function.

I still like the simple idea a world is the element of a mixed state after decoherence - but its not the only way of looking at it.

Thanks
Bill

 

[A. Neumaier]

a world is the element of a mixed state after decoherence

What is an ''element of a mixed state''?

 

[A. Neumaier]

the definition of "worlds" I gave before--worlds are terms in the superposition--can't be the one that MWI advocates normally use, since MWI advocates describe the process I described in the OP as one world splitting into two.

For the process I described in the OP, I think an MWI advocate would say that "worlds" are defined by the "apparatus" kets; there is one world before the interaction because the state only contains $|R>$, one apparatus ket, and there are two worlds after the interaction because the state has two apparatus kets, $|U>$ and $|D>$. Or perhaps a better way of saying it is that [...] entanglement interaction is the "splitting" of one world into two.

This does not change my conclusions: Separability is lost immediately after the experiment is set up (since even at a distance, the interactions are present though tiny).

Each term in the superposition evolves independently of the others.

No. The unitary evolution introduces terms in the other worlds, since the eigenstates 1 and 2 are not eigenstates of the interacting Hamiltonian (if it were, measurement would be impossible).

 

[Derek P]

This does not change my conclusions: Separability is lost immediately after the experiment is set up (since even at a distance, the interactions are present though tiny).

No. The unitary evolution introduces terms in the other worlds, since the eigenstates 1 and 2 are not eigenstates of the interacting Hamiltonian (if it were, measurement would be impossible).

Surely they are eigenstates of the total interaction including interaction with the "environment"? The states are already superposed and therefore evolving independently before decoherence starts. Or maybe I'm just not getting the point.

 

[A. Neumaier]

Surely they are eigenstates of the total interaction including interaction with the "environment"?

No. Why should they be? There is only one Hamiltonian, that for the whole universe, and there is no reason at all why this Hamiltonian should have separable eigenstates. Almost no Hamiltonians, except very simple contrived ones, have this property.

 

[Derek P]

No. Why should they be? There is only one Hamiltonian, that for the whole universe, and there is no reason at all why this Hamiltonian should have separable eigenstates. Almost no Hamiltonians, except very simple contrived ones, have this property.

Sure. Even Zurek has said that MWI may need to postulate that "there are systems". The model of detection begins with a microscopic interaction that is then amplified and decohered. If the universe can't be modeled this way then pretty well all of science falls apart.

 

[A. Neumaier]

Sure. Even Zurek has said that MWI may need to postulate that "there are systems". The model of detection begins with a microscopic interaction that is then amplified and decohered. If the universe can't be modeled this way then pretty well all of science falls apart.

Why does ''there are systems'' (or what you state about decoherence) imply that the total Hamiltonian has separable eigenstates? It doesn't. Only the initial state may be assumed (by preparation) to be separable. Apart from that, system, detector and environment are represented as terms in the total Hamiltonian, but the existence of interaction terms makes it very unlikely that the eigenstates of the total Hamiltonian are separable.

You are far too vague for a serious discussion. You neither define your terms nor point to sources where they are clearly defined.

 

[PeterDonis]

the eigenstates 1 and 2 are not eigenstates of the interacting Hamiltonian

In the model I was describing, the Hamiltonian is time-dependent; the interaction term is only "turned on" for a short period (while the particle is inside the apparatus). Before and after that, there is no interaction term in the Hamiltonian, so the kets I wrote down should be eigenstates during those times.

This is similar, as I understand it, to the way scattering experiments are modeled. Are you saying that this can only be an approximation (similar to the way you have to take the times to minus or plus infinity, heuristically speaking, to get definite "in" and "out" states in scattering--at any finite time there is still a nonzero interaction)?

 

[Derek P]

Why does ''there are systems'' (or what you state about decoherence) imply that the total Hamiltonian has separable eigenstates? It doesn't. Only the initial state may be assumed (by preparation) to be separable. Apart from that, system, detector and environment are represented as terms in the total Hamiltonian, but the existence of interaction terms makes it very unlikely that the eigenstates of the total Hamiltonian are separable.

You are far too vague for a serious discussion. You neither define your terms nor point to sources where they are clearly defined.

Yeah, you're probably right. Ta-ta!

 

[A. Neumaier]

the interaction term is only "turned on" for a short period (while the particle is inside the apparatus). Before and after that, there is no interaction term in the Hamiltonian, so the kets I wrote down should be eigenstates during those times.

This still doesn't change my analysis; it just restricts it to the short period where the interaction is on. Afterwards the system is stationary so nothing changes.

This is similar, as I understand it, to the way scattering experiments are modeled. Are you saying that this can only be an approximation (similar to the way you have to take the times to minus or plus infinity, heuristically speaking, to get definite "in" and "out" states in scattering--at any finite time there is still a nonzero interaction)?

Indeed, it is only an approximation. But even when the interaction time is taken to be finite, the analysis still leads to the same strange result. Separability is lost once the interaction is turned on, hence (in microscopic terms) long before the measurement is completed. Thus with your revised definition, the worlds split at the moment the measurement begins, and the result in each world is predetermined.

Moreover, if one thinks of reversing the situation (unitary dynamics is reversible), worlds should disappear (merge) whenever two states of the detector happen to become equal. This is quite unreasonable from a formal point of view. The natural thing to expect is that the two worlds were always there, and will always be there, which is the reversible situation. The other, even more natural interpretation is that the worlds are an artifact of imposing a particular tensor product basis on the universe, and appear and disappear together with the coordinate system. This is what I had in mind when asking about the ''point of view'' interpretation.

Note also that nothing is said by this version of the MWI about how the worlds are selected by a real observer - they all have fully democratic existence of the same kind. Labeling them by a formal number called probability is of course possible, but nothing explains why this formal label actually has the property of an observed relative frequency by an observer moving along a particular (coarse-grained) world line.