> On 9 Aug 2018, at 14:03, Bruce Kellett <[email protected]> wrote:
>
> From: Bruno Marchal <[email protected] <mailto:[email protected]>>
>>> On 9 Aug 2018, at 06:55, Bruce Kellett <[email protected]
>>> <mailto:[email protected]>> wrote:
>>>
>>> From: Bruno Marchal <[email protected] <mailto:[email protected]>>
>>>>> On 8 Aug 2018, at 13:50, Bruce Kellett <[email protected]
>>>>> <mailto:[email protected]>> wrote:
>>>>> The real problem I see with many-minds theory is that it does not
>>>>> actually explain the observed correlations. The correlations are presumed
>>>>> not to exist in reality -- all possible combinations of experimental
>>>>> outcomes happen, but when Alice and Bob meet, their bodies are still in
>>>>> indefinite states -- no actual results are recorded by entanglement with
>>>>> their bodies -- but their minds will be in definite states that agree
>>>>> with the quantum correlations. This step seems to introduce yet more
>>>>> unreasonable magic into the 'explanation'. Why are the minds like this
>>>>> when they communicate?
>>>>
>>>> Because all Alice and Bob are coupled in that way, by the singlet state.
>>>> That works if we keep in mind that the singlet state (when not already
>>>> observed by neither Alice nor Bob) describes an infinity of Alice and Bob,
>>>> with the spin in all directions, but always correlated.
>>>
>>> This is not quantum mechanics. The Hilbert space for the singlet is
>>> two-dimensional. This space is spanned by two (mutually orthogonal) vector,
>>> but these basis vector may be chose in an infinity of different ways, all
>>> describing the same state. There is no way that the singlet is a
>>> superposition of all possible basis vectors: it is 2-dim, not
>>> infinite-dimensional.
>>
>> Due to the fact that a^2 and b^2 can be irrational number, I interpret even
>> any au+bd has involving infinities of histories possible. The singlet is the
>> superposition of two base vectors, I agree, but the number of worlds is not
>> related to the dimension of the Hilbert space.
>
> But the number of elements in any quantum superposition is related to the
> dimension of the associated Hilbert space. If you want to stay within the
> bounds of more-or-less conventional quantum mechanics (that which is taught
> in text books),
Most textbook still invoke the collapse.
> the only way you can get your infinite number of copies of Alice who can
> self-locate in some branch is to have the branches form a quantum
> superposition. If it is just a matter of an infinite number of worlds in some
> multiverse, then there is no coherent superposition, and the concept of
> self-location in the quantum sense loses meaning.
I agree.That is why we might better use the Omnes-Griffith formalism, and treat
directly the superposition of histories in some (relativistic) space-time.
>
>
>>> I think you are describing a particular hidden variable model, in which
>>> there is actually an infinite dimensional space of the hidden variables,
>>> one dimension for each hidden vector that picks out a particular spin
>>> orientation. If we now place an infinite of Alices in this space (by her
>>> interacting with the singlet, say), the Alice's measurement locates her in
>>> this infinite space according to the direction in which she made her
>>> measurement. The other particle of the singlet in this space has an
>>> "element of reality" corresponding to spin component in the direction
>>> opposite to Alice's result. When Bob measures this second particle, he gets
>>> 100% correlated results provided he measures along the same axis as Alice.
>>
>> OK. (I would not called that a hidden variable theory though, as it is the
>> description of the multiverse that I associate to the singlet state). If Bob
>> choses an arbitrary direction, the perfect correlation is transformed into
>> correlation leading to the violation of Bell’s inequality.
>
> It has to be interpreted as a hidden variable model if you are to use the
> machinery of Hilbert space to form a coherent superposition within which one
> can self-locate. Just as working in 2-dim Hilbert space for the spin
> projections of a spin-1/2 particle, one is working with a superposition of
> up-down in any basis at all. Then one self-locates in the measurement
> basis by making a measurement. This is quantum physics, after all. We are not
> doing classical duplications as in your H-->W/M thought experiments.
Indeed.
>
>
>>> Before his measurement, Bob is in the same infinite-dimensional space as
>>> Alice was, so by measuring along the same axis, he self-locates into the
>>> same subspace, when his particle was already primed to have its spin
>>> pointing along that axis in the direction opposite to Alice's particle.
>>> This hidden variable account will work to give the correct correlation for
>>> measurements along the same axis at both ends.
>>
>> OK, but same subspace do not entail that the original Alice and Bob will get
>> the correlated results. It means that they will share the correlated result
>> only with their respective counterparts.
>
> The original Alice and Bob are those in the same branch of the wave function
> all the way along. There are no unmatched Alices or Bobs.
In each branch, I agree. But to get the reasoning right, and treat the case of
the violation of Bell’s inequality, we need to take into account the unmatched
Alice and Bob who exist “transworldly” if I can say. They belong to different
branches, and can never meet again. That is important to make the FTL
eventually into an illusion, still keeping the violation of the Bell’s
inequality.
>
>
>>> But it immediately fails if Bob makes a random rotation before he makes his
>>> measurement. That will then locate him in a different subspace to that
>>> inhabited by Alice, so the only possible results he can obtain are up or
>>> down with 50% probability for either. This does not give the correct
>>> correlation with Alice for any angles other that 0 or 180 degrees.
>>
>> But there is no reason that Bob and Alice will ever be able to share those
>> incorrect correlation result. They don’t know to which worlds they belong.
>
> There can be no incorrect correlation results. Did you not understand the
> argument backwards from the final meeting of the pair with their lab books.
> That proves that they must have been in the same branch of the superposition
> all along. All possible branches involve such a meeting. And all such
> meetings confirm the correct statistics. There are no loose ends -- unmatched
> operators.
In each branches. But when space separated, that makes no sense to me, as the
branches evolved locally.
> This is actually an essential feature of this whole scenario in the
> many-worlds (no collapse) picture.
>
>
>>> I had thought through this possible explanation for your insistence that
>>> the measurements of both Alice and Bob serve only to locate them in the
>>> relevant subspaces some time ago. But I realized very quickly that this
>>> crudest of hidden variable models could not work for general magnet
>>> orientations.
>>
>> I will think about this. My feeling is that you are using the usual
>> mind/brain identity thesis.
>
> I have no idea why you might think this. As far as I can see I am not using
> any thesis at all about either the brain or the mind. All of these
> measurement scenarios can be carried out entirely automatically by
> appropriately programmed computers. The programs will always halt with the
> correct quantum statistics.
Yes, and if you use programmed computer, you see that Mechanism can be used,
albeit without collapsing any wave at any time, if you agree to apply QM to
this classical machine/observer. (But then you cannot attach a mind to one
machine, but only an infinity of machines, no to for quantum mechanical reason,
but simpler arithmetical one).
>
>
>>> The point about general relative orientations is that the probabilities for
>>> Alice and Bob getting the same or different results depends on the relative
>>> angle of their measurements.
>>
>> Yes.
>>
>>> In any single trial on an entangled pair, they can get any one of four
>>> combinations of the possible results.
>>
>> OK.
>>
>>> It is the relative probabilities of these results that are crucial for
>>> reproducing the quantum correlations. And these probabilities depend on the
>>> relative magnet orientation, a fact which is available only non-locally.
>>
>> Yes.
>
> That is interesting. So you agree that the relative magnet orientation
> information required to get the correct statistics is available only
> non-locally? In which case, physics is proven to be non-local, and we are
> done.
Yes I agree. I would say that it is the essence of QM at play here. But the non
locality is not a result of some FTL in a universe, but from the fact that
whatever results we find, it get spread-out in the building of new branches
that we are necessarily ignorant of.
>
>
>>> Besides, the above is a hidden variable model that has nothing to do with
>>> conventional quantum mechanics.
>>
>> I am not sure what you mean by conventional QM.
>
> Quantum mechanics as taught in text books. That need not be Copenhagen or a
> collapse interpretation. It can just as easily be an Everettian many-worlds
> interpretation. As you might have seen, I am happy to move freely between the
> two models, because I have sufficient familiarity with quantum mechanics to
> be able to tell which I am using at any particular time, or in translating
> results from one picture to the other.
>
>
>> Without collapse and FTL potential, or FTL (non-local) hidden variable
>> theory, how do you interpret the singlet state?
>
> That is actually a rather strange question. How do you think I might
> interpret the singlet state? I think that I have been talking about it here
> for long enough for you to have worked it out. The singlet state is a
> non-separable state that is symmetric under rotations about the axis between
> the particles. However, that symmetry will generally be broken by any
> interaction with one or other of the constituent particles.
But it is here that I suspect you introduce some collapse.
> In particular, the symmetry is broken by the imposition of a directional
> magnetic field, as in a Stern-Gerlach magnet used to measure the spin
> component of one of the particles in the direct defined by that external
> magnetic field.
>
> The singlet is strongly non-separable, so this external interaction with one
> of the constituents is instantaneously felt by the other component particle.
How could this be verified? Any verification possible will need further
interaction, and we can see only the branche of the universe our own result
have spread on.
> That non-local influence is the essence of the non-separability of the state
> -- it is a unit, and any interaction with a part is an interaction with the
> whole.
That looks magical to me, and as I said, I am not sure this can be verified.
Aspect-like experiment do not verify this for sure.
>
>> Before any measurement I interpret it as a multiverse where the spin can be
>> in any direction, for both Alice and Bob, but in all branches their spin are
>> opposite. Then I use the fact that if their measure in non “orthogonal
>> direction”, they have only partial knowledge, and it concerns only what they
>> can share with their respective counterpart.
>
> That account of the result of measurements at arbitrary relative angles is
> too vague and hand-waving. If you want to convince me that you can actually
> get the right answers, you will have to spell this "only partial knowledge"
> out in considerably more detail, with precise mathematics.
You are the one telling that the Bell’s inequality violation entails FTL
influence. Personally, I do not dig on that issue, because I use only Everett
QM to evaluate what mechanism predicts. I might try to send a post why I do not
follow your critic of Tipler and Baylock, some day.
>
>
>>>> When Alice and Bob make their measurement, if they are space separated, it
>>>> makes no sense to ask if they are or not in the same world or branches.
>>>> The result they obtained only entangle each of them with the environment,
>>>> locally, and that spread on the whole universe (at subliminal speed) so
>>>> that both of them will encounter only their “correlated” counterparts.
>>>
>>> This is just nonsense. A world is defined as a branch of the wave function
>>> that is fully decohered and disjoint from all other branches.
>>
>> I disagree.
>
> The definition of a "world" that I have given is coherent and unambiguous.
> You appear to want to equivocate between several different meanings of the
> word.
It might be that I disagree indeed with your definition of worlds. I do have
some difficulties with QM + SR + “one world”.
>
>
>>> Alice and Bob are semi-classical objects, and they do not oscillate between
>>> branches of the wave function according to random quantum fluctuations of
>>> their constituent molecules.
>>
>> Even if that is true, it is not relevant. It is not a question of
>> oscillating between branches, but of ignoring to which branches their belong
>> before they do a measurement. I suspect again some naive brain/mind identity
>> link.
>
> Again you raise this. It makes no sense to me because I am not assuming
> anything about brains and/or minds. I am doing objective physics.
Physics always use a theory of mind. But for most of its progress up to QM, a
simple perspectival approach, using the traditional mind/memory link, was
enough, like with Galilee and Einstein.
With QM, the choice is almost between a consciousness which reduces the wave at
“infinite speed”, or a wave of relative histories amplitudes.
Interaction/entanglement makes the experience first person *plural*, and local
for all collection of observers able to interact with each others.
>
>> (Note that even Everett is ambiguous on this, and some of its motivation
>> consists in assuring that brain-mind identity link. That cannot work with
>> mechanism, and eventually we need to take mechanism into account.
>
> Whether or not anything can work with mechanism is entirely irrelevant to
> this discussion. We are discussing quantum mechanics, not your "mechanism”.
The whole point of abandoning the collapse is to avoid a dualist theory of
observation. You are right in the sense that Everett assumes only that the
observer obey QM. That works even with a non computable halmitonian. But to
justify the collapse phenomenologically, Everett use more or less explicitly
(depending of its texts) a mechanist hypothesis. This entails also that the
observer has to obey QM when implemented in the physical reality.
But, then, as I have explained (and can give references for more detailed
explanation), the assumption of mechanism makes Everett’s work incomplete. We
need to extract the wave itself from the universal machine self-reference.
Physics is simply no more the fundamental science. It becomes a branch of
number biology/psychology/theology, itself already branch of number theory (or
combinator, etc.).
>
>
>>> Perhaps the best way to understand the inevitable quantum substructure of
>>> and macroscopic object is in analogy with statistical mechanics. We do not
>>> change worlds according to the thermal fluctuations of the molecules in a
>>> gas -- these fluctuations are averaged over (coarse-grained) in the bulk
>>> properties that characterize the gas, such as volume, temperature and
>>> pressure, etc. The same is the case for people and other macroscopic
>>> objects. They are obtained by averaging over the quantum fluctuations of
>>> their constituents, and it makes no sense to pretend otherwise. We live in
>>> a (semi-)classical world in which quantum effects are averaged out to
>>> insignificance for macroscopic bodies. So, if Alice and Bob are together,
>>> or share a pair of entangled states, they are in the same world by any
>>> reasonable definition of a "world”.
>>
>> I do not think such notion of worlds makes sense, already with just
>> Mechanism. I suspect we have quite different notion of worlds.
>
> That is possible. I work in quantum theory, whether in Everett's many-worlds
> or Copenhagen collapse. Everett, after all, is just Copenhagen with collapse.
I guess you mean “without” collapse.
> Both are based on the Schrödinger equation. One just retains the non-observed
> branches that the other disregards. There is actually very little difference
> between them, since both give the same predictions for all observable
> results. Why get so hung up on the differences?
Using only the SWE avoids the pitfall of dualism with clear monism, avoid
3p-indeterminacy, avoid (Imo) spooky action at a distance. It solves the
measurement problem (or makes it more easy: I don’t claim everything is solved).
Then, the SWE without collapse describes a type of physical reality predicted
by mechanism. The "many worlds” looks up to know, qualitatively and
qualitatively like the physics that we must extract from machine or number
self-reference.
>
>> I would even prefer to abandon that term, and use the notion of consistent
>> histories instead, but of course that too should be handle with caution. I
>> don’t think there is a viable notion of quasi classical world in any
>> non-collapse theory
>
> I disagree. There has been a lot of work on the transition from the quantum
> to the classical worlds. After all, our experience is of the classical world,
Well, not really. It is just that the quantum phenomenon where too much subtle
to be noticed. Note that Newton and Huygens disputed already the nature of
light. It was particles for Newton and thy were wave for Huygens. QM is just
saying they were both correct and wrong, and than that is true for all piece of
matter.
> and it is that experience that we have to explain via its quantum substrate.
I think you presuppose physicalism. With mechanism, we need to explain the
illusion of substratum from elementary arithmetic (or Turing equivalent).
> It is not a viable option just to claim that the classical does not exist.
I was just claiming that it is an open question if we can make sense of a
classical world. Of course, I can explain why this notion does not make sense
when we postulate mechanism.
>
>>>>> In his 2011 thinking I can only imagine that he would have seen
>>>>> many-minds in much the same way as he later saw many-worlds -- if appeal
>>>>> is made to the wave function to make sense of the correlations in
>>>>> many-worlds, then we have to recognize that this is not a local account
>>>>> since the wave function is not a local object.
>>>>
>>>> I don’t really understand what you mean by that.
>>>
>>> That is, in many ways, the crux of the matter. The singlet state of two
>>> entangled spinors is a paradigm non-separable state: it cannot be written
>>> as the product of two components, one referring exclusively to particle 1,
>>> and the other referring exclusively to particle 2.
>>
>> Right.
>>
>>> And such a separable two particle state is required if one is to
>>> incorporate Einstein's concept of local realism.
>>
>> I disagree here.
>
> I don't think that this is a debatable point. Einstein's concept of classical
> realism is enunciated by people such as Bob Griffiths in exactly these terms,
> as well as by almost all other authors I can think of.
But without the FTL. Einstein’s local realism makes sense only if we embrace
the MW. That is the point I am arguing.
>
>>> But the singlet cannot be written in that way. The only state that is
>>> symmetric under rotations about the axis joining the particles is the
>>> non-separable state:
>>>
>>> |psi> = (|+>|-> - |->|+>)/sqrt(2).
>>>
>>> No rotationally symmetric separable state can be constructed.
>>
>> I agree.
>>
>>> This non-separable state has no dependence on the separation between the
>>> particles,
>>
>> I agree. But before any measurement is done, the couple Bob and Alice belong
>> to each possible situation, including possible non correlated one, in which
>> case, they will never be able to meet again: only to meet their counterparts.
>
> That is the point at which your account becomes hopelessly confused. There
> are no couples of Alice and Bob that never meet again.
That explains why you believe in FTL.
It should be clear that we interpret the wave differently. If Alice and Bob are
space-separated, locality implies that they can find uncorrelated results, but
that will never contradict the quantum non locality, because both Alice and Bob
will only meet the Bob and Alice they can interact with, and each will meet the
relevant counterpart. That is how I interpret the singlet state. The
splitting/differentiation of the histories is subliminal.
> You confuse a pre-existing ensemble of experimenters with the actual case of
> a possible superposition in a hidden variable theory.
?
Whatever hidden variable is used will need to be non-local.
> Quantum physics can cope with superpositions, but it does not have to assume
> infinite ensembles of anything.
The behaviour of just one particle asks for an infinity of histories. The
assumed infinite ensemble *is* the multiverse, roughly speaking.
> The Schrödinger equation does not require a pre-existing ensemble in order to
> get many-worlds (or relative state branches).
I don’t understand.
>
>
>>> so the same entangled state persists for arbitrary separations.
>>
>> From the point of view of all observers. OK. That is a relative notion.
>>
>>> But because particle 2 is intrinsically entangled with particle 1, any
>>> interaction with one particle necessarily affects the other particle.
>>
>> I don’t see why you say this, except that you talk like if Bob and Alice
>> where related to the same branch, which makes non sense to me if they are
>> space separated.
>
> I think you need to think some more about this. Two people can be in the same
> branch if they are time-like separated, and if they are space-like separated:
> Just meet up and walk a few paces away from each other! Neither party has
> moved to a different world, but they are now space-like separated.
So you will need FTL to interpret the singlet state indeed. I have no need of
FTL. I know that it is bit sad from the romantic point of view, but
space-separated people are always in their own world. They can share their
experiences only through local interaction or entanglement.
>
>>> Thus the non-separable state is intrinsically non-local.
>>
>> Yes, but without the need to the FTL if you use the only available
>> mind-brain identity link possible with Mechanism (in cognitive science).
>
> You are clearly obsessed with FTL and the mind-brain link. Forget these. They
> are strictly irrelevant to the discussion.
?
That was the subject of the conversation.
I claim that Bell’s inequality violation does not entail FTL. You do.
So we try to pinpoint where we differ in the interpretation of the wave. And
the mind-brain links are important because they brings the nuances between a
variety of non-collapse (and non hidden variable) interpretations of the SWE.
>
>>> If you wanted to make it local, you would have to break the entanglement
>>> and make it into a separable state.
>>
>> That cannot work. That violate QM, but that is not needed, it is simpler to
>> abandon the brain mind identity link.
>
> No, you cannot make the singlet separable,
That is exactly what I am saying. Superposition and entanglement never leads to
mixture. It is always the relative state which makes thing looking separated,
but that is a “local” or “first person” view.
> and separability is the essence of locality. There is no brain-mind identity
> link here. Perhaps only in your mind, but that is not the issue at hand.
?
>
>> Alice and Bob always belong to infinities of histories, and totally or
>> partially localise themselves in the branch of the universal wave.
>
> Start with one Bob and one Alice. If you say this is impossible, then you are
> not doing quantum mechanics, but some weird theory of your own devising.
I just show you that we can interpret the Bell’s inequality violation without
the need to introduce magical FTL.
> Ostensive definitions of these two people will do just fine.
Not at all, and that is why they are empty hulk in the many minds theory of
Albert and Loewer. With Everett relative state, there are worlds providing
sense to all observers, but that can no more to be identical for space
separated person, although they will still share their impression at the speed
of light.
>
>
>>> And that is not possible in quantum mechanics. If you have a hidden
>>> variable, or any other system, that does this, then you do not have quantum
>>> mechanics but some other theory. You would then have to re-establish all
>>> the well confirmed results of quantum theory in your new theory. Not
>>> impossible, perhaps, but highly unlikely.
>>>
>>>> I am reading your paper, which is nice and well written, but too quick for
>>>> me on both Tipler and Baylock. It helps me to better see how you interpret
>>>> the wave, and where we might differ.
>>>
>>> Thank you. I hope that I have managed to express myself more clearly than
>>> is often possible in emails.
>>
>> You are still quick, but I think we differ, not on QM, but on the theory of
>> mind used in the non collapse theory. I think we might progress on where we
>> depart.
>>
>>
>>>> It seems to me that when Alice and Bob prepare the singlet state, even
>>>> before their long distance separation, there is no sense to say that they
>>>> are still in the same world.
>>>
>>> Of course they can be in the same world.
>>
>> Alice ((|+>|-> - |->|+>)/sqrt(2)) = Alice((|+'>|-'> - |-'>|+'>)/sqrt(2)), so
>> I don’t know which worlds you are talking about.
>
> Your equation is true only if Alice does not interact with the singlet. Any
> interaction breaks the symmetry. Or do you still not understand this?
I have never doubt this. But that does not answer my question. In the
experience, Alice do not measure her particle at once, and has still the choice
of changing the angle of her apparatus. So, in which world belong Alice before
she makes here measurement?
>
>> They know only that whatever their measure, will be correlated with what the
>> other is measuring, but the “other’ refer to infinities of “Bob” before any
>> measurement is done.
>
> Not necessarily, if you are doing conventional quantum physics. (And don't
> raise your usual point that you don't know what conventional physics is!)
?
>
>
>>> The singlet is prepared at some point between them. The entangled particles
>>> then separate and reach Alice and Bob respectively.
>>
>> That makes no sense to me. You talk like if the singlet state fixes the
>> base{ |+>, |->}.
>
> I do no such thing. The two particles of the singlet can separate -- that is
> the whole point of this set-up. If nothing interacts with the singlet, it is
> preserved in its full rotational symmetry up to infinite separations.
I still fail to understand what is your MW interpretation of the singlet state.
>
>> I take into account a possible phase factor, which makes the multiverse
>> rotationally invariant,
>
> The multiverse is not rotationally invariant because it contain non-symmetric
> objects -- such as trees, rocks, and people.
OK. Here I was perhaps wrong. Sometimes I think that all non-symmetries in
physics is a perspectival view. I am not sure. It is not relevant for the point
here.
>
>> and this remains true forever in the non collapse view. The breaking of the
>> rotational invariance below only to the relative person views, like
>> decoherence.
>
> No, the multiverse is not symmetric at the start, nor at any subsequent time.
> It is only the isolated singlet state that is symmetric. And that is an
> objective fact, not dependent on any relative person view.
OK
> Measurement objectively breaks this invariance.
I am not sure. I have to think about this.
>
>
>>> Since the particles were prepared together, they separate in the same world
>>> in which they were entangled.
>>
>> As long as no-one look at the particles, I don’t see how you select the
>> worlds.
>
> As I say below, I can talk of the world to which they belong because they
> cannot jump to another world.
You can do that, from the bird perspective, but *they* (Alice, Bob, any frog)
cannot talk about which world they belong, before choosing some angle and doing
the measurement.
>
>>> They cannot jump between disjoint worlds.
>>
>> They both just belongs to infinity of “worlds".
>
> That is not the concept of "world" that is applicable here.
You use the usual brain mind identity link, or better, the world-mind, identity
link, which makes no sense in Everett relative state. If I do not observe the
position of an (supposedly well isolated) particle, after some time there is as
much “me” than position (and momenta) of that particle.
> As usual, you equivocate on the meaning of the term in order to confuse the
> argument in your attempt to prove me wrong. You are not going to succeed,
> because I am not wrong on this.
If you say so.
>
>
>>> So when they meet Alice and Bob (or copies of these two from multiple
>>> possibilities) their meeting and interacting with the entangled particles
>>> ensures that Alice and Bob are in the same world when they make their
>>> respective measurements.
>>
>> Well, I understand this will re-introduce FTL. I see no QM reason to
>> interpret the singlet state in the way you do.
>
> Then you do not understand quantum mechanics well enough. You are mixing in
> too much of your own "mechanism", which is simply confusing you.
On the contrary, despite non-locality would not been a threat to Mechanism per
se, it just do not make sense with SR or GR. And you have fail to give a reason
why your interpretation would be more correct that mind. So it is enough that
my interpretation is consistent to illustrate that Bell’s inequality violation
does not necessarily introduce FTL in the physical reality.
>
>
> [deleting a lot of repetitive stuff...]
>
>
>
>> So, just tell me how you interpret the singlet state without singling a base
>> out of all possible base. That should be possible in the non-collapse case.
>
> I think I have already done this in sufficient detail -- in both collapse and
> non-collapse interpretations -- to convince even the most committed skeptic.
Then do a copy and paste, please, as I have yet to figure out what it is.
Bruno
>
> Bruce
>
>
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