On Friday, March 1, 2019 at 8:49:54 AM UTC-6, Bruno Marchal wrote:
>
>
> On 1 Mar 2019, at 01:42, Lawrence Crowell <goldenfield...@gmail.com
> <javascript:>> wrote:
>
>
>
> On Monday, February 25, 2019 at 9:42:01 AM UTC-6, Bruno Marchal wrote:
>>
>>
>> On 25 Feb 2019, at 12:39, Lawrence Crowell <goldenfield...@gmail.com>
>> wrote:
>>
>> On Monday, February 25, 2019 at 2:44:14 AM UTC-6, Bruno Marchal wrote:
>>>
>>>
>>> On 24 Feb 2019, at 15:24, Lawrence Crowell <goldenfield...@gmail.com>
>>> wrote:
>>>
>>> On Friday, February 22, 2019 at 3:18:01 PM UTC-6, Brent wrote:
>>>>
>>>>
>>>>
>>>> On 2/22/2019 11:39 AM, Lawrence Crowell wrote:
>>>>
>>>> This sounds almost tautological. I have not read Masanes' paper, but he
>>>> seems to be saying the Born rule is a matter of pure logic. In some ways
>>>> that is what Born said.
>>>>
>>>> The Born rule is not hard to understand. If you have a state space with
>>>> vectors |u_i> then a quantum state can be written as sum_ic_i|u_i>. For an
>>>> observable O with eigenvectors o_i the expectation values for that
>>>> observable is
>>>>
>>>> sum_{ij}<u_j|O|u_i> = sum_{ij}<u_j|o_i|u_i> = sum_ip_io_i.
>>>>
>>>> So the expectations of each eigenvalue is multiple of the probability
>>>> for the system to be found in that state. It is not hard to understand,
>>>> but
>>>> the problem is there is no general theorem and proof that the eigenvalues
>>>> of an operator or observable are diagonal in the probabilities.
>>>>
>>>>
>>> I am not sure I understand this.
>>>
>>>
>>>
>>>
>>> In fact this has some subtle issues with degeneracies.
>>>>
>>>>
>>>> Doesn't Gleason's theorem show that there is no other consistent way to
>>>> assign probabilities to subspaces of a Hilbert space?
>>>>
>>>> Brent
>>>>
>>>
>>> It is close. Gleason's theorem tells us that probabilities are a
>>> consequence of certain measurements. So for a basis Q = {q_n} then in a
>>> span in Q = P{q_n}, for P a projection operator that a measure μ(Q} is
>>> given by a trace over projection operators. This is close, but it does not
>>> address the issue of eigenvalues of an operator or observable. Gleason
>>> tried to make this work for operators, but was ultimately not able to.
>>>
>>>
>>> It should work for the projection operator, that this is the
>>> yes-no-experiment, but that extends to the other measurement, by reducing
>>> (as usual) the question “what is the value of A” into the (many) question
>>> “does A measurement belong to this interval” … Gleason’s theorem assures
>>> that the measure is unique (on the subspaces of H with dim bigger or equal
>>> to 3), so the Born rule should be determined, at least in non degenerate
>>> case (but also in the degenerate case when the degeneracy is due to tracing
>>> out a subsystem from a bigger system. I will verify later as my mind
>>> belongs more to the combinator and applicative algebra that QM for now.
>>>
>>>
>>>
>>>
>>> Many years ago I had an idea that since the trace of a density matrix
>>> may be thought of as constructed from projection operators with tr(ρ_n) =
>>> sum_n |c_n|^2P_n, that observables that commute with the density matrix
>>> might have a derived Born rule following Gleason. Further, maybe operators
>>> that do not commute then have some dual property that still upholds Born
>>> rule. I was not able to make this work.
>>>
>>>
>>> I will think about this. Normally the measure is determine by the
>>> “right" quantum logic, and the right quantum logic is determined by the any
>>> “provability” box accompanied by consistency condition (like []p & p, []p &
>>> <>t, …). The main difference to be expected, is that eventually we get a
>>> “quantum credibility measure”, not really a probability. It is like
>>> probability, except that credibility is between 0 and infinity (not 0 and
>>> 1).
>>>
>>> Bruno
>>>
>>>
>> I think I ran into the issue of why Gleason's theorem does not capture
>> the Born rule. Not all operators are commutative with the density matrix.
>> So if you construct the diagonal of the density matrix, or its trace
>> elements, with projector operators and off diagonal elements with left and
>> right acting projectors (left acting hit bra vectors and right acting hit
>> ket vectors) the problem is many operators are non-commutative. In
>> particular the usual situation is for the Hamiltonian to have nontrivial
>> commutation with the density matrix.
>>
>>
>>
>> It seems to me that Gleason theorem takes this into account. It only
>> means that the probabilities does not make the same partition of the
>> multiverse, but that is not a problem for someone who use physics to see if
>> it confirms or refute the “observable” available to the universal
>> numbers/machines in arithmetic.
>>
>
> Gleason's theorem applies for just one set of commuting operators,
>
>
>
> I am astonished by this. Are you sure you refer Gleason’s original work? I
> have seen many “simplified” proof, which sometimes add simplifying
> hypothesis.
>
> I’m afraid you will have to wait that I find the time to revise my proof
> of Gleason theorem ...
>
Gleason's theorem applies to the trace of states or a spectra that can be
given by the density matrix. I do not think it works in general for other
commuting sets of operators that do not commute with the density matrix. I
think for this reason Gleason's theorem is close to giving the Born rule,
but is not sufficiently general.
>
>
>
>
> and in particular those that commute with the density matrix. The Born
> rule holds for all operators, and especially the Hamiltonian that does not
> commute with the density matrix.
>
>
>>
>> I am not completely sure. You raise a doubt, and I’m afraid it will take
>> some time I come back to Gleason theorem. But I appreciate. My conversation
>> with Bruce and Brent makes me think that the notion of multiverse is far
>> from clear. At least with mechanism things are crystal clear! There is only
>> the sigma_1 sentences, and the nuances imposed by incompleteness for the
>> “Löbian number” who “lives” through them (them for the sigma_sentences,
>> which “realises” the computations).
>>
>
> I would not confuse the multiverse with this. There are several levels of
> multioverse. The first is just the world beyond what we can ever observe
> due to the cosmic horizon.
>
>
>
> If mechanism, that is only a sharable dream/video games played by numbers.
>
> That a tiny part of arithmetic realise all computation is entirely proved
> in Gödel 1931 already, except that Gödel missed the Church-Turing thesis,
> and so this will only be explicitly seen by Turing, Kleene, etc.
>
> But that is enough to doubt that “there is” a primary physical universe,
> and with Mechanism there is no choice: we have to retrieve physics from
> number (Turing universal) relations.
>
> Have you study my papers? I can explain this here if you are interested.
> To get the quanta, we can extrapolate relations from our observation, but
> to get both the quanta and the qualia, we need to extract the quanta from
> the Gödel-Löb-Solovay “true” modal logic of self-reference. It seems to
> work. Would it not work, we would get some empirical evidences that
> Mechanism (in cognitive science) is wrong. But up to now, thanks to QM, it
> seems that Mechanism fits very well. In fact QM without collapse is very
> close to what a solution of the mind-body problem should resemble if
> Mechanism is true.
>
>
>
I read a couple of short papers you wrote, I will have to confess this is
somewhat removed from my area of work, though 25 years ago I was somewhat
knowledgeable in this subject. I do have this idea of using incompletenss
of Diophantine equations as a route to showing how measurement as a sort of
Gödel loop has no formal description or in physics a dynamics. I would need
to find time to bend metal on this. This sounds different than your idea
which has mechanism as all. My sense is that mechanism is a sort of
causality chain that may be formalized as a type of computation. If
arithmetic is not formally complete, then from a physics perspective this
would seem to my mind to imply that self-reference leads to a strange
situation where a system can assume states or configurations according to
certain observables for no dynamical reason at all.
>
>
>
> The second is the vacuum pocket worlds in an inflationary de Sitter
> spacetime. A third may be how these are connected to anti-de Sitter
> spacetimes and how the landscape or swampland is generated. The fourth is
> the idea that many worlds interpretation is the grand or ultimate many
> worlds. This last one I would not take that seriously. Many worlds
> interpretation, as with all interpretations, is an addition to quantum
> mechanics that is less about physics and more about metaphysics.
>
>
> I disagree. Here I am OK with Deutsch. Quantum theory without collapse is
> automatically a “many-relative state theory”. I avoid the word “world”
> because that one *is* metaphysically charged.
>
> Anyway, elementary arithmetic is a many computations theory, too, without
> any added metaphysics. Then, what the machines perceive from inside
> arithmetic, taking into account the fact that they cannot distinguish their
> computation (of themselves) with a quasi-continuum of computations, we can
> extract the appearance of the physical reality, and its
> stability/persistence, from their sharable first person points of view.
>
> With mechanism, both matter and consciousness are explained entirely from
> just two equations:
>
> Kxy = x
> Sxyz = xy(yz)
>
> And three rules:
>
> If A = B and A = C then A = C
> If B = C then AB = AC
> If B = C then BA = CA
>
> Together with some definitions, motivated by the Mechanist hypothesis
> and/or Plato’s analysis of knowledge.
>
> We cannot add anything more. The extensionality axioms (like If AC = BC
> then A = B, equivalent with ([x](Ax) =A (x not occurring in A); not to be
> confused with the definition of elimination ([x]A)x) = x (true for all
> combination A) are already phenomenological.
>
>
>
But ..., as with Peano arithmetic this is incomplete.
I think many worlds as with Copenhagen are not really physical theories. I
think they are additional axioms, and in some ways the proliferation of
quantum interpretations seems to mirror the issue of completeness, where
you end up with a bouquet of axiomatic systems that are not consistent with
each other. The problems with all of them are evidence. The
classical-quantum dichotomy of Bohr's Copenhagen runs into trouble with
quantum gravitation, where quantum gravitation implies the quantization of
"everything." Many worlds runs into trouble with what is means by when
worlds split apart with a measurement, for with relativity there is no
global meaning to simultaneity. These are the two main interpretations, and
others such as Qubism and the Montevideo (with Penrose's related idea)
interpretation have problems.
LC
>
>
>
>>
>> Of course I come from the other side, but if mechanism is correct, I can
>> only cross physics when and where physics is correct. For now, physics is
>> not yet a solved problem, as GR does not fit with QM. The very notion of
>> “force” or “interaction” seems conceptually very different in GR and QM. We
>> can expect surprise, but with Mechanism, the quantum weirdness is welcomed,
>> and we are far from having any notion of physical space, and why 3D or 11D
>> or 26D. Mechanism is a 0 dimension theory of the mind, à la Plato, where
>> the ideas are the numbers i, and the partial recursive function phi_i, and
>> the operator phi_phi_i, etc.
>>
>
> Spacetime is likely emergent from quantum entanglements. Quantum
> entanglements are entirely nonlocal, so it seems strange that something
> that is local should be so defined. However the Einstein field equation
> R_{ab} - 1/2Rg_{ab} = T_{ab} has a curious duality about it. It says that
> high energy quantum gravity on the left is equal to low energy ordinary
> quantum fields. Further, the T_{ab} is for local quantum fields and these
> are dual to nonlocal physics as gravitation in the spacetime bulk.
>
>
>
> Very interesting and rather compelling. OK. But to solve the mind body
> problem, both space and time must be recovered from self-reference, itself
> deducible from the little theory above.
>
> Bruno
>
>
>
>
> LC
>
>
>>
>> Space, like in Kant, is a universal pattern of the universal machine,
>> although this is not yet proved, only suspected, as it could still be that
>> even space is “geographical” and that consciousness can survive without it.
>> Well, the theology of the numbers is in its infancy, if not still an
>> embryo: but the propositional parts is given by the two arithmetical
>> completeness theorem of Solovay, leading to G and G* describing all what
>> can be said on this. G gives the part that all sound machine can justify,
>> and G* gives the true, but non justifiable part. In between the rational
>> and the irrational there is a “surrational part”: what science can learn
>> from experience but never rationally justify.
>>
>> Bruno
>>
>>
>>
>>
>> LC
>>
>>
>>>
>>>
>>> LC
>>>
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