> On 1 Mar 2019, at 01:42, Lawrence Crowell <[email protected]>
> 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 <[email protected]
>> <javascript:>> 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 <[email protected] <>>
>>> 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 ...
> 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.
> 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.
>
>
> 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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