On Friday, March 1, 2019 at 8:49:54 AM UTC-6, Bruno Marchal wrote: > > > On 1 Mar 2019, at 01:42, Lawrence Crowell <[email protected] > <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 <[email protected]> >> 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 > > > I still think this semantics of qualia is ultimately "joy eliminativism" (a great term introduced here by someone before).
George Eliot [(link: https://books.google.com/books?id=NskZTEr8Td8C&pg=PA520&lpg=PA520) books.google.com/books?id=NskZT… <https://t.co/ZgU6zMLiiF?amp=1>] [image: Image] <https://twitter.com/philipthrift/status/1101565464373805062/photo/1> 1:30 PM · Mar 1, 2019 · Twitter Web App <https://help.twitter.com/using-twitter/how-to-tweet#source-labels> - pt -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
