> On 10 Aug 2019, at 20:34, Jason Resch <[email protected]> wrote: > > > > On Fri, Aug 9, 2019 at 10:20 AM Bruno Marchal <[email protected] > <mailto:[email protected]>> wrote: > >> On 9 Aug 2019, at 13:09, Jason Resch <[email protected] >> <mailto:[email protected]>> wrote: >> >> <snip> >>> >>> Bruno, >>> >>> Forgive me if I have asked this before, but can you elaborate on the >>> how/why the math suggests negative interference? >>> >>> I currently have no intuition for why this should be. >>> >>> I recall reading something on continuous probability as being more natural >>> and leading to something much like the probability formulas in quantum >>> mechanics. Is that related? >> >> >> It is not intuitive at all. With the UDA, we can have have the intuition >> coming from the first person indeterminacy on all all computational >> continuation in arithmetic, but in the AUDA (the Arithmetical UDA), the >> probabilities are constrained by the logic of self-reference G and G*. So >> the reason why we can hope for negative amplitude of probability comes from >> the fact that modal variant of the first person on the (halting) >> computations, which is given by the arithmetical interpretation of: >> >> []p & p >> >> or >> >> []p & <>t >> >> or >> >> []p & <>t & p >> >> With, as usual, [] = Beweisbar, and p is an arbitrary sigma_1 sentences >> (partial computable formula). >> >> They all give a quantum logic enough close to Dalla Chiara’s presentation of >> them, to have the quantum features like complimentary observable, and what I >> have called a sort of abstract linear evolution build on a highly >> symmetrical core (than to LASE: the little Schroeder equation: p -> []<>p, >> which provides a quantisation of the sigma_1 arithmetical reality. >> >> It is mainly the presence of this quantisation which justify that the >> probabilities behave in a quantum non boolean way, but this is hard to >> verify because the nesting of boxes in the G* translation makes those >> formula … well, probably in need of a quantum computer to be evaluated. But >> normally, if mechanism (and QM) are correct this should work. >> >> This is explained with more detail in “Conscience et Mécanisme”. >> >> Bruno >> >> >> Thank you Bruno for your explanation and references. > > Y’re welcome. > > >> Regarding “Conscience et Mécanisme”, is there a web/html or English version >> available? Unfortunately my browser cannot do translations of PDFs but can >> translate web pages. If not don't worry, I can copy and paste into a >> translator. > > Yes, There is no HTML page for the long text. But you can consult also my > paper: > > Marchal B. The Universal Numbers. From Biology to Physics, Progress in > Biophysics and Molecular Biology, 2015, Vol. 119, Issue 3, 368-381. > https://www.ncbi.nlm.nih.gov/pubmed/26140993 > <https://www.ncbi.nlm.nih.gov/pubmed/26140993> > > You will still need some background in quantum logic, like the paper by > Goldblatt which makes the link between minimal quantum logic and the B modal > logic. > > There is also a paper by Rawling and Selesnick which shows how to build a > quantum NOT gate, from the Kripke semantics of the B logic. It is not > entirely clear if this can be used in arithmetic, because we loss the > necessitation rule in “our” B logic. Open problem. A positive solution on > this would be a great step toward an explanation that the universal machine > has necessarily a quantum structure and can exploit the “parallel > computations in arithmetic” in the limit of the 1p indeterminacy.. > > Rawling JP and Selesnick SA, 2000, Orthologic and Quantum Logic: Models and > Computational Elements, Journal of the ACM, Vol. 47, n° 4, pp. 721-T51. > > Ask question, online or here. It *is* rather technical at some point. > > Bruno > > > > I've been reading those references, and have found a few more which might be > related and of interest. Effectively, they provide arguments for the quantum > probability theory based on the requirement for continuous reversible > operations, or the juxtaposition between finite information-carry capacity > and smoothness. > > > Lucien Hardy's "Quantum Theory From Five Reasonable Axioms" > https://arxiv.org/abs/quant-ph/0101012 > <https://arxiv.org/abs/quant-ph/0101012> > > The usual formulation of quantum theory is based on rather obscure axioms > (employing complex Hilbert spaces, Hermitean operators, and the trace rule > for calculating probabilities). In this paper it is shown that quantum theory > can be derived from five very reasonable axioms. The first four of these are > obviously consistent with both quantum theory and classical probability > theory. Axiom 5 (which requires that there exists continuous reversible > transformations between pure states) rules out classical probability theory. > If Axiom 5 (or even just the word "continuous" from Axiom 5) is dropped then > we obtain classical probability theory instead. This work provides some > insight into the reasons quantum theory is the way it is. For example, it > explains the need for complex numbers and where the trace formula comes from. > We also gain insight into the relationship between quantum theory and > classical probability theory. > > and Jochen Rau's "On quantum vs. classical probability" > https://arxiv.org/abs/0710.2119v2 <https://arxiv.org/abs/0710.2119v2> > > The key (and novel) technical result, on the other hand, will pertain to the > second objective: I will show that the single distinguishing property of > quantum theory is the juxtaposition of finite information-carrying capacity > and smoothness, where the concept of smoothness will be carefully defined and > motivated. The mathematical derivation of this result will involve close > inspection of the symmetry group, with successive constraints leading > unequivocally to the unitary group of transformations in complex Hilbert > space. As for the final objective, I will provide arguments why there is > likely no further probabilistic theory that satisfies basic physical > desiderata.
Interesting papers, but I agree with the second that the first assume too much, from the continuum, the states, the tensorial structure, etc. Then both assumes more or less explicitly some physical reality, and are unaware of the need to derive it from the “universal machine’s consciousness theory”, if relevant for relating coherently the quale logic with the quantum logic. Such paper gives hope for making easier the last step of the derivation of physics from arithmetic though. I did not know the second one, which seems very interesting, but I read it only very quickly. It is has lady in common the necessity of the continuum, some quantum logic which could not be expanded for physics (but perhaps for “psychology”!). > > Would you say these properties are inherent in the computations of the UD? As far as they are relevant to the correct physics, those properties have to be derived from the right mixture of the 3p structures on all computations, or the UD*, and the relative first person (plural) indeterminacy for the average universal numbers with respect to all universal numbers running them. Yes, that has to be the case … as far as both Mechanism in the cognitive science, and the Quantum principles (Hilbert Space, or von Neuman Algebra). I might appreciate also to derive the unitary group from few principles. I suspect braids and Temperely-Lieb algebra, coming from the grade strcuture of the material modes: []p & <>t (&p) becoming []^n p & <>^m t with n < m Which gives different but related quantum logic. Some sorts of dualities between the quantisations []<>p and its dual <>[]p should “braid" the “material mode” and I suspect space and time, or space-time, to start from this, or similar. The infinities of universal systems under “our” substitution level might be a universal topological braiding, a sort of universal quantum dovetailer. > In so far as any computational thread representing an observer or a system > the observer interacts with is finite in its information carrying capacity, > but all the threads of similar indistinguishable computations for a continuum? Right. > Is there a reason to suppose operations are reversible (could this be due > to some conservation of information principal in non-halting programs?). We can cheat, and say that as Mechanism imposes the existence of a measure, we impose symmetry (and continuity) to have a nice rich group structure with know rich Measure theory (and then compact Lie groups + exceptional structure) can pave the way. I can only pray of this to happen, but the material mode suggest this makes sense by showing that the (true) sigma_1 sentences do impose symmetry at the bottom, as the three first person (plural) modes imposes the "Brouwersche axiom of symmetry”: p -> []<>p (when you get p, you can get p back from any world in the neighbourhood. That introduces symmetry, a notion of perpendicularity, a proximity relation of the type of a scalar product, if not necessarily its square. With the combinators I like to sum up physics by “No Kestrels! No Starling!”. We cannot eliminate things/information, and we cannot duplicate things/information … at the bottom. The core of the physical reality is a BCI-algebra (Bxyz = x(yz), Cxyz = xzy, Ix = x). You can compose/apply things and permute them, at the bottom. Note that such a “bottom” is not Turing universal, but the relative breaking of the symmetries are brought by what needed to be added here, which is easy for the mind (add just K and S!), but hard for the physical (why a tensor, why space-time waves/strings, why vertex operator, etc.). > > Is the appearance of complex numbers in the quantum probability sufficient to > get interference? Embed the real line in the plane, then a multiplication of numbers, or of a couple of numbers, by -1, becomes a rotation of 180°, so to get (-1) = i^2, a rotation of 90° provides a natural interpretation, and 1 and i becomes perpendicular, which is is the key notion in the type of probabilities we could hope to make sense in physics. a+ bi = re^it = cos(t) + i*sin(t), t real, the complex numbers are just little waves at the start, they interfere all the time, so to speak, it is more the interference which suggest the use of the complex numbers, then, crazily enough, nature seems to be “complex” (wave like) at the bottom. Maybe this is due to the fact that the first order theory of the real is not Turing universal, but the first order theory of the complex numbers is! (A wave is a continuum trick to get the natural numbers, as you can define the numbers by where the sinus get null (up to even multiple of pi)). The limit on the first person indeterminacy on all computations, is expected to be Turing universal and continuous, that might be the simplest reason. Note that the parallel worlds are given by perpendicular states. They should be called the perpendicular universes. Once two “universes/histories" are not perpendicular they can interfere “statistically”, and they are inter-reachable “probabilistically” through appropriate measurements/interactions. That imposes also some symmetries. 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