On Sun, Aug 11, 2019 at 12:24 PM Bruno Marchal <[email protected]> wrote:
> > 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]> wrote: > >> >> On 9 Aug 2019, at 13:09, Jason Resch <[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 >> >> 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 > > 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 > > 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. > I am glad you find them interesting. Regarding you comment about the first one assuming too much, I just learned that Markus Muller put out a paper similar to Lucien Hardy's but without assuming the simplicity axiom: http://arxiv.org/abs/1004.1483 I haven't had a chance to go through it yet, I am doing so now. > > 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: > I know nothing of Braids nor Temperely-Lieb algebra. In doing some searching I came across this paper ( https://arxiv.org/abs/quant-ph/0601050 ) which claims to link the two with quantum phenomenon, including quantum computation. > > []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. > Very interesting. Thank you. Jason > > 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. > > > Bruno > > > -- 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 view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CA%2BBCJUhL6we1S5Uiq9x_hJAN3PCih_M5-HZBTqgUEDav_1rMcQ%40mail.gmail.com.
