# Re: STEP 3

`On Sun, Aug 11, 2019 at 12:24 PM Bruno Marchal <marc...@ulb.ac.be> wrote:`
```
>
> On 10 Aug 2019, at 20:34, Jason Resch <jasonre...@gmail.com> wrote:
>
>
>
> On Fri, Aug 9, 2019 at 10:20 AM Bruno Marchal <marc...@ulb.ac.be> wrote:
>
>>
>> On 9 Aug 2019, at 13:09, Jason Resch <jasonre...@gmail.com> 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
>
>
>

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