Re: COMP, Quantum Logic and Gleason's Theorem

From: Günther Greindl <>
Date: Mon, 19 Jan 2009 22:55:58 +0100

Hi Bruno,

ok, I have not yet had the time to study modal logic (it is on my list,
but intermediate future). Thanks for the Goldblatt reference.

The paper is not online, but I found it in this book which is at our
University Library, maybe interesting also for other people:

Goldblatt, Mathematics of Modality

(the book contains the full paper)


Bruno Marchal wrote:
> On 16 Jan 2009, at 22:04, Günther Greindl wrote:
>> Hi all,
>> the question goes primarily to Bruno but all other input is
>> welcome :-))
>> Bruno, you said you have already arrived at a quantum logic in your
>> technical work?
> Yes. The hypostases, with p restrict to the Sigma-1 sentences (the
> UD) given by
> Bp & p (the knower certainty)
> Bp & Dp (the observer certainty)
> Bp & Dp & p (the "feeler" certainty), with B the Godel Beweisbar
> predicate, and Da = ~B~a.
> gives rise to Brouwersche like modal logics with natural quantization
> (BDp) which act like quantum projector, except that I loose the
> Brouwersche necessitation rule, which formally makes things more
> complex, more rich also.
>> May I refer to the following two paragraphs?:
>> We can read here:
>> The Reconstruction of QM
>> From the single premise that the “experimental propositions”
>> associated
>> with a physical system are encoded by projections in the way indicated
>> above, one can reconstruct the rest of the formal apparatus of quantum
>> mechanics. The first step, of course, is Gleason's theorem, which
>> tells
>> us that probability measures on L(H) correspond to density operators.
>> There remains to recover, e.g., the representation of “observables” by
>> self-adjoint operators, and the dynamics (unitary evolution). The
>> former
>> can be recovered with the help of the Spectral theorem and the latter
>> with the aid of a deep theorem of E. Wigner on the projective
>> representation of groups. See also R. Wright [1980]. A detailed
>> outline
>> of this reconstruction (which involves some distinctly non-trivial
>> mathematics) can be found in the book of Varadarajan [1985]. The point
>> to bear in mind is that, once the quantum-logical skeleton L(H) is in
>> place, the remaining statistical and dynamical apparatus of quantum
>> mechanics is essentially fixed. In this sense, then, quantum
>> mechanics —
>> or, at any rate, its mathematical framework — reduces to quantum logic
>> and its attendant probability theory.
> Very nice text. I agree, but it is a difficult matter. You can extract
> the quantum of 1 bit, but the quibit needs a good tensor product,
> which is not easy to derive (unless in ad hoc way) from quantum logic.
> With comp, I think we will need the first order extension of the
> "hypostases", and it could be that special feature of computability
> theory will need to be discovered to complete the derivation. In my
> 1991 paper I sum by saying that comp is in search of its Gleason
> theorem". A lot of work remains, of course.

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Received on Mon Jan 19 2009 - 17:00:15 PST

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