Re: COMP, Quantum Logic and Gleason's Theorem

From: Bruno Marchal <>
Date: Sat, 17 Jan 2009 18:46:12 +0100

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.

> And here we read:
> Quantum logic treats quantum events (or measurement outcomes) as
> logical
> propositions, and studies the relationships and structures formed by
> these events, with specific emphasis on quantum measurement. More
> formally, a quantum logic is a set of events that is closed under a
> countable disjunction of countably many mutually exclusive events. The
> representation theorem in quantum logic shows that these logics form a
> lattice which is isomorphic to the lattice of subspaces of a vector
> space with a scalar product.
> It remains an open problem in quantum logic to prove that the field K
> over which the vector space is defined, is either the real numbers,
> complex numbers, or the quaternions. This is a necessary result for
> Gleason's theorem to be applicable, since in all these cases we know
> that the definition of the inner product of a non-zero vector with
> itself will satisfy the requirements to make the vector space in
> question a Hilbert space.
> Application
> The representation theorem allows us to treat quantum events as a
> lattice L = L(H) of subspaces of a real or complex Hilbert space.
> Gleason's theorem allows us to assign probabilities to these events.
> So I wonder - how much are you still missing to construct QM out of
> the
> logical results you have arrived at?

I have the formal systems. In a sense, nothing is missing. Except
enough competent and interested people in those weird self-referential
logics. It is a sequence of open math problems. It is normal. When the
research is driven by high level question, you don't choose the
mathematical objects you have to handle. You discover them.

I could later give more explanation, but here we are at the end of the
AUDA (!). It would be too much technical right now.
you can take a look at Goldblatt 1974, one or the clearest paper on
the Brouwersche Modal "quantum" logic.



You received this message because you are subscribed to the Google Groups "Everything List" group.
To post to this group, send email to
To unsubscribe from this group, send email to
For more options, visit this group at
Received on Sat Jan 17 2009 - 12:46:19 PST

This archive was generated by hypermail 2.3.0 : Fri Feb 16 2018 - 13:20:15 PST