On 17 Jan 2009, at 07:52, Brent Meeker wrote:
>
> 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?
>>
>> May I refer to the following two paragraphs?:
>>
>> We can read here:
>> http://plato.stanford.edu/entries/qt-quantlog/
>>
>> 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.
>>
>>
>> And here we read:
>>
>> http://en.wikipedia.org/wiki/Gleason%27s_theorem
>>
>> 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.
>>
>>
>> END QUOTE
>>
>> So I wonder - how much are you still missing to construct QM out of
>> the
>> logical results you have arrived at?
>>
>> Best Wishes,
>> Günther
>>
> I don't think this form of QM is consistent with Bruno's ideas.
> Quantum
> logic takes the projection operation as be fundamental which is
> inconsistent with unitary evolution and the MWI.
But in QM the unitary evolution gives a third person point of view.
UDA shows (or is supposed to show) that Physics is first person
(plural). A logic of projection is interesting for just that reason.
Quantum logic and many world/dream are related by a relation akin to
the difference between a ket Ia>, and a projection on that ket Ia><aI.
The relation of proximity on the worlds is the anti-relation of
perpendicularity among the states (this transform Kripke semantics of
quantum logic into Kripke semantics of the Brouwersche modal logic).
I know some have used QL to "solve" (or hide) the conceptual problems
of QM, like if QL could evacuate the many worlds, but this is not the
case. The modal (à-la-Goldblatt) view of QL invites the many
"alternate" realties.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Sat Jan 17 2009 - 12:58:37 PST