On 17 Jan 2009, at 07:52, Brent Meeker wrote:
>
> G�nther 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,
>> G�nther
>>
> 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 (�-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