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:
> 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 . A detailed outline
> of this reconstruction (which involves some distinctly non-trivial
> mathematics) can be found in the book of Varadarajan . 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:
> 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.
> 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,
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.
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