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 . 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: > > 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.
Brent --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---