# COMP, Quantum Logic and Gleason's Theorem

```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?:

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
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.

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

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