sci.physics.research

Thus spake Gen Zhang
>This is a plea for the experts to refine my understand:
>
>As I understand it, the current state-of-the-art on reconstruction of
>QM from postulates, in the vein of von Neumann, has gotten so far:
>
>1. The fundamental measurement of a system, or possible information
>about a sytem, is a binary valued question.
>2. Each system has a finite number of such questions which maximally
>describe it.
>3. There are still more questions, the answers to which are not
>determined from a maximal description.
>
>I believe that it is a theorem that this gives an orthomodular lattice
>containing subsets which are boolean algebras. Now the next step is
>hazy: that it implies a representation as a Hilbert space with
>projectors. Words seem to be minced over this point. Is this really
>true? Is it only true if we understand "representation" to be some
>homomorphism to a vector space? If so, why a vector space?

I think you may find the mathematical relationship between a lattice of
propositions and Hilbert space explained here

Karl Svozil arXiv:quant-ph/9902042

If you are looking for a more intuitive kind of understanding, think of
vector addition as a form of weighted logical OR, and think of the
probability (the square of the norm) as a truth value.

>Now, we assume that it is a vector space (Piron suggests that a
>Hilbert space over R, C or H would do). Following Chris Fuchs, we
>introduce another postulate,
>
>4. The answer to a question is independent of the other questions
>asked -- non-contextuality.
>
>Then we recover the tensor product structure for combining systems
>together, POVM for measurements, density matrices and the Born rule
>for interpretation as probabilities. As a corollary, we exclude R and
>H for the field over which our Hilbert space is defined, as only C
>gives the tensor product rule uniquely; to be rigorous, we've only
>excluded H as it over-constrains the choice -- R would need some other
>postulate.
>
>More speculatively, we can recover unitary time evolution, if we're
>willing to accept:
>
>5. It is not possible to physically perform NP-complete problems in P
>time.
>
Usually unitary time evolution is required by conservation of
probabilities.


Regards

--
Charles Francis
moderator .
substitute charles for NotI to email