Hi Bruno,
This is very exciting to me as it gives me some hope that the Stone
duality idea will work as it is envisioned. I found several papers that
discussed Scattered Topologies and the definition fits the idea that a dust is
the dual of a Boolean algebra. Such as:
www.jstor.org/stable/2274571
www.normalesup.org/~cornulier/CGPab.pdf
www.phil.uu.nl/preprints/preprints/PREPRINTS/preprint282.pdf
www.mrlonline.org/jot/2010-063-002/2010-063-002-012.pdf
My intension is to see if we can run Pratt's idea to get a way to model
dynamics in this way. Think of dust scattered in empty space as moving
randomly; how does the change in the relative distances and so forth of the
dusts dualize to transitions from one form of a Boolean algebra to another?
Pratt gives us a toy model that I need to generalize. He wrote in
http://boole.stanford.edu/pub/ratmech.pdf :
"When we unravel the primitive causal links contributing to secondary causal
interaction we find that two events, or two states, communicate with each other
by interrogating all entities of the opposite type. Thus event a deduces that
it precedes event b not by broaching the matter with b directly, but instead by
consulting the record of every state to see if there is any state volunteering a
counterexample. When none is found, the precedence is established. Conversely
when a Chu space is in state x and desires to pass to state y, it inquires as to
whether this would undo any event that has already occurred. If not then the
transition is allowed."
We then need to work out the kinds of transitions that could occur
following this method and the invariants under those transitions to recover the
symmetry groups that relate to those transitions and invariances. The eventual
goal is to show that all of the symmetry groups that we associate with physical
systems, even general covariance of General relativity, can be derived this
way. There is a lot of hard work to do.
BTW, I think that it might be a mistake to assume that it is necessary
to have a tensor product for the quantum orthoalgebras as discussed in the
quote below. I think that all that might be necessary is some finite
approximation to it over a large collection of "semi-separate" quantum systems.
My idea for that involves finding a way to define something like strata or
layers within which the notion of "separate systems" is valid to some finite
approximation. I will get back to this when we talk about the problem of matter
(and space!) that you mentioned in your reply to my paper post.
Onward!
Stephen P. King
-----Original Message-----
From: everything-list@googlegroups.com
[mailto:everything-l...@googlegroups.com] On Behalf Of Bruno Marchal
Sent: Thursday, September 30, 2010 12:53 PM
To: everything-list@googlegroups.com
Subject: Re: As to the problem of tensor products for quantum logic
Hi Stephen,
I see you mention the work of my friend Diederik Aerts. We discussed those
problems a very long time ago. Actually it was the difficulty of quantum logic
which motivated me in part for searching a non physical origin of the quantum,
and this leads to the intensional variants of self-reference (through UDA-like
reasoning). Now it is almost fun that the quantum logics of self-reference
(S4Grz1, X1*, Z1*) seems to have very similar problems, notably with the notion
of subsystem, composite systems, etc. This is due to the easy to prove
indeterminacy, non locality, and non clonability of 'matter' from the first
person views (assuming DM). (I recall that S4Grz1 = S4Grz + 1, and 1 is "p ->
Bp", the "1" is for Sigma_1. All true Sigma_1 sentences are provable by
Sigma_1 complete entities, and löbianity is characterized by being able to
prove that p->Bp for all Sigma_1 sentences.
As for the covering laws that you mention, I just cannot translate them in
arithmetic, despite dovetailing gives an intuitively simple way to encode
parallelism and interaction. This cannot be lifted easily on the "consistent
extensions".
Btw, I was wrong in a previous post about the Esakia logic wK4. It is not the
logic G in that paper, although Esakia studied relation between wK4 and G. wK4
is K + ((p & Bp) -> BBp), and the usual modal rules (modus ponens,
necessitation).
This can be used to prove completeness of G with respect to Cantor scattered
topological ordinals. But that's a technical result a bit beyond the topic, I
think.
- Bruno
On 27 Sep 2010, at 17:21, Stephen P. King wrote:
> Hi Folks,
>
> I have been researching the notion of tensor products in quantum
> logic and found the following for your consideration and comment:
>
> http://plato.stanford.edu/entries/qt-quantlog/
> ***
> 7. Composite Systems
> Some of the most puzzling features of quantum mechanics arise in
> connection with attempts to describe compound physical systems. It is
> in this context, for instance, that both the measurement problem and
> the non-locality results centered on Bell's theorem arise. It is
> interesting that coupled systems also present a challenge to the
> quantum-logical programme. I will conclude this article with a
> description of two results that show that the coupling of quantum-
> logical models tends to move us further from the realm of Hilbert
> space quantum mechanics.
> The Foulis-Randall Example
> A particularly striking result in this connection is the observation
> of Foulis and Randall [1981] that any reasonable (and reasonably
> general) tensor product of orthoalgebras will fail to preserve ortho-
> coherence. Let A5 denote the test space {{a,x,b}, {b,y,c}, {c,z,d},
> {d,w,e}, {e,v,s}} consisting of five three-outcome tests pasted
> together in a loop.
> This test space is by no means pathological; it is both ortho-
> coherent and algebraic. Moreover, it admits a separating set of
> dispersion-free states and hence, a classical interpretation. Now
> consider how we might model a compound system consisting of two
> separated sub-systems each modeled by A5. We would need to construct a
> test space B and a mapping ⊗ : X × X → Y = ∪B satisfying, minimally,
> the following;
> a. For all outcomes x, y, z ∈ X, if x⊥y, then x⊗z ⊥ y⊗z and
> z⊗x ⊥ z⊗y,
> b. For each pair of states α, β ∈ ω(A5), there exists at least
> one state ω on B such that ω(x⊗y) = α(x)β(y), for all outcomes x, y ∈
> X.
> Foulis and Randall show that no such embedding exists for which B is
> orthocoherent.
> Aerts' Theorem
> Another result having a somewhat similar force is that of Aerts
> [1982]. If L1 and L2 are two Piron lattices, Aerts constructs in a
> rather natural way a lattice L representing two separated systems,
> each modeled by one of the given lattices. Here “separated” means that
> each pure state of the larger system L is entirely determined by the
> states of the two component systems L1 and L2. Aerts then shows that L
> is again a Piron lattice iff at least one of the two factors L1 and L2
> is classical. (This result has recently been strengthened by Ischi
> [2000] in several ways.) The thrust of these no-go results is that
> straightforward constructions of plausible models for composite
> systems destroy regularity conditions (ortho-coherence in the case of
> the Foulis- Randall result, orthomodularity and the covering law in
> that of Aerts' result) that have widely been used to underwrite
> reconstructions of the usual quantum-mechanical formalism. This puts
> in doubt whether any of these conditions can be regarded as having the
> universality that the most optimistic version of Mackey's programme
> asks for. Of course, this does not rule out the possibility that these
> conditions may yet be motivated in the case of especially simple
> physical systems.
> ***
> What does this imply? Bruno wrote that " If *you* can demonstrate
> that arithmetical quantum logic have no tensor product enough
> "coherent" for allowing concurrency, then you have refuted DM." but
> given the results discussed here I think that there might be some way
> to wiggle out of this by attacking the notion of "separated".
>
>
> Onward!
>
> Stephen P. King
>
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