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
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to [email protected]
.
For more options, visit this group at http://groups.google.com/group/everything-list?hl=en
.
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to
[email protected].
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
