categories: quantum cats

["Al Vilcius" <avilcius@webpearls.com>]

Can topos theory be used to enhance (rework?) our models of state space
and its dynamics in quantum theory?

There are two approaches I am aware of, specifically:

1) M. Adelman and J.V. Corbett. "A Sheaf Model for Intuitionistic 
Quantum
Mechanics" Applied Categorical Structures. (1995)(3)(1) ref:
ftp://ftp.mpce.mq.edu.au/pub/maths/murray
and other related papers.

2) C.J. Isham and J.Butterfield, "Some Possible Roles for Topos Theory 
in
Quantum Theory and Quantum Gravity" ref:
http://xxx.lanl.gov/abs/gr-qc/9910005
and related papers.

Plus there is of course the fantastic n-category approach of John Baez
 http://www.math.ucr.edu/home/baez/
but I have no idea yet of how this relates to QM (any hints?).

Has anyone done a comparison of these approaches ?, perhaps relating
them to the idea of pasting together Boolean algebras in some "partial"
structures, as outlined briefly in "Charting the labyrinth of quantum 
logics"
by Hardegree / Frazer (1979).

An intriguing picture (to me) that includes state space S is
a ----> S -----> c
where the (contravarient "over" S) left side represents actions, shapes
(points), constructions
and the (covarient "under" S) right side represents observations,
destructions, attributes
in an algebra/co-algebra framework.
(there should also be an endo-arrow on S for dynamics, I guess,
but I could not draw it here)
Has anyone studied algebra/co-algebra models for QM?

My interest is strictly personal (ie. I have no organized program)
and actually arose from my attempts to understand what is being
called quantum computing.  I was compelled to dig deeper because
I just cannot come to grips with "irreducible uncertainty", or more 
precisely:
epistemological vs. ontological uncertainty (as differentiated by David 
Cohen '89),
and why it is that we should model measurement using a (classical) 
continuum
of real numbers (pointing to an SDG alternative perhaps).

Your gentle guidance is appreciated.
Al Vilcius