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 17-Sep-2001 10:16:26 -0300,2387;000000000001-00000015
Quantum cats? I assume we're not talking about Schrodinger's.... "Al Vilcius" <avilcius@webpearls.com> writes:
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?).
I got interested in n-categories precisely because I think they'll shed a lot of light on quantum gravity! Here are some papers where I try to explain why: Higher-dimensional algebra and topological quantum field theory, with James Dolan, Jour. Math. Phys. 36 (1995), 6073-6105. (Not available electronically, since it contains lots of hand-drawn pictures.) Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189. Available as http://xxx.lanl.gov/abs/q-alg/9609018 Higher-dimensional algebra and Planck-scale physics, in Physics Meets Philosophy at the Planck Scale, eds. Craig Callender and Nick Huggett, Cambridge U. Press, 2001. Available as http://xxx.lanl.gov/abs/gr-qc/9902017
From finite sets to Feynman diagrams, with James Dolan, in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Bj\"orn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50. Available as http://arXiv.org/abs/math.QA/0004133
2-categories are also fundamental to my work on spin foam models of quantum gravity, but I have done my best to keep that fact secret, to avoid scaring the physicists: An introduction to spin foam models of BF theory and quantum gravity, in Geometry and Quantum Physics, eds. Helmut Gausterer and Harald Grosse, Lecture Notes in Physics, Springer-Verlag, Berlin. Available as http://xxx.lanl.gov/abs/gr-qc/9905087 Best, John Baez 17-Sep-2001 15:59:36 -0300,7087;000000000001-00000016
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