Big toy models: Representing physical systems as Chu spaces

We pursue a model-oriented rather than axiomatic approach to the foundations of Quantum Mechanics, with the idea that new models can often suggest new axioms. This approach has often been fruitful in Logic and Theoretical Computer Science. Rather than seeking to construct a simplified toy model, we...

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Bibliographic Details
Published in:Synthese (Dordrecht) 2012-06, Vol.186 (3), p.697-718
Main Author: Abramsky, Samson
Format: Article
Language:English
Subjects:
Category theory
Education
Epistemology
Functors
Hilbert spaces
Logic
Logical theorems
Mathematical theorems
Mathematical vectors
Metaphysics
Morphisms
Philosophy
Philosophy of Language
Philosophy of Science
Projective geometry
Quantum mechanics
Toy models
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Summary:We pursue a model-oriented rather than axiomatic approach to the foundations of Quantum Mechanics, with the idea that new models can often suggest new axioms. This approach has often been fruitful in Logic and Theoretical Computer Science. Rather than seeking to construct a simplified toy model, we aim for a 'big toy model', in which both quantum and classical systems can be faithfully represented—as well as, possibly, more exotic kinds of systems. To this end, we show how Chu spaces can be used to represent physical systems of various kinds. In particular, we show how quantum systems can be represented as Chu spaces over the unit interval in such a way that the Chu morphisms correspond exactly to the physically meaningful symmetries of the systems—the unitaries and antiunitaries. In this way we obtain a full and faithful functor from the groupoid of Hubert spaces and their symmetries to Chu spaces. We also consider whether it is possible to use a finite value set rather than the unit interval; we show that three values suffice, while the two standard possibilistic reductions to two values both fail to preserve fullness.
ISSN:0039-7857
1573-0964
DOI:10.1007/s11229-011-9912-x