Topos Models for Physics and Topos Theory
arXiv:1309.5640 · doi:10.1063/1.4892100
Abstract
What is the role of topos theory in the topos models for quantum theory as used by Isham, Butterfield, Doring, Heunen, Landsman, Spitters and others? In other words, what is the interplay between physical motivation for the models and the mathematical framework used in these models? Concretely, we show that the presheaf topos model of Butterfield, Isham and Doring resembles classical physics when viewed from the internal language of the presheaf topos, similar to the copresheaf topos model of Heunen, Landsman and Spitters. Both the presheaf and copresheaf models provide a `quantum logic' in the form of a complete Heyting algebra. Although these algebras are natural from a topos theoretic stance, we seek a physical interpretation for the logical operations. Finally, we investigate dynamics. In particular we describe how an automorphism on the operator algebra induces a homeomorphism (or isomorphism of locales) on the associated state spaces of the topos models, and how elementary propositions and truth values transform under the action of this homeomorphism. Also with dynamics the focus is on the internal perspective of the topos.
54 pages, 0 figures
References in corpus (10)
- A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
- A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
- A Topos Foundation for Theories of Physics: IV. Categories of Systems
- A topos for algebraic quantum theory
- A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows
- Intuitionistic quantum logic of an n-level system
- Classical and Quantum Probabilities as Truth Values
- Flows on Generalised Gelfand Spectra of Nonabelian Unital C*-Algebras and Time Evolution of Quantum Systems
- Generalised Gelfand Spectra of Nonabelian Unital C*-Algebras
- Bohrification of local nets of observables