quantum logic toposes
This is a topic entry on extensions^{} of standard and elementary toposes to quantum topoi founded upon many-valued logics.
Definition 0.1.
A quantum logic^{} topos (QLT) is defined as an extension of the concept of a topos in which the Heyting algebra or subobject classifier of the standard elementary topos is replaced by a quantum logic that is axiomatically defined by a non-commutative lattice^{} structure^{} such as that of a many valued, $L{M}_{n}$-logic algebra^{}, modified to a non-distributive lattice structure corresponding to that of the quantum physics events.
Remark 0.1.
Quantum logics topoi are thus generalizations^{} of the Birkhoff and von Neumann definition of quantum state spaces^{} based on their definition of a quantum logic (lattice), as well as a non-Abelian^{}, higher dimensional extension of the recently proposed concept of a ‘quantum’ topos which employs the (commutative^{}) Heyting logic algebra as a subobject classifier.
Some specific examples are considered in the following two recent references.
References
- 1 Butterfield, J. and C. J. Isham: 2001, space-time and the philosophical challenges of quantum gravity., in C. Callender and N. Hugget (eds. ) Physics Meets Philosophy at the Planck scale., Cambridge University Press,pp.33–89.
- 2 Butterfield, J. and C. J. Isham: 1998, 1999, 2000–2002, A topos perspective on the Kochen–Specker theorem I - IV, Int. J. Theor. Phys, 37 No 11., 2669–2733 38 No 3., 827–859, 39 No 6., 1413–1436, 41 No 4., 613–639.
Title | quantum logic toposes |
Canonical name | QuantumLogicToposes |
Date of creation | 2013-03-22 18:13:18 |
Last modified on | 2013-03-22 18:13:18 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 29 |
Author | bci1 (20947) |
Entry type | Topic |
Classification | msc 81P68 |
Classification | msc 18B25 |
Classification | msc 81P15 |
Classification | msc 81P10 |
Synonym | extended quantum topos |
Related topic | AxiomsOfTopoi |
Related topic | GeneralizedToposesTopoiWithManyValuedLogicSubobjectClassifiers |
Related topic | AlgebraicCategoryOfLMnLogicAlgebras |
Related topic | QuantumLogic |
Related topic | QuantumOperatorAlgebrasInQuantumFieldTheories |
Related topic | Topos |
Defines | quantum logics topoi |
Defines | quantum logics toposes |
Defines | toposes with quantum logics subobject classifiers |