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,
-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 |