quantum logic toposes

This is a topic entry on extensionsPlanetmathPlanetmathPlanetmath of standard and elementary toposes to quantum topoi founded upon many-valued logics.

Definition 0.1.

A quantum logicPlanetmathPlanetmath 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 latticeMathworldPlanetmath structureMathworldPlanetmath such as that of a many valued, LMn-logic algebraPlanetmathPlanetmath, modified to a non-distributive lattice structure corresponding to that of the quantum physics events.

Remark 0.1.

Quantum logics topoi are thus generalizationsPlanetmathPlanetmath of the Birkhoff and von Neumann definition of quantum state spacesPlanetmathPlanetmath based on their definition of a quantum logic (lattice), as well as a non-AbelianMathworldPlanetmathPlanetmath, higher dimensional extension of the recently proposed concept of a ‘quantum’ topos which employs the (commutativePlanetmathPlanetmath) Heyting logic algebra as a subobject classifier.

Some specific examples are considered in the following two recent 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