# 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, $LM_{n}$-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