# PlanetPhysics/Quantum Logic Topoi

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## Quantum logic topoi

A \htmladdnormallink{quantum logic {http://planetphysics.us/encyclopedia/TheoryOfHilbertLattices.html} topos} (QLT ) is defined as an extension of the concept of topos in which the Heyting logic algebra (or subobject classifier) of the standard elementary topos is replaced by a quantum logic which is axiomatically defined by \htmladdnormallink{non-commutative {http://planetphysics.us/encyclopedia/AbelianCategory3.html} and non-distributive} lattice structures.

### Remark

Quantum logic 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.