quantum logic topoi

Quantum logic topoi

Definition 1.1   A quantum logic 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 logicwhich is axiomatically defined by non-commutative 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.

Bibliography

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.



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