quantum logic toposes
This is a topic entry on extensions of standard and elementary toposes toquantum 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 quantumphysics 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 thephilosophical challenges of quantum gravity., in C. Callender andN. Hugget (eds. ) Physics Meets Philosophy at the Planckscale., Cambridge University Press,pp.33–89.
- 2 Butterfield, J. and C. J. Isham: 1998, 1999, 2000–2002, A toposperspective 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.