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单词 QuantumLogicToposes
释义

quantum logic toposes


This is a topic entry on extensionsPlanetmathPlanetmathPlanetmath of standard and elementary toposes toquantum 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 quantumphysics 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.

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.
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