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quantum topos
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(Definition)
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Preliminary Data.
There are several distinct definitions of quantum topos in the Mathematical Physics literature attempting to redefine the quantum logic that was first introduced by von Neumann and Birkhoff for the foundation of Quantum Mechanics. The definitions of quantum topoi published so far are not, however, those of `quantum' categories (previously introduced as rigid monoidal categories) - with finite limits and power objects.
Definition 0.3 A Heyting algebra is a p-algebra (as defined next in Definition 1.3 ) with the relative pseudocomplentation operation $\to$ (which replaces the propositional implication $\Rightarrow$ ).
Given an element $a$ in a bounded lattice $L$ , a complement of $a$ is defined to be an element $b\in L$ , if such an element exists, such that $$a\wedge b=0,\qquad{ and }\qquad a\vee b =1.$$
To surmount the non-uniqueness of the complement, an alternative to the latter was defined- the pseudocomplement of an element.
An element $b$ in a lattice $L$ with $0$ is a pseudocomplement of $a\in L$ if
- $b\wedge a=0$
- for any $c$ such that $c\wedge a=0$ then $c\le b$ .
In other words, $b$ is the maximal element in the set $\lbrace c\in L\mid c\wedge a=0\rbrace$ .
Definition 0.4 A convenient modification of the pseudocomplemented (pc) lattice concept is a p-algebra (or pseudocomplemented algebra) which is a pc-lattice where $^*$ is regarded as an algebraic operator. Thus, a morphism of pc-lattices is a proper lattice homomorphism, whereas a morphism between two p-algebras is a lattice homomorphism $f$ that also preserves the pc-algebraic operation $^*$ , i.e., $f(a^*)= f(a)^*$ . One can therefore define a category of p-algebras by specifying the morphism between any pair of p-algebras (considered as objects of this algebraic logic category) as the $\lbrace 0,1\rbrace$ -lattice homomorphism, with the following condition $f(1)=f(0^*)=f(0)^*=0^*=1$ being also satisfied.
Remark Unlike the Heyting lattice, an $LM_n$ -logic algebra has a non-commutative lattice structure and is therefore considered as a stronger candidate for quantum logics, including those based on the orthomodular lattices of the original quantum logic of Birkhoff and von Neumann. Thus, a generalized topos defined with a subobject classifier based on $LM_n$ -logic algebra may provide suitable representations of arbitrary quantum state spaces.
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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.
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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"quantum topos" is owned by bci1.
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See Also: non-commutative structure, commutative, quantum category, Heyting algebra, quantum logic, categorical dynamics, lattice, intuitionistic logic
| Other names: |
quantum category |
| Also defines: |
quantum state space |
| Keywords: |
quantum state spaces, commutative lattice, subobject classifier, pc-lattice, category of pc-lattices, quantum topoi, quantum logics, Heyting logic algebra, Heyting lattice, quantum categories, quantum spaces, quantum system, the fundamental concept of quantum theories, QuantumFundamentalGroupoid, QuantumGroupoids, QuantumGroups, QuantumSystem, Quantum Electrodynamics, Hamiltonian Operator, QCD or QuantumChromodynamics |
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Cross-references: subobject classifier, stronger, non-commutative, homomorphism, objects, preserves, lattice homomorphism, morphism, algebraic, modification, maximal element, pseudocomplement, complement, bounded lattice, implication, operation, p-algebra, Boolean, propositional logic, operator, material implication, relatively pseudocomplemented, element, lattice, logic, section, pseudocomplemented lattice, Brouwer logic, postulate, orthomodular lattice, variance, structure, Heyting lattice, commutative, Heyting algebra, algebra, category, categorical, differences, subobject, logic algebra, quantum state spaces, representation, power objects, limits, finite, monoidal categories, rigid, topoi, foundation, quantum logic, definitions
There are 9 references to this entry.
This is version 22 of quantum topos, born on 2008-08-03, modified 2008-10-25.
Object id is 10913, canonical name is QuantumTopos.
Accessed 1276 times total.
Classification:
| AMS MSC: | 81Q05 (Quantum theory :: General mathematical topics and methods in quantum theory :: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations) | | | 81P05 (Quantum theory :: Axiomatics, foundations, philosophy :: General and philosophical) | | | 81-00 (Quantum theory :: General reference works ) | | | 55U99 (Algebraic topology :: Applied homological algebra and category theory :: Miscellaneous) | | | 18-00 (Category theory; homological algebra :: General reference works ) | | | 18D25 (Category theory; homological algebra :: Categories with structure :: Strong functors, strong adjunctions) |
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Pending Errata and Addenda
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