 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 Valuations for the quantum propositional structures...
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Valuations for the quantum propositional structures and hidden variables for quantum mechanics Chernavska, Ariadna
Abstract
The thesis investigates the possibility of a classical semantics for quantum propositional structures. A classical semantics is defined as a set of mappings each of which is (i) bivalent, i.e., the value 1 (true) or 0 (false) is assigned to each proposition, and (ii) truthfunctional, i.e., the logical operations are preserved. In addition, this set must be "full", i.e., any pair of distinct propositions is assigned different values by some mapping in the set. When the propositions make assertions about the properties of classical or of quantum systems, the mappings should also be (iii) "stateinduced", i.e., values assigned by the semantics should accord with values assigned by classical or by quantum mechanics. In classical propositional logic, (equivalence classes of) propositions form a Boolean algebra, and each semantic mapping assigns the value 1 to the members of a certain subset of the algebra, namely, an ultrafilter, and assigns 0 to the members of the dual ultraideal, where the union of these two subsets is the entire algebra. The propositional structures of classical mechanics are likewise Boolean algebras, so one can straightforwardly provide a classical semantics, which also satisfies (iii). However, quantum propositional structures are nonBoolean, so it is an open question whether a semantics satisfying (i), (ii) and (iii) can be provided. Von Neumann first proposed (1932) that the algebraic structures of the subspaces (or projectors) of Hilbert space be regarded as the propositional structures PQM of quantum mechanics. These structures have been formalized in two ways: as orthomodular lattices which have the binary operations "and", "or", defined among all elements, compatible [symbol omitted] and incompatible [symbol omitted]; and as partialBoolean algebras which have the binary operations defined among only compatible elements. In the thesis, two basic senses in which these structures are nonBoolean are discriminated. And two notions of truthfunctionality are distinguished: truthfunctionality [symbols omitted] applicable to the PQM lattices; and truthfunctionality [symbol omitted] applicable to both the lattices and partialBoolean algebras. Then it is shown how the lattice definitions of "and", "or", among incompatibles rule out a bivalent, truthfunctional [symbols omitted] semantics for any lattice containing incompatible, elements. In contrast, the Gleason and KochenSpecker proofs of the impossibility of hiddenvariables for quantum mechanics show the impossibility of a bivalent, truthfunctional [symbol omitted] semantics for threeorhigher dimensional Hilbert space structures; and the presence of incompatible elements is necessary but is not sufficient to rule out such a semantics. As for (iii), each quantum stateinduced expectationfunction on a PQM truthfunctionally assigns 1 and 0 values to the elements in a ultrafilter and dual ultraideal of PQM³ where, in general the union of an ultrafilter and its dual ultraideal is smaller than the entire structure. Thus it is argued that each expectationfunction is the quantum analog of a classical semantic mapping, even though the domain where each expectationfunction is bivalent and truthfunctional is usually a nonBoolean substructure of PQM. The final portion of the thesis surveys proposals for the introduction of hidden variables into quantum mechanics, proofs of the impossibility of such hiddenvariable proposals, and criticisms of these impossibility proofs. And arguments in favour of the partialBoolean algebra, rather than the orthomodular lattice, formalization of the quantum propositional structures are reviewed.
Item Metadata
Title 
Valuations for the quantum propositional structures and hidden variables for quantum mechanics

Creator  
Publisher 
University of British Columbia

Date Issued 
1980

Description 
The thesis investigates the possibility of a classical semantics for quantum propositional structures. A classical semantics is defined as a set of mappings each of which is (i) bivalent, i.e., the value 1 (true) or 0 (false) is assigned to each proposition, and (ii) truthfunctional, i.e., the logical operations are preserved. In addition, this set must be "full", i.e., any pair of distinct propositions is assigned different values by some mapping in the set. When the propositions make assertions about the properties of classical or of quantum systems, the mappings should also be (iii) "stateinduced", i.e., values assigned by the semantics should accord with values assigned by classical or by quantum mechanics. In classical propositional logic, (equivalence classes of) propositions form a Boolean algebra, and each semantic mapping assigns the value 1 to the members of a certain subset of the algebra, namely, an ultrafilter, and assigns 0 to the members of the dual ultraideal, where the union of these two subsets is the entire algebra. The propositional structures of classical mechanics are likewise Boolean algebras, so one can straightforwardly provide a classical semantics, which also satisfies (iii). However, quantum propositional structures are nonBoolean, so it is an open question whether a semantics satisfying (i), (ii) and (iii) can be provided.
Von Neumann first proposed (1932) that the algebraic structures of the subspaces (or projectors) of Hilbert space be regarded as the propositional structures PQM of quantum mechanics. These structures have been formalized in two ways: as orthomodular lattices which have the binary operations "and", "or", defined among all elements, compatible [symbol omitted] and incompatible [symbol omitted]; and as partialBoolean algebras which have the binary operations defined among only compatible elements. In the thesis, two basic senses in which these structures are nonBoolean are discriminated. And two notions of truthfunctionality are distinguished: truthfunctionality [symbols omitted] applicable to the PQM lattices; and truthfunctionality [symbol omitted] applicable to both the lattices and partialBoolean algebras. Then it is shown how the lattice definitions of "and", "or", among incompatibles rule out a bivalent, truthfunctional [symbols omitted] semantics for any lattice containing incompatible, elements. In contrast, the Gleason and KochenSpecker proofs of the impossibility of hiddenvariables for quantum mechanics show the impossibility of a bivalent, truthfunctional [symbol omitted] semantics for threeorhigher dimensional Hilbert space structures; and the presence of incompatible elements is necessary but is not sufficient to rule out such a semantics. As for (iii), each quantum stateinduced expectationfunction on a PQM truthfunctionally assigns 1 and 0 values to the elements in a ultrafilter and dual ultraideal of PQM³ where, in general the union of an ultrafilter and its dual ultraideal is smaller than the entire structure. Thus it is argued that each expectationfunction is the quantum analog of a classical semantic mapping, even though the domain where each expectationfunction is bivalent and truthfunctional is usually a nonBoolean substructure of PQM. The final portion of the thesis surveys proposals for the introduction of hidden variables into quantum mechanics, proofs of the impossibility of such hiddenvariable proposals, and criticisms of these impossibility proofs. And arguments in favour of the partialBoolean algebra, rather than the orthomodular lattice, formalization of the quantum propositional structures are reviewed.

Genre  
Type  
Language 
eng

Date Available 
20100323

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0095142

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.