23 research outputs found
Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory (Extended Abstract)
The notion of equality between two observables will play many important roles
in foundations of quantum theory. However, the standard probabilistic
interpretation based on the conventional Born formula does not give the
probability of equality relation for a pair of arbitrary observables, since the
Born formula gives the probability distribution only for a commuting family of
observables. In this paper, quantum set theory developed by Takeuti and the
present author is used to systematically extend the probabilistic
interpretation of quantum theory to define the probability of equality relation
for a pair of arbitrary observables. Applications of this new interpretation to
measurement theory are discussed briefly.Comment: In Proceedings QPL 2014, arXiv:1412.810
Quantum Reality and Measurement: A Quantum Logical Approach
The recently established universal uncertainty principle revealed that two
nowhere commuting observables can be measured simultaneously in some state,
whereas they have no joint probability distribution in any state. Thus, one
measuring apparatus can simultaneously measure two observables that have no
simultaneous reality. In order to reconcile this discrepancy, an approach based
on quantum logic is proposed to establish the relation between quantum reality
and measurement. We provide a language speaking of values of observables
independent of measurement based on quantum logic and we construct in this
language the state-dependent notions of joint determinateness, value identity,
and simultaneous measurability. This naturally provides a contextual
interpretation, in which we can safely claim such a statement that one
measuring apparatus measures one observable in one context and simultaneously
it measures another nowhere commuting observable in another incompatible
context.Comment: 16 pages, Latex. Presented at the Conference "Quantum Theory:
Reconsideration of Foundations, 5 (QTRF5)," Vaxjo, Sweden, 15 June 2009. To
appear in Foundations of Physics
Orthomodular-Valued Models for Quantum Set Theory
In 1981, Takeuti introduced quantum set theory by constructing a model of set
theory based on quantum logic represented by the lattice of closed linear
subspaces of a Hilbert space in a manner analogous to Boolean-valued models of
set theory, and showed that appropriate counterparts of the axioms of
Zermelo-Fraenkel set theory with the axiom of choice (ZFC) hold in the model.
In this paper, we aim at unifying Takeuti's model with Boolean-valued models by
constructing models based on general complete orthomodular lattices, and
generalizing the transfer principle in Boolean-valued models, which asserts
that every theorem in ZFC set theory holds in the models, to a general form
holding in every orthomodular-valued model. One of the central problems in this
program is the well-known arbitrariness in choosing a binary operation for
implication. To clarify what properties are required to obtain the generalized
transfer principle, we introduce a class of binary operations extending the
implication on Boolean logic, called generalized implications, including even
non-polynomially definable operations. We study the properties of those
operations in detail and show that all of them admit the generalized transfer
principle. Moreover, we determine all the polynomially definable operations for
which the generalized transfer principle holds. This result allows us to
abandon the Sasaki arrow originally assumed for Takeuti's model and leads to a
much more flexible approach to quantum set theory.Comment: 25 pages, v2: to appear in Rev. Symb. Logic, v3: corrected typo
Towards a Paraconsistent Quantum Set Theory
In this paper, we will attempt to establish a connection between quantum set
theory, as developed by Ozawa, Takeuti and Titani, and topos quantum theory, as
developed by Isham, Butterfield and Doring, amongst others. Towards this end,
we will study algebraic valued set-theoretic structures whose truth values
correspond to the clopen subobjects of the spectral presheaf of an orthomodular
lattice of projections onto a given Hilbert space. In particular, we will
attempt to recreate, in these new structures, Takeuti's original isomorphism
between the set of all Dedekind real numbers in a suitably constructed model of
set theory and the set of all self adjoint operators on a chosen Hilbert space.Comment: In Proceedings QPL 2015, arXiv:1511.0118
Sheaf Logic, Quantum Set Theory and the Interpretation of Quantum Mechanics
Based on the Sheaf Logic approach to set theoretic forcing, a hierarchy of
Quantum Variable Sets is constructed which generalizes and simplifies the
analogous construction developed by Takeuti on boolean valued models of set
theory. Over this model two alternative proofs of Takeuti's correspondence,
between self adjoint operators and the real numbers of the model, are given.
This approach results to be more constructive showing a direct relation with
the Gelfand representation theorem, revealing also the importance of these
results with respect to the interpretation of Quantum Mechanics in close
connection with the Deutsch-Everett multiversal interpretation. Finally, it is
shown how in this context the notion of genericity and the corresponding
generic model theorem can help to explain the emergence of classicality also in
connection with the Deutsch- Everett perspective.Comment: 34 pages, 2 figure