3,714 research outputs found
A presentation of Quantum Logic based on an "and then" connective
When a physicist performs a quantic measurement, new information about the
system at hand is gathered. This paper studies the logical properties of how
this new information is combined with previous information. It presents Quantum
Logic as a propositional logic under two connectives: negation and the "and
then" operation that combines old and new information. The "and then"
connective is neither commutative nor associative. Many properties of this
logic are exhibited, and some small elegant subset is shown to imply all the
properties considered. No independence or completeness result is claimed.
Classical physical systems are exactly characterized by the commutativity, the
associativity, or the monotonicity of the "and then" connective. Entailment is
defined in this logic and can be proved to be a partial order. In orthomodular
lattices, the operation proposed by Finch (1969) satisfies all the properties
studied in this paper. All properties satisfied by Finch's operation in modular
lattices are valid in Hilbert Space Quantum Logic. It is not known whether all
properties of Hilbert Space Quantum Logic are satisfied by Finch's operation in
modular lattices. Non-commutative, non-associative algebraic structures
generalizing Boolean algebras are defined, ideals are characterized and a
homomorphism theorem is proved.Comment: 28 pages. Submitte
An alternative Gospel of structure: order, composition, processes
We survey some basic mathematical structures, which arguably are more
primitive than the structures taught at school. These structures are orders,
with or without composition, and (symmetric) monoidal categories. We list
several `real life' incarnations of each of these. This paper also serves as an
introduction to these structures and their current and potentially future uses
in linguistics, physics and knowledge representation.Comment: Introductory chapter to C. Heunen, M. Sadrzadeh, and E. Grefenstette.
Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse.
Oxford University Press, 201
Algebras of Measurements: the logical structure of Quantum Mechanics
In Quantum Physics, a measurement is represented by a projection on some
closed subspace of a Hilbert space. We study algebras of operators that
abstract from the algebra of projections on closed subspaces of a Hilbert
space. The properties of such operators are justified on epistemological
grounds. Commutation of measurements is a central topic of interest. Classical
logical systems may be viewed as measurement algebras in which all measurements
commute. Keywords: Quantum measurements, Measurement algebras, Quantum Logic.
PACS: 02.10.-v.Comment: Submitted, 30 page
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