4,623 research outputs found
Quantum mechanics as a theory of probability
We develop and defend the thesis that the Hilbert space formalism of quantum
mechanics is a new theory of probability. The theory, like its classical
counterpart, consists of an algebra of events, and the probability measures
defined on it. The construction proceeds in the following steps: (a) Axioms for
the algebra of events are introduced following Birkhoff and von Neumann. All
axioms, except the one that expresses the uncertainty principle, are shared
with the classical event space. The only models for the set of axioms are
lattices of subspaces of inner product spaces over a field K. (b) Another axiom
due to Soler forces K to be the field of real, or complex numbers, or the
quaternions. We suggest a probabilistic reading of Soler's axiom. (c) Gleason's
theorem fully characterizes the probability measures on the algebra of events,
so that Born's rule is derived. (d) Gleason's theorem is equivalent to the
existence of a certain finite set of rays, with a particular orthogonality
graph (Wondergraph). Consequently, all aspects of quantum probability can be
derived from rational probability assignments to finite "quantum gambles". We
apply the approach to the analysis of entanglement, Bell inequalities, and the
quantum theory of macroscopic objects. We also discuss the relation of the
present approach to quantum logic, realism and truth, and the measurement
problem.Comment: 37 pages, 3 figures. Forthcoming in a Festschrift for Jeffrey Bub,
ed. W. Demopoulos and the author, Springer (Kluwer): University of Western
Ontario Series in Philosophy of Scienc
The indexed time table approach for planning and acting
A representation is discussed of symbolic temporal relations, called IxTeT, that is both powerful enough at the reasoning level for tasks such as plan generation, refinement and modification, and efficient enough for dealing with real time constraints in action monitoring and reactive planning. Such representation for dealing with time is needed in a teleoperated space robot. After a brief survey of known approaches, the proposed representation shows its computational efficiency for managing a large data base of temporal relations. Reactive planning with IxTeT is described and exemplified through the problem of mission planning and modification for a simple surveying satellite
Nonabelian Duality and Solvable Large N Lattice Systems
We introduce the basics of the nonabelian duality transformation of SU(N) or
U(N) vector-field models defined on a lattice. The dual degrees of freedom are
certain species of the integer-valued fields complemented by the symmetric
groups' \otimes_{n} S(n) variables. While the former parametrize relevant
irreducible representations, the latter play the role of the Lagrange
multipliers facilitating the fusion rules involved. As an application, I
construct a novel solvable family of SU(N) D-matrix systems graded by the rank
1\leq{k}\leq{(D-1)} of the manifest [U(N)]^{\oplus k} conjugation-symmetry.
Their large N solvability is due to a hidden invariance (explicit in the dual
formulation) which allows for a mapping onto the recently proposed
eigenvalue-models \cite{Dub1} with the largest k=D symmetry. Extending
\cite{Dub1}, we reconstruct a D-dimensional gauge theory with the large N free
energy given (modulo the volume factor) by the free energy of a given proposed
1\leq{k}\leq{(D-1)} D-matrix system. It is emphasized that the developed
formalism provides with the basis for higher-dimensional generalizations of the
Gross-Taylor stringy representation of strongly coupled 2d gauge theories.Comment: TeX, 46 page
Effects and Propositions
The quantum logical and quantum information-theoretic traditions have exerted
an especially powerful influence on Bub's thinking about the conceptual
foundations of quantum mechanics. This paper discusses both the quantum logical
and information-theoretic traditions from the point of view of their
representational frameworks. I argue that it is at this level, at the level of
its framework, that the quantum logical tradition has retained its centrality
to Bub's thought. It is further argued that there is implicit in the quantum
information-theoretic tradition a set of ideas that mark a genuinely new
alternative to the framework of quantum logic. These ideas are of considerable
interest for the philosophy of quantum mechanics, a claim which I defend with
an extended discussion of their application to our understanding of the
philosophical significance of the no hidden variable theorem of Kochen and
Specker.Comment: Presented to the 2007 conference, New Directions in the Foundations
of Physic
An algorithmic proof for the completeness of two-dimensional Ising model
We show that the two dimensional Ising model is complete, in the sense that
the partition function of any lattice model on any graph is equal to the
partition function of the 2D Ising model with complex coupling. The latter
model has all its spin-spin coupling equal to i\pi/4 and all the parameters of
the original model are contained in the local magnetic fields of the Ising
model. This result has already been derived by using techniques from quantum
information theory and by exploiting the universality of cluster states. Here
we do not use the quantum formalism and hence make the completeness result
accessible to a wide audience. Furthermore our method has the advantage of
being algorithmic in nature so that by following a set of simple graphical
transformations, one is able to transform any discrete lattice model to an
Ising model defined on a (polynomially) larger 2D lattice.Comment: 18 pages, 15 figures, Accepted for publication in Physical Review
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