112 research outputs found
Contextuality and Noncommutative Geometry in Quantum Mechanics
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values; dually, states can be modelled as functions from the algebra of observables to outcome values. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be jointly measurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state. Here, we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical (deterministic) state spaces to the quantum setting and argue that these generalised state spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton–Isham–Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry à la Connes et al. To each unital C*- algebra A we associate a geometric object—a diagram of topological spaces collating quotient spaces of the noncommutative space underlying A —that performs the role of a generalised Gel'fand spectrum. We show how any functor F from compact Hausdorff spaces to a suitable target category C can be applied directly to these geometric objects to automatically yield an extension F~ acting on all unital C*-algebras. This procedure is used to give a novel formulation of the operator K0-functor via a finitary variant K~f of the extension K~ of the topological K-functor. We then delineate a C*-algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital C*-algebra the Zariski topological lattice of its primitive ideal spectrum, i.e. its lattice of closed two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture
Probabilistic foundations of quantum mechanics and quantum information
We discuss foundation of quantum mechanics (interpretations, superposition,
principle of complementarity, locality, hidden variables) and quantum
information theory.Comment: Contextual probabilistic viewpoint to quantum cryptography projec
Noncommutative quantum mechanics and Bohm's ontological interpretation
We carry out an investigation into the possibility of developing a Bohmian
interpretation based on the continuous motion of point particles for
noncommutative quantum mechanics. The conditions for such an interpretation to
be consistent are determined, and the implications of its adoption for
noncommutativity are discussed. A Bohmian analysis of the noncommutative
harmonic oscillator is carried out in detail. By studying the particle motion
in the oscillator orbits, we show that small-scale physics can have influence
at large scales, something similar to the IR-UV mixing
Naive Realism about Operators
A source of much difficulty and confusion in the interpretation of quantum
mechanics is a ``naive realism about operators.'' By this we refer to various
ways of taking too seriously the notion of operator-as-observable, and in
particular to the all too casual talk about ``measuring operators'' that occurs
when the subject is quantum mechanics. Without a specification of what should
be meant by ``measuring'' a quantum observable, such an expression can have no
clear meaning. A definite specification is provided by Bohmian mechanics, a
theory that emerges from Sch\"rodinger's equation for a system of particles
when we merely insist that ``particles'' means particles. Bohmian mechanics
clarifies the status and the role of operators as observables in quantum
mechanics by providing the operational details absent from standard quantum
mechanics. It thereby allows us to readily dismiss all the radical claims
traditionally enveloping the transition from the classical to the quantum
realm---for example, that we must abandon classical logic or classical
probability. The moral is rather simple: Beware naive realism, especially about
operators!Comment: 18 pages, LaTex2e with AMS-LaTeX, to appear in Erkenntnis, 1996 (the
proceedings of the international conference ``Probability, Dynamics and
Causality,'' Luino, Italy, 15-17 June 1995, a special issue edited by D.
Costantini and M.C. Gallavotti and dedicated to Prof. R. Jeffrey
Partial and Total Ideals of Von Neumann Algebras
A notion of partial ideal for an operator algebra is a weakening the notion
of ideal where the defining algebraic conditions are enforced only in the
commutative subalgebras. We show that, in a von Neumann algebra, the
ultraweakly closed two-sided ideals, which we call total ideals, correspond to
the unitarily invariant partial ideals. The result also admits an equivalent
formulation in terms of central projections. We place this result in the
context of an investigation into notions of spectrum of noncommutative
-algebras.Comment: 14 page
Contextuality and the fundamental theorems of quantum mechanics
Contextuality is a key feature of quantum mechanics, as was first brought to
light by Bohr and later realised more technically by Kochen and Specker. Isham
and Butterfield put contextuality at the heart of their topos-based formalism
and gave a reformulation of the Kochen-Specker theorem in the language of
presheaves. Here, we broaden this perspective considerably (partly drawing on
existing, but scattered results) and show that apart from the Kochen-Specker
theorem, also Wigner's theorem, Gleason's theorem, and Bell's theorem relate
fundamentally to contextuality. We provide reformulations of the theorems using
the language of presheaves over contexts and give general versions valid for
von Neumann algebras. This shows that a very substantial part of the structure
of quantum theory is encoded by contextuality.Comment: v2: minor revisions, added definition of Bell presheaf, adjustment of
Bell's theorem in contextual for
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