33,585 research outputs found
Discrete Riemannian Geometry
Within a framework of noncommutative geometry, we develop an analogue of
(pseudo) Riemannian geometry on finite and discrete sets. On a finite set,
there is a counterpart of the continuum metric tensor with a simple geometric
interpretation. The latter is based on a correspondence between first order
differential calculi and digraphs. Arrows originating from a vertex span its
(co)tangent space. If the metric is to measure length and angles at some point,
it has to be taken as an element of the left-linear tensor product of the space
of 1-forms with itself, and not as an element of the (non-local) tensor product
over the algebra of functions. It turns out that linear connections can always
be extended to this left tensor product, so that metric compatibility can be
defined in the same way as in continuum Riemannian geometry. In particular, in
the case of the universal differential calculus on a finite set, the Euclidean
geometry of polyhedra is recovered from conditions of metric compatibility and
vanishing torsion. In our rather general framework (which also comprises
structures which are far away from continuum differential geometry), there is
in general nothing like a Ricci tensor or a curvature scalar. Because of the
non-locality of tensor products (over the algebra of functions) of forms,
corresponding components (with respect to some module basis) turn out to be
rather non-local objects. But one can make use of the parallel transport
associated with a connection to `localize' such objects and in certain cases
there is a distinguished way to achieve this. This leads to covariant
components of the curvature tensor which then allow a contraction to a Ricci
tensor. In the case of a differential calculus associated with a hypercubic
lattice we propose a new discrete analogue of the (vacuum) Einstein equations.Comment: 34 pages, 1 figure (eps), LaTeX, amssymb, epsfi
Intentionality versus Constructive Empiricism
By focussing on the intentional character of observation in science, we argue that Constructive Empiricism – B.C. van Fraassen’s much debated and explored view of science – is inconsistent. We then argue there are at least two ways out of our Inconsistency Argument, one of which is more easily to square with Constructive Empiricism than the other
Discerning Elementary Particles
We extend the quantum-mechanical results of Muller & Saunders (2008)
establishing the weak discernibility of an arbitrary number of similar fermions
in finite-dimensional Hilbert-spaces in two ways: (a) from fermions to bosons
for all finite-dimensional Hilbert-spaces; and (b) from finite-dimensional to
infinite-dimensional Hilbert-spaces for all elementary particles. In both cases
this is performed using operators whose physical significance is beyond
doubt.This confutes the currently dominant view that (A) the quantum-mechanical
description of similar particles conflicts with Leibniz's Principle of the
Identity of Indiscernibles (PII); and that (B) the only way to save PII is by
adopting some pre-Kantian metaphysical notion such as Scotusian haecceittas or
Adamsian primitive thisness. We take sides with Muller & Saunders (2008)
against this currently dominant view, which has been expounded and defended by,
among others, Schr\"odinger, Margenau, Cortes, Dalla Chiara, Di Francia,
Redhead, French, Teller, Butterfield, Mittelstaedt, Giuntini, Castellani,
Krause and Huggett.Comment: Final Version. To appear in Philosophy of Science, July 200
Tensor product representations of the quantum double of a compact group
We consider the quantum double D(G) of a compact group G, following an
earlier paper. We use the explicit comultiplication on D(G) in order to build
tensor products of irreducible *-representations. Then we study their behaviour
under the action of the R-matrix, and their decomposition into irreducible
*-representations. The example of D(SU(2)) is treated in detail, with explicit
formulas for direct integral decomposition (`Clebsch-Gordan series') and
Clebsch-Gordan coefficients. We point out possible physical applications.Comment: LaTeX2e, 27 pages, corrected references, accepted by Comm.Math.Phy
Withering Away,Weakly
One of the reasons provided for the shift away from an ontology for physical reality of material objects & properties towards one of physical structures & relations (Ontological Structural Realism: OntSR) is that the quantum-mechanical description of composite physical systems of similar elementary particles entails they are indiscernible. As material objects, they ‘whither away’, and when they wither away, structures emerge in their stead. We inquire into the question whether recent results establishing the weak discernibility of elementary particles pose a threat for this quantum mechanical reason for OntSR, because precisely their newly discovered discernibility prevents them from ‘whithering away’. We argue there is a straightforward manner to consider the recent results as a reason in favour of OntSR rather than against it
Six Measurement Problems of Quantum Mechanics
The notorious ‘measurement problem’ has been roving around quantum mechanics for nearly a century since its inception, and has given rise to a variety of ‘interpretations’ of quantum mechanics, which are meant to evade it. We argue that no less than six problems need to be distinguished, and that several of them classify as different types of problems. One of them is what traditionally is called ‘the measurement problem’. Another of them has nothing to do with measurements but is a profound metaphysical problem. We also analyse critically Maudlin’s (Topoi 14:7–15, 1995) well-known statement of ‘three measurements problems’, and the clash of the views of Brown (Found Phys 16:857–870, 1986) and Stein (Maximal of an impossibility theorem concerning quantum measurement. In: R. S. Cohen et al. (Eds.), Potentiality, entanglement and passion-at-a-distance, 1997) on one of the six measurement problems. Finally, we summarise a solution to one measurement problem which has been largely ignored but tacitly if not explicitly acknowledged.</p
Withering Away,Weakly
One of the reasons provided for the shift away from an ontology for physical reality of material objects & properties towards one of physical structures & relations (Ontological Structural Realism: OntSR) is that the quantum-mechanical description of composite physical systems of similar elementary particles entails they are indiscernible. As material objects, they ‘whither away’, and when they wither away, structures emerge in their stead. We inquire into the question whether recent results establishing the weak discernibility of elementary particles pose a threat for this quantum mechanical reason for OntSR, because precisely their newly discovered discernibility prevents them from ‘whithering away’. We argue there is a straightforward manner to consider the recent results as a reason in favour of OntSR rather than against it
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