2,256 research outputs found
A geometry of information, I: Nerves, posets and differential forms
The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial
Representation: Continuous vs. Discrete'. Spatial representation has two
contrasting but interacting aspects (i) representation of spaces' and (ii)
representation by spaces. In this paper, we will examine two aspects that are
common to both interpretations of the theme, namely nerve constructions and
refinement. Representations change, data changes, spaces change. We will
examine the possibility of a `differential geometry' of spatial representations
of both types, and in the sequel give an algebra of differential forms that has
the potential to handle the dynamical aspect of such a geometry. We will
discuss briefly a conjectured class of spaces, generalising the Cantor set
which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl
seminar portal http://drops.dagstuhl.de/portals/04351
Spacetime-Free Approach to Quantum Theory and Effective Spacetime Structure
Motivated by hints of the effective emergent nature of spacetime structure,
we formulate a spacetime-free algebraic framework for quantum theory, in which
no a priori background geometric structure is required. Such a framework is
necessary in order to study the emergence of effective spacetime structure in a
consistent manner, without assuming a background geometry from the outset.
Instead, the background geometry is conjectured to arise as an effective
structure of the algebraic and dynamical relations between observables that are
imposed by the background statistics of the system. Namely, we suggest that
quantum reference states on an extended observable algebra, the free algebra
generated by the observables, may give rise to effective spacetime structures.
Accordingly, perturbations of the reference state lead to perturbations of the
induced effective spacetime geometry. We initiate the study of these
perturbations, and their relation to gravitational phenomena
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