27,219 research outputs found
Algebraic description of spacetime foam
A mathematical formalism for treating spacetime topology as a quantum
observable is provided. We describe spacetime foam entirely in algebraic terms.
To implement the correspondence principle we express the classical spacetime
manifold of general relativity and the commutative coordinates of its events by
means of appropriate limit constructions.Comment: 34 pages, LaTeX2e, the section concerning classical spacetimes in the
limit essentially correcte
Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity
Previous work on applications of Abstract Differential Geometry (ADG) to
discrete Lorentzian quantum gravity is brought to its categorical climax by
organizing the curved finitary spacetime sheaves of quantum causal sets
involved therein, on which a finitary (:locally finite), singularity-free,
background manifold independent and geometrically prequantized version of the
gravitational vacuum Einstein field equations were seen to hold, into a topos
structure. This topos is seen to be a finitary instance of both an elementary
and a Grothendieck topos, generalizing in a differential geometric setting, as
befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies.
The paper closes with a thorough discussion of four future routes we could take
in order to further develop our topos-theoretic perspective on ADG-gravity
along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references
polished) Submitted to the International Journal of Theoretical Physic
Lattices and Their Continuum Limits
We address the problem of the continuum limit for a system of Hausdorff
lattices (namely lattices of isolated points) approximating a topological space
. The correct framework is that of projective systems. The projective limit
is a universal space from which can be recovered as a quotient. We dualize
the construction to approximate the algebra of continuous
functions on . In a companion paper we shall extend this analysis to systems
of noncommutative lattices (non Hausdorff lattices).Comment: 11 pages, 1 Figure included in the LaTeX Source New version, minor
modifications (typos corrected) and a correction in the list of author
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