1,417 research outputs found
Matter from Space
General Relativity offers the possibility to model attributes of matter, like
mass, momentum, angular momentum, spin, chirality etc. from pure space, endowed
only with a single field that represents its Riemannian geometry. I review this
picture of `Geometrodynamics' and comment on various developments after
Einstein.Comment: 37 Pages, 17 figures. Based on a talk delivered at the conference
"Beyond Einstein: Historical Perspectives on Geometry, Gravitation, and
Cosmology in the Twentieth Century", September 2008 at the University of
Mainz in Germany. To appear in the Einstein-Studies Series, Birkhaeuser,
Boston. v2: Reference [7] added and typo in formula [42] correcte
On Computability and Triviality of Well Groups
The concept of well group in a special but important case captures
homological properties of the zero set of a continuous map on a
compact space K that are invariant with respect to perturbations of f. The
perturbations are arbitrary continuous maps within distance r from f
for a given r>0. The main drawback of the approach is that the computability of
well groups was shown only when dim K=n or n=1.
Our contribution to the theory of well groups is twofold: on the one hand we
improve on the computability issue, but on the other hand we present a range of
examples where the well groups are incomplete invariants, that is, fail to
capture certain important robust properties of the zero set.
For the first part, we identify a computable subgroup of the well group that
is obtained by cap product with the pullback of the orientation of R^n by f. In
other words, well groups can be algorithmically approximated from below. When f
is smooth and dim K<2n-2, our approximation of the (dim K-n)th well group is
exact.
For the second part, we find examples of maps with all well
groups isomorphic but whose perturbations have different zero sets. We discuss
on a possible replacement of the well groups of vector valued maps by an
invariant of a better descriptive power and computability status.Comment: 20 pages main paper including bibliography, followed by 22 pages of
Appendi
The Spin-Statistics Connection in Quantum Gravity
It is well-known that is spite of sharing some properties with conventional
particles, topological geons in general violate the spin-statistics theorem. On
the other hand, it is generally believed that in quantum gravity theories
allowing for topology change, using pair creation and annihilation of geons,
one should be able to recover this theorem. In this paper, we take an
alternative route, and use an algebraic formalism developed in previous work.
We give a description of topological geons where an algebra of "observables" is
identified and quantized. Different irreducible representations of this algebra
correspond to different kinds of geons, and are labeled by a non-abelian
"charge" and "magnetic flux". We then find that the usual spin-statistics
theorem is indeed violated, but a new spin-statistics relation arises, when we
assume that the fluxes are superselected. This assumption can be proved if all
observables are local, as is generally the case in physical theories. Finally,
we also show how our approach fits into conventional formulations of quantum
gravity.Comment: LaTeX file, 31 pages, 5 figure
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