54 research outputs found
Quantum Geometry Phenomenology: Angle and Semiclassical States
The phenomenology for the deep spatial geometry of loop quantum gravity is
discussed. In the context of a simple model of an atom of space, it is shown
how purely combinatorial structures can affect observations. The angle operator
is used to develop a model of angular corrections to local, continuum
flat-space 3-geometries. The physical effects involve neither breaking of local
Lorentz invariance nor Planck scale suppression, but rather reply on only the
combinatorics of SU(2) recouping theory. Bhabha scattering is discussed as an
example of how the effects might be observationally accessible.Comment: 5 pages, slightly extended version of the contribution to the
Loops'11 conference proceeding
Asymptotics of classical spin networks
A spin network is a cubic ribbon graph labeled by representations of
. Spin networks are important in various areas of Mathematics
(3-dimensional Quantum Topology), Physics (Angular Momentum, Classical and
Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin
network is an integer number. The main results of our paper are: (a) an
existence theorem for the asymptotics of evaluations of arbitrary spin networks
(using the theory of -functions), (b) a rationality property of the
generating series of all evaluations with a fixed underlying graph (using the
combinatorics of the chromatic evaluation of a spin network), (c) rigorous
effective computations of our results for some -symbols using the
Wilf-Zeilberger theory, and (d) a complete analysis of the regular Cube
spin network (including a non-rigorous guess of its Stokes constants), in the
appendix.Comment: 24 pages, 32 figure
An algebraic interpretation of the Wheeler-DeWitt equation
We make a direct connection between the construction of three dimensional
topological state sums from tensor categories and three dimensional quantum
gravity by noting that the discrete version of the Wheeler-DeWitt equation is
exactly the pentagon for the associator of the tensor category, the
Biedenharn-Elliott identity. A crucial role is played by an asymptotic formula
relating 6j-symbols to rotation matrices given by Edmonds.Comment: 10 pages, amstex, uses epsf.tex. New version has improved
presentatio
Embedded graph invariants in Chern-Simons theory
Chern-Simons gauge theory, since its inception as a topological quantum field
theory, has proved to be a rich source of understanding for knot invariants. In
this work the theory is used to explore the definition of the expectation value
of a network of Wilson lines - an embedded graph invariant. Using a slight
generalization of the variational method, lowest-order results for invariants
for arbitrary valence graphs are derived; gauge invariant operators are
introduced; and some higher order results are found. The method used here
provides a Vassiliev-type definition of graph invariants which depend on both
the embedding of the graph and the group structure of the gauge theory. It is
found that one need not frame individual vertices. Though, without a global
projection of the graph, there is an ambiguity in the relation of the
decomposition of distinct vertices. It is suggested that framing may be seen as
arising from this ambiguity - as a way of relating frames at distinct vertices.Comment: 20 pages; RevTex; with approx 50 ps figures; References added,
introduction rewritten, version to be published in Nuc. Phys.
Shape in an Atom of Space: Exploring quantum geometry phenomenology
A phenomenology for the deep spatial geometry of loop quantum gravity is
introduced. In the context of a simple model, an atom of space, it is shown how
purely combinatorial structures can affect observations. The angle operator is
used to develop a model of angular corrections to local, continuum flat-space
3-geometries. The physical effects involve neither breaking of local Lorentz
invariance nor Planck scale suppression, but rather reply on only the
combinatorics of SU(2) recoupling. Bhabha scattering is discussed as an example
of how the effects might be observationally accessible.Comment: 14 pages, 7 figures; v2 references adde
Simple model for quantum general relativity from loop quantum gravity
New progress in loop gravity has lead to a simple model of `general-covariant
quantum field theory'. I sum up the definition of the model in self-contained
form, in terms accessible to those outside the subfield. I emphasize its
formulation as a generalized topological quantum field theory with an infinite
number of degrees of freedom, and its relation to lattice theory. I list the
indications supporting the conjecture that the model is related to general
relativity and UV finite.Comment: 8 pages, 3 figure
A Lorentzian Signature Model for Quantum General Relativity
We give a relativistic spin network model for quantum gravity based on the
Lorentz group and its q-deformation, the Quantum Lorentz Algebra.
We propose a combinatorial model for the path integral given by an integral
over suitable representations of this algebra. This generalises the state sum
models for the case of the four-dimensional rotation group previously studied
in gr-qc/9709028.
As a technical tool, formulae for the evaluation of relativistic spin
networks for the Lorentz group are developed, with some simple examples which
show that the evaluation is finite in interesting cases. We conjecture that the
`10J' symbol needed in our model has a finite value.Comment: 22 pages, latex, amsfonts, Xypic. Version 3: improved presentation.
Version 2 is a major revision with explicit formulae included for the
evaluation of relativistic spin networks and the computation of examples
which have finite value
Holographic hierarchy in the Gaussian matrix model via the fuzzy sphere
81 pages, 33 figuresS.R. is supported by STFC Grant ST/J000469/1, String theory, gauge theory, and duality
Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
We reconsider the conceptual foundations of the renormalization-group (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the RG
map, defined on a suitable space of interactions (= formal Hamiltonians), is
always single-valued and Lipschitz continuous on its domain of definition. This
rules out a recently proposed scenario for the RG description of first-order
phase transitions. On the pathological side, we make rigorous some arguments of
Griffiths, Pearce and Israel, and prove in several cases that the renormalized
measure is not a Gibbs measure for any reasonable interaction. This means that
the RG map is ill-defined, and that the conventional RG description of
first-order phase transitions is not universally valid. For decimation or
Kadanoff transformations applied to the Ising model in dimension ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . We discuss in detail
the distinction between Gibbsian and non-Gibbsian measures, and give a rather
complete catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also
ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.
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