17 research outputs found
An Effective Model for Crumpling in Two Dimensions?
We investigate the crumpling transition for a dynamically triangulated random
surface embedded in two dimensions using an effective model in which the
disordering effect of the variables on the correlations of the normals is
replaced by a long-range ``antiferromagnetic'' term. We compare the results
from a Monte Carlo simulation with those obtained for the standard action which
retains the 's and discuss the nature of the phase transition.Comment: 5 page
Spin Networks and Quantum Gravity
We introduce a new basis on the state space of non-perturbative quantum
gravity. The states of this basis are linearly independent, are well defined in
both the loop representation and the connection representation, and are labeled
by a generalization of Penrose's spin netoworks. The new basis fully reduces
the spinor identities (SU(2) Mandelstam identities) and simplifies calculations
in non-perturbative quantum gravity. In particular, it allows a simple
expression for the exact solutions of the Hamiltonian constraint
(Wheeler-DeWitt equation) that have been discovered in the loop representation.
Since the states in this basis diagnolize operators that represent the three
geometry of space, such as the area and volumes of arbitrary surfaces and
regions, these states provide a discrete picture of quantum geometry at the
Planck scale.Comment: 42 page
Quantization of Point Particles in 2+1 Dimensional Gravity and Space-Time Discreteness
By investigating the canonical commutation rules for gravitating quantized
particles in a 2+1 dimensional world it is found that these particles live on a
space-time lattice. The space-time lattice points can be characterized by three
integers. Various representations are possible, the details depending on the
topology chosen for energy-momentum space. We find that an
topology yields a physically most interesting lattice within which first
quantization of Dirac particles is possible. An topology also gives a
lattice, but does not allow first quantized particles.Comment: 23 pages Plain TeX, 3 Figure
The Well-Defined Phase of Simplicial Quantum Gravity in Four Dimensions
We analyze simplicial quantum gravity in four dimensions using the Regge
approach. The existence of an entropy dominated phase with small negative
curvature is investigated in detail. It turns out that observables of the
system possess finite expectation values although the Einstein-Hilbert action
is unbounded. This well-defined phase is found to be stable for a one-parameter
family of measures. A preliminary study indicates that the influence of the
lattice size on the average curvature is small. We compare our results with
those obtained by dynamical triangulation and find qualitative correspondence.Comment: 29 pages, uuencoded postscript file; to appear in Phys. Rev.
On the stability of renormalizable expansions in three-dimensional gravity
Preliminary investigations are made for the stability of the expansion
in three-dimensional gravity coupled to various matter fields, which are
power-counting renormalizable. For unitary matters, a tachyonic pole appears in
the spin-2 part of the leading graviton propagator, which implies the unstable
flat space-time, unless the higher-derivative terms are introduced. As another
possibility to avoid this spin-2 tachyon, we propose Einstein gravity coupled
to non-unitary matters. It turns out that a tachyon appears in the spin-0 or -1
part for any linear gauges in this case, but it can be removed if non-minimally
coupled scalars are included. We suggest an interesting model which may be
stable and possess an ultraviolet fixed point.Comment: 32 pages. (A further discussion to avoid tachyons is included. To be
Published in Physical Review D.
Quantum symmetry, the cosmological constant and Planck scale phenomenology
We present a simple algebraic argument for the conclusion that the low energy
limit of a quantum theory of gravity must be a theory invariant, not under the
Poincare group, but under a deformation of it parameterized by a dimensional
parameter proportional to the Planck mass. Such deformations, called
kappa-Poincare algebras, imply modified energy-momentum relations of a type
that may be observable in near future experiments. Our argument applies in both
2+1 and 3+1 dimensions and assumes only 1) that the low energy limit of a
quantum theory of gravity must involve also a limit in which the cosmological
constant is taken very small with respect to the Planck scale and 2) that in
3+1 dimensions the physical energy and momenta of physical elementary particles
is related to symmetries of the full quantum gravity theory by appropriate
renormalization depending on Lambda l^2_{Planck}. The argument makes use of the
fact that the cosmological constant results in the symmetry algebra of quantum
gravity being quantum deformed, as a consequence when the limit \Lambda
l^2_{Planck} -> 0 is taken one finds a deformed Poincare invariance. We are
also able to isolate what information must be provided by the quantum theory in
order to determine which presentation of the kappa-Poincare algebra is relevant
for the physical symmetry generators and, hence, the exact form of the modified
energy-momentum relations. These arguments imply that Lorentz invariance is
modified as in proposals for doubly special relativity, rather than broken, in
theories of quantum gravity, so long as those theories behave smoothly in the
limit the cosmological constant is taken to be small.Comment: LaTex, 19 page
The basis of the Ponzano-Regge-Turaev-Viro-Ooguri model is the loop representation basis
We show that the Hilbert space basis that defines the Ponzano-Regge-
Turaev-Viro-Ooguri combinatorial definition of 3-d Quantum Gravity is the same
as the one that defines the Loop Representation. We show how to compute lengths
in Witten's 3-d gravity and how to reconstruct the 2-d geometry from a state of
Witten's theory. We show that the non-degenerate geometries are contained in
the Witten's Hilbert space. We sketch an extension of the combinatorial
construction to the physical 4-d case, by defining a modification of Regge
calculus in which areas, rather than lengths, are taken as independent
variables. We provide an expression for the scalar product in the Loop
representation in 4-d. We discuss the general form of a nonperturbative quantum
theory of gravity, and argue that it should be given by a generalization of
Atiyah's topological quantum field theories axioms.Comment: 16 page
The Harris-Luck criterion for random lattices
The Harris-Luck criterion judges the relevance of (potentially) spatially
correlated, quenched disorder induced by, e.g., random bonds, randomly diluted
sites or a quasi-periodicity of the lattice, for altering the critical behavior
of a coupled matter system. We investigate the applicability of this type of
criterion to the case of spin variables coupled to random lattices. Their
aptitude to alter critical behavior depends on the degree of spatial
correlations present, which is quantified by a wandering exponent. We consider
the cases of Poissonian random graphs resulting from the Voronoi-Delaunay
construction and of planar, ``fat'' Feynman diagrams and precisely
determine their wandering exponents. The resulting predictions are compared to
various exact and numerical results for the Potts model coupled to these
quenched ensembles of random graphs.Comment: 13 pages, 9 figures, 2 tables, REVTeX 4. Version as published, one
figure added for clarification, minor re-wordings and typo cleanu