16 research outputs found
Holonomy observables in Ponzano-Regge type state sum models
We study observables on group elements in the Ponzano-Regge model. We show
that these observables have a natural interpretation in terms of Feynman
diagrams on a sphere and contrast them to the well studied observables on the
spin labels. We elucidate this interpretation by showing how they arise from
the no-gravity limit of the Turaev-Viro model and Chern-Simons theory.Comment: 15 pages, 2 figure
4-Dimensional BF Theory as a Topological Quantum Field Theory
Starting from a Lie group G whose Lie algebra is equipped with an invariant
nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory
with cosmological term gives rise to a TQFT satisfying a generalization of
Atiyah's axioms to manifolds equipped with principal G-bundle. The case G =
GL(4,R) is especially interesting because every 4-manifold is then naturally
equipped with a principal G-bundle, namely its frame bundle. In this case, the
partition function of a compact oriented 4-manifold is the exponential of its
signature, and the resulting TQFT is isomorphic to that constructed by Crane
and Yetter using a state sum model, or by Broda using a surgery presentation of
4-manifolds.Comment: 15 pages in LaTe
so(4) Plebanski Action and Relativistic Spin Foam Model
In this note we study the correspondence between the ``relativistic spin
foam'' model introduced by Barrett, Crane and Baez and the so(4) Plebanski
action. We argue that the Plebanski model is the continuum analog of
the relativistic spin foam model. We prove that the Plebanski action possess
four phases, one of which is gravity and outline the discrepancy between this
model and the model of Euclidean gravity. We also show that the Plebanski model
possess another natural dicretisation and can be associate with another, new,
spin foam model that appear to be the counterpart of the spin foam
model describing the self dual formulation of gravity.Comment: 12 pages, REVTeX using AMS fonts. Some minor corrections and
improvement
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
Positivity of Spin Foam Amplitudes
The amplitude for a spin foam in the Barrett-Crane model of Riemannian
quantum gravity is given as a product over its vertices, edges and faces, with
one factor of the Riemannian 10j symbols appearing for each vertex, and simpler
factors for the edges and faces. We prove that these amplitudes are always
nonnegative for closed spin foams. As a corollary, all open spin foams going
between a fixed pair of spin networks have real amplitudes of the same sign.
This means one can use the Metropolis algorithm to compute expectation values
of observables in the Riemannian Barrett-Crane model, as in statistical
mechanics, even though this theory is based on a real-time (e^{iS}) rather than
imaginary-time (e^{-S}) path integral. Our proof uses the fact that when the
Riemannian 10j symbols are nonzero, their sign is positive or negative
depending on whether the sum of the ten spins is an integer or half-integer.
For the product of 10j symbols appearing in the amplitude for a closed spin
foam, these signs cancel. We conclude with some numerical evidence suggesting
that the Lorentzian 10j symbols are always nonnegative, which would imply
similar results for the Lorentzian Barrett-Crane model.Comment: 15 pages LaTeX. v3: Final version, with updated conclusions and other
minor changes. To appear in Classical and Quantum Gravity. v4: corrects # of
samples in Lorentzian tabl
Cosmological Deformation of Lorentzian Spin Foam Models
We study the quantum deformation of the Barrett-Crane Lorentzian spin foam
model which is conjectured to be the discretization of Lorentzian Plebanski
model with positive cosmological constant and includes therefore as a
particular sector quantum gravity in de-Sitter space. This spin foam model is
constructed using harmonic analysis on the quantum Lorentz group. The
evaluation of simple spin networks are shown to be non commutative integrals
over the quantum hyperboloid defined as a pile of fuzzy spheres. We show that
the introduction of the cosmological constant removes all the infrared
divergences: for any fixed triangulation, the integration over the area
variables is finite for a large class of normalization of the amplitude of the
edges and of the faces.Comment: 37 pages, 7 figures include
Quantum Gravity and the Algebra of Tangles
In Rovelli and Smolin's loop representation of nonperturbative quantum
gravity in 4 dimensions, there is a space of solutions to the Hamiltonian
constraint having as a basis isotopy classes of links in R^3. The physically
correct inner product on this space of states is not yet known, or in other
words, the *-algebra structure of the algebra of observables has not been
determined. In order to approach this problem, we consider a larger space H of
solutions of the Hamiltonian constraint, which has as a basis isotopy classes
of tangles. A certain algebra T, the ``tangle algebra,'' acts as operators on
H. The ``empty state'', corresponding to the class of the empty tangle, is
conjectured to be a cyclic vector for T. We construct simpler representations
of T as quotients of H by the skein relations for the HOMFLY polynomial, and
calculate a *-algebra structure for T using these representations. We use this
to determine the inner product of certain states of quantum gravity associated
to the Jones polynomial (or more precisely, Kauffman bracket).Comment: 16 pages (with major corrections
Spin Foam Models of Riemannian Quantum Gravity
Using numerical calculations, we compare three versions of the Barrett-Crane
model of 4-dimensional Riemannian quantum gravity. In the version with face and
edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we
show the partition function diverges very rapidly for many triangulated
4-manifolds. In the version with modified face and edge amplitudes due to Perez
and Rovelli, we show the partition function converges so rapidly that the sum
is dominated by spin foams where all the spins labelling faces are zero except
for small, widely separated islands of higher spin. We also describe a new
version which appears to have a convergent partition function without drastic
spin-zero dominance. Finally, after a general discussion of how to extract
physics from spin foam models, we discuss the implications of convergence or
divergence of the partition function for other aspects of a spin foam model.Comment: 23 pages LaTeX; this version to appear in Classical and Quantum
Gravit