21 research outputs found
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
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
State Sum Models and Simplicial Cohomology
We study a class of subdivision invariant lattice models based on the gauge
group , with particular emphasis on the four dimensional example. This
model is based upon the assignment of field variables to both the - and
-dimensional simplices of the simplicial complex. The property of
subdivision invariance is achieved when the coupling parameter is quantized and
the field configurations are restricted to satisfy a type of mod- flatness
condition. By explicit computation of the partition function for the manifold
, we establish that the theory has a quantum Hilbert space
which differs from the classical one.Comment: 28 pages, Latex, ITFA-94-13, (Expanded version with two new sections
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
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
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
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