5 research outputs found
Three Dimensional Quantum Geometry and Deformed Poincare Symmetry
We study a three dimensional non-commutative space emerging in the context of
three dimensional Euclidean quantum gravity. Our starting point is the
assumption that the isometry group is deformed to the Drinfeld double D(SU(2)).
We generalize to the deformed case the construction of the flat Euclidean space
as the quotient of its isometry group ISU(2) by SU(2). We show that the algebra
of functions becomes the non-commutative algebra of SU(2) distributions endowed
with the convolution product. This construction gives the action of ISU(2) on
the algebra and allows the determination of plane waves and coordinate
functions. In particular, we show that: (i) plane waves have bounded momenta;
(ii) to a given momentum are associated several SU(2) elements leading to an
effective description of an element in the algebra in terms of several physical
scalar fields; (iii) their product leads to a deformed addition rule of momenta
consistent with the bound on the spectrum. We generalize to the non-commutative
setting the local action for a scalar field. Finally, we obtain, using harmonic
analysis, another useful description of the algebra as the direct sum of the
algebra of matrices. The algebra of matrices inherits the action of ISU(2):
rotations leave the order of the matrices invariant whereas translations change
the order in a way we explicitly determine.Comment: latex, 37 page
A Note on B-observables in Ponzano-Regge 3d Quantum Gravity
We study the insertion and value of metric observables in the (discrete) path
integral formulation of the Ponzano-Regge spinfoam model for 3d quantum
gravity. In particular, we discuss the length spectrum and the relation between
insertion of such B-observables and gauge fixing in the path integral.Comment: 17 page
Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model
A dual formulation of group field theories, obtained by a Fourier transform
mapping functions on a group to functions on its Lie algebra, has been proposed
recently. In the case of the Ooguri model for SO(4) BF theory, the variables of
the dual field variables are thus so(4) bivectors, which have a direct
interpretation as the discrete B variables. Here we study a modification of the
model by means of a constraint operator implementing the simplicity of the
bivectors, in such a way that projected fields describe metric tetrahedra. This
involves a extension of the usual GFT framework, where boundary operators are
labelled by projected spin network states. By construction, the Feynman
amplitudes are simplicial path integrals for constrained BF theory. We show
that the spin foam formulation of these amplitudes corresponds to a variant of
the Barrett-Crane model for quantum gravity. We then re-examin the arguments
against the Barrett-Crane model(s), in light of our construction.Comment: revtex, 24 page
Matrix Models as Non-commutative Field Theories on R^3
In the context of spin foam models for quantum gravity, group field theories
are a useful tool allowing on the one hand a non-perturbative formulation of
the partition function and on the other hand admitting an interpretation as
generalized matrix models. Focusing on 2d group field theories, we review their
explicit relation to matrix models and show their link to a class of
non-commutative field theories invariant under a quantum deformed 3d Poincare
symmetry. This provides a simple relation between matrix models and
non-commutative geometry. Moreover, we review the derivation of effective 2d
group field theories with non-trivial propagators from Boulatov's group field
theory for 3d quantum gravity. Besides the fact that this gives a simple and
direct derivation of non-commutative field theories for the matter dynamics
coupled to (3d) quantum gravity, these effective field theories can be
expressed as multi-matrix models with non-trivial coupling between matrices of
different sizes. It should be interesting to analyze this new class of
theories, both from the point of view of matrix models as integrable systems
and for the study of non-commutative field theories.Comment: 13 pages, v3: details added and title change