4 research outputs found
Non-commutative flux representation for loop quantum gravity
The Hilbert space of loop quantum gravity is usually described in terms of
cylindrical functionals of the gauge connection, the electric fluxes acting as
non-commuting derivation operators. It has long been believed that this
non-commutativity prevents a dual flux (or triad) representation of loop
quantum gravity to exist. We show here, instead, that such a representation can
be explicitly defined, by means of a non-commutative Fourier transform defined
on the loop gravity state space. In this dual representation, flux operators
act by *-multiplication and holonomy operators act by translation. We describe
the gauge invariant dual states and discuss their geometrical meaning. Finally,
we apply the construction to the simpler case of a U(1) gauge group and compare
the resulting flux representation with the triad representation used in loop
quantum cosmology.Comment: 12 pages, matches published versio
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
Towards classical geometrodynamics from Group Field Theory hydrodynamics
We take the first steps towards identifying the hydrodynamics of group field
theories (GFTs) and relating this hydrodynamic regime to classical
geometrodynamics of continuum space. We apply to GFT mean field theory
techniques borrowed from the theory of Bose condensates, alongside standard GFT
and spin foam techniques. The mean field configuration we study is, in turn,
obtained from loop quantum gravity coherent states. We work in the context of
2d and 3d GFT models, in euclidean signature, both ordinary and colored, as
examples of a procedure that has a more general validity. We also extract the
effective dynamics of the system around the mean field configurations, and
discuss the role of GFT symmetries in going from microscopic to effective
dynamics. In the process, we obtain additional insights on the GFT formalism
itself.Comment: revtex4, 32 pages. Contribution submitted to the focus issue of the
New Journal of Physics on "Classical and Quantum Analogues for Gravitational
Phenomena and Related Effects", R. Schuetzhold, U. Leonhardt and C. Maia,
Eds; v2: typos corrected, references updated, to match the published versio
Effective Hamiltonian Constraint from Group Field Theory
Spinfoam models provide a covariant formulation of the dynamics of loop
quantum gravity. They are non-perturbatively defined in the group field theory
(GFT) framework: the GFT partition function defines the sum of spinfoam
transition amplitudes over all possible (discretized) geometries and
topologies. The issue remains, however, of explicitly relating the specific
form of the group field theory action and the canonical Hamiltonian constraint.
Here, we suggest an avenue for addressing this issue. Our strategy is to expand
group field theories around non-trivial classical solutions and to interpret
the induced quadratic kinematical term as defining a Hamiltonian constraint on
the group field and thus on spin network wave functions. We apply our procedure
to Boulatov group field theory for 3d Riemannian gravity. Finally, we discuss
the relevance of understanding the spectrum of this Hamiltonian operator for
the renormalization of group field theories.Comment: 14 page