54 research outputs found
Quantum gravity as a group field theory: a sketch
We give a very brief introduction to the group field theory approach to
quantum gravity, a generalisation of matrix models for 2-dimensional quantum
gravity to higher dimension, that has emerged recently from research in spin
foam models.Comment: jpconf; 8 pages, 9 figures; to appear in the Proceedings of the
Fourth Meeting on Constrained Dynamics and Quantum Gravity, Cala Gonone,
Italy, September 12-16, 200
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
Encoding simplicial quantum geometry in group field theories
We show that a new symmetry requirement on the GFT field, in the context of
an extended GFT formalism, involving both Lie algebra and group elements,
leads, in 3d, to Feynman amplitudes with a simplicial path integral form based
on the Regge action, to a proper relation between the discrete connection and
the triad vectors appearing in it, and to a much more satisfactory and
transparent encoding of simplicial geometry already at the level of the GFT
action.Comment: 15 pages, 2 figures, RevTeX, references adde
Coupling of spacetime atoms and spin foam renormalisation from group field theory
We study the issue of coupling among 4-simplices in the context of spin foam
models obtained from a group field theory formalism. We construct a
generalisation of the Barrett-Crane model in which an additional coupling
between the normals to tetrahedra, as defined in different 4-simplices that
share them, is present. This is realised through an extension of the usual
field over the group manifold to a five argument one. We define a specific
model in which this coupling is parametrised by an additional real parameter
that allows to tune the degree of locality of the resulting model,
interpolating between the usual Barrett-Crane model and a flat BF-type one.
Moreover, we define a further extension of the group field theory formalism in
which the coupling parameter enters as a new variable of the field, and the
action presents derivative terms that lead to modified classical equations of
motion. Finally, we discuss the issue of renormalisation of spin foam models,
and how the new coupled model can be of help regarding this.Comment: RevTeX, 18 pages, no figure
Group field theory formulation of 3d quantum gravity coupled to matter fields
We present a new group field theory describing 3d Riemannian quantum gravity
coupled to matter fields for any choice of spin and mass. The perturbative
expansion of the partition function produces fat graphs colored with SU(2)
algebraic data, from which one can reconstruct at once a 3-dimensional
simplicial complex representing spacetime and its geometry, like in the
Ponzano-Regge formulation of pure 3d quantum gravity, and the Feynman graphs
for the matter fields. The model then assigns quantum amplitudes to these fat
graphs given by spin foam models for gravity coupled to interacting massive
spinning point particles, whose properties we discuss.Comment: RevTeX; 28 pages, 21 figure
Group field theory and simplicial quantum gravity
We present a new Group Field Theory for 4d quantum gravity. It incorporates
the constraints that give gravity from BF theory, and has quantum amplitudes
with the explicit form of simplicial path integrals for 1st order gravity. The
geometric interpretation of the variables and of the contributions to the
quantum amplitudes is manifest. This allows a direct link with other simplicial
gravity approaches, like quantum Regge calculus, in the form of the amplitudes
of the model, and dynamical triangulations, which we show to correspond to a
simple restriction of the same.Comment: 14 pages, no figures; RevTex4; v2: definition of the model modified,
discussion extended and improve
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
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
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
The 1/N expansion of colored tensor models in arbitrary dimension
In this paper we extend the 1/N expansion introduced in [1] to group field
theories in arbitrary dimension and prove that only graphs corresponding to
spheres S^D contribute to the leading order in the large N limit.Comment: 4 pages, 3 figure
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