71 research outputs found
(Broken) Gauge Symmetries and Constraints in Regge Calculus
We will examine the issue of diffeomorphism symmetry in simplicial models of
(quantum) gravity, in particular for Regge calculus. We find that for a
solution with curvature there do not exist exact gauge symmetries on the
discrete level. Furthermore we derive a canonical formulation that exactly
matches the dynamics and hence symmetries of the covariant picture. In this
canonical formulation broken symmetries lead to the replacements of constraints
by so--called pseudo constraints. These considerations should be taken into
account in attempts to connect spin foam models, based on the Regge action,
with canonical loop quantum gravity, which aims at implementing proper
constraints. We will argue that the long standing problem of finding a
consistent constraint algebra for discretized gravity theories is equivalent to
the problem of finding an action with exact diffeomorphism symmetries. Finally
we will analyze different limits in which the pseudo constraints might turn
into proper constraints. This could be helpful to infer alternative
discretization schemes in which the symmetries are not broken.Comment: 32 pages, 15 figure
From the discrete to the continuous - towards a cylindrically consistent dynamics
Discrete models usually represent approximations to continuum physics.
Cylindrical consistency provides a framework in which discretizations mirror
exactly the continuum limit. Being a standard tool for the kinematics of loop
quantum gravity we propose a coarse graining procedure that aims at
constructing a cylindrically consistent dynamics in the form of transition
amplitudes and Hamilton's principal functions. The coarse graining procedure,
which is motivated by tensor network renormalization methods, provides a
systematic approximation scheme towards this end. A crucial role in this coarse
graining scheme is played by embedding maps that allow the interpretation of
discrete boundary data as continuum configurations. These embedding maps should
be selected according to the dynamics of the system, as a choice of embedding
maps will determine a truncation of the renormalization flow.Comment: 22 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
Regge calculus from a new angle
In Regge calculus space time is usually approximated by a triangulation with
flat simplices. We present a formulation using simplices with constant
sectional curvature adjusted to the presence of a cosmological constant. As we
will show such a formulation allows to replace the length variables by 3d or 4d
dihedral angles as basic variables. Moreover we will introduce a first order
formulation, which in contrast to using flat simplices, does not require any
constraints. These considerations could be useful for the construction of
quantum gravity models with a cosmological constant.Comment: 8 page
Area-angle variables for general relativity
We introduce a modified Regge calculus for general relativity on a
triangulated four dimensional Riemannian manifold where the fundamental
variables are areas and a certain class of angles. These variables satisfy
constraints which are local in the triangulation. We expect the formulation to
have applications to classical discrete gravity and non-perturbative approaches
to quantum gravity.Comment: 7 pages, 1 figure. v2 small changes to match published versio
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
Phase space descriptions for simplicial 4d geometries
Starting from the canonical phase space for discretised (4d) BF-theory, we
implement a canonical version of the simplicity constraints and construct phase
spaces for simplicial geometries. Our construction allows us to study the
connection between different versions of Regge calculus and approaches using
connection variables, such as loop quantum gravity. We find that on a fixed
triangulation the (gauge invariant) phase space associated to loop quantum
gravity is genuinely larger than the one for length and even area Regge
calculus. Rather, it corresponds to the phase space of area-angle Regge
calculus, as defined by Dittrich and Speziale in [arXiv:0802.0864] (prior to
the imposition of gluing constraints, that ensure the metricity of the
triangulation). We argue that this is due to the fact that the simplicity
constraints are not fully implemented in canonical loop quantum gravity.
Finally, we show that for a subclass of triangulations one can construct first
class Hamiltonian and Diffeomorphism constraints leading to flat 4d
space-times.Comment: corrected structure constants, several references ad
Operator Spin Foam Models
The goal of this paper is to introduce a systematic approach to spin foams.
We define operator spin foams, that is foams labelled by group representations
and operators, as the main tool. An equivalence relation we impose in the set
of the operator spin foams allows to split the faces and the edges of the
foams. The consistency with that relation requires introduction of the
(familiar for the BF theory) face amplitude. The operator spin foam models are
defined quite generally. Imposing a maximal symmetry leads to a family we call
natural operator spin foam models. This symmetry, combined with demanding
consistency with splitting the edges, determines a complete characterization of
a general natural model. It can be obtained by applying arbitrary (quantum)
constraints on an arbitrary BF spin foam model. In particular, imposing
suitable constraints on Spin(4) BF spin foam model is exactly the way we tend
to view 4d quantum gravity, starting with the BC model and continuing with the
EPRL or FK models. That makes our framework directly applicable to those
models. Specifically, our operator spin foam framework can be translated into
the language of spin foams and partition functions. We discuss the examples: BF
spin foam model, the BC model, and the model obtained by application of our
framework to the EPRL intertwiners.Comment: 19 pages, 11 figures, RevTex4.
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
Coupling gauge theory to spinfoam 3d quantum gravity
We construct a spinfoam model for Yang-Mills theory coupled to quantum
gravity in three dimensional riemannian spacetime. We define the partition
function of the coupled system as a power series in g_0^2 G that can be
evaluated order by order using grasping rules and the recoupling theory. With
respect to previous attempts in the literature, this model assigns the
dynamical variables of gravity and Yang-Mills theory to the same simplices of
the spinfoam, and it thus provides transition amplitudes for the spin network
states of the canonical theory. For SU(2) Yang-Mills theory we show explicitly
that the partition function has a semiclassical limit given by the Regge
discretization of the classical Yang-Mills action.Comment: 18 page
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