5,651 research outputs found
Local spinfoam expansion in loop quantum cosmology
The quantum dynamics of the flat Friedmann-Lemaitre-Robertson-Walker and
Bianchi I models defined by loop quantum cosmology have recently been
translated into a spinfoam-like formalism. The construction is facilitated by
the presence of a massless scalar field which is used as an internal clock. The
implicit integration over the matter variable leads to a nonlocal spinfoam
amplitude. In this paper we consider a vacuum Bianchi I universe and show that
by choosing an appropriate regulator a spinfoam expansion can be obtained
without selecting a clock variable and that the resulting spinfoam amplitude is
local.Comment: 12 page
Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory
We study the large-j asymptotics of the Euclidean EPRL/FK spin foam amplitude
on a 4d simplicial complex with arbitrary number of simplices. We show that for
a critical configuration (j_f, g_{ve}, n_{ef}) in general, there exists a
partition of the simplicial complex into three regions: Non-degenerate region,
Type-A degenerate region and Type-B degenerate region. On both the
non-degenerate and Type-A degenerate regions, the critical configuration
implies a non-degenerate Euclidean geometry, while on the Type-B degenerate
region, the critical configuration implies a vector geometry. Furthermore we
can split the Non-degenerate and Type-A regions into sub-complexes according to
the sign of Euclidean oriented 4-simplex volume. On each sub-complex, the spin
foam amplitude at critical configuration gives a Regge action that contains a
sign factor sgn(V_4(v)) of the oriented 4-simplices volume. Therefore the Regge
action reproduced here can be viewed as a discretized Palatini action with
on-shell connection. The asymptotic formula of the spin foam amplitude is given
by a sum of the amplitudes evaluated at all possible critical configurations,
which are the products of the amplitudes associated to different type of
geometries.Comment: 27 pages, 5 figures, references adde
Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Lorentzian Theory
The present paper studies the large-j asymptotics of the Lorentzian EPRL
spinfoam amplitude on a 4d simplicial complex with an arbitrary number of
simplices. The asymptotics of the spinfoam amplitude is determined by the
critical configurations. Here we show that, given a critical configuration in
general, there exists a partition of the simplicial complex into three type of
regions R_{Nondeg}, R_{Deg-A}, R_{Deg-B}, where the three regions are
simplicial sub-complexes with boundaries. The critical configuration implies
different types of geometries in different types of regions, i.e. (1) the
critical configuration restricted into R_{Nondeg} is degenerate of type-A in our definition of degeneracy, but implies
a nondegenerate discrete Euclidean geometry on R_{Deg-A}, (3) the critical
configuration restricted into R_{Deg-B} is degenerate of type-B, and implies a
vector geometry on R_{Deg-B}. With the critical configuration, we further make
a subdivision of the regions R_{Nondeg} and R_{Deg-A} into sub-complexes (with
boundary) according to their Lorentzian/Euclidean oriented 4-simplex volume
V_4(v), such that sgn(V_4(v)) is a constant sign on each sub-complex. Then in
the each sub-complex, the spinfoam amplitude at the critical configuration
gives the Regge action in Lorentzian or Euclidean signature respectively on
R_{Nondeg} or R_{Deg-A}. The Regge action reproduced here contains a sign
factor sgn(V_4(v)) of the oriented 4-simplex volume. Therefore the Regge action
reproduced here can be viewed a discretized Palatini action with on-shell
connection. Finally the asymptotic formula of the spinfoam amplitude is given
by a sum of the amplitudes evaluated at all possible critical configurations,
which are the products of the amplitudes associated to different type of
geometries.Comment: 54 pages, 2 figures, reference adde
Loop quantum gravity: the first twenty five years
This is a review paper invited by the journal "Classical ad Quantum Gravity"
for a "Cluster Issue" on approaches to quantum gravity. I give a synthetic
presentation of loop gravity. I spell-out the aims of the theory and compare
the results obtained with the initial hopes that motivated the early interest
in this research direction. I give my own perspective on the status of the
program and attempt of a critical evaluation of its successes and limits.Comment: 24 pages, 3 figure
Euclidean three-point function in loop and perturbative gravity
We compute the leading order of the three-point function in loop quantum
gravity, using the vertex expansion of the Euclidean version of the new spin
foam dynamics, in the region of gamma<1. We find results consistent with Regge
calculus in the limit gamma->0 and j->infinity. We also compute the tree-level
three-point function of perturbative quantum general relativity in position
space, and discuss the possibility of directly comparing the two results.Comment: 16 page
Feynman diagrammatic approach to spin foams
"The Spin Foams for People Without the 3d/4d Imagination" could be an
alternative title of our work. We derive spin foams from operator spin network
diagrams} we introduce. Our diagrams are the spin network analogy of the
Feynman diagrams. Their framework is compatible with the framework of Loop
Quantum Gravity. For every operator spin network diagram we construct a
corresponding operator spin foam. Admitting all the spin networks of LQG and
all possible diagrams leads to a clearly defined large class of operator spin
foams. In this way our framework provides a proposal for a class of 2-cell
complexes that should be used in the spin foam theories of LQG. Within this
class, our diagrams are just equivalent to the spin foams. The advantage,
however, in the diagram framework is, that it is self contained, all the
amplitudes can be calculated directly from the diagrams without explicit
visualization of the corresponding spin foams. The spin network diagram
operators and amplitudes are consistently defined on their own. Each diagram
encodes all the combinatorial information. We illustrate applications of our
diagrams: we introduce a diagram definition of Rovelli's surface amplitudes as
well as of the canonical transition amplitudes. Importantly, our operator spin
network diagrams are defined in a sufficiently general way to accommodate all
the versions of the EPRL or the FK model, as well as other possible models. The
diagrams are also compatible with the structure of the LQG Hamiltonian
operators, what is an additional advantage. Finally, a scheme for a complete
definition of a spin foam theory by declaring a set of interaction vertices
emerges from the examples presented at the end of the paper.Comment: 36 pages, 23 figure
Dynamics of Scalar Field in Polymer-like Representation
In recent twenty years, loop quantum gravity, a background independent
approach to unify general relativity and quantum mechanics, has been widely
investigated. We consider the quantum dynamics of a real massless scalar field
coupled to gravity in this framework. A Hamiltonian operator for the scalar
field can be well defined in the coupled diffeomorphism invariant Hilbert
space, which is both self-adjoint and positive. On the other hand, the
Hamiltonian constraint operator for the scalar field coupled to gravity can be
well defined in the coupled kinematical Hilbert space. There are 1-parameter
ambiguities due to scalar field in the construction of both operators. The
results heighten our confidence that there is no divergence within this
background independent and diffeomorphism invariant quantization approach of
matter coupled to gravity. Moreover, to avoid possible quantum anomaly, the
master constraint programme can be carried out in this coupled system by
employing a self-adjoint master constraint operator on the diffeomorphism
invariant Hilbert space.Comment: 24 pages, accepted for pubilcation in Class. Quant. Gra
Commuting Simplicity and Closure Constraints for 4D Spin Foam Models
Spin Foam Models are supposed to be discretised path integrals for quantum
gravity constructed from the Plebanski-Holst action. The reason for there being
several models currently under consideration is that no consensus has been
reached for how to implement the simplicity constraints. Indeed, none of these
models strictly follows from the original path integral with commuting B
fields, rather, by some non standard manipulations one always ends up with non
commuting B fields and the simplicity constraints become in fact anomalous
which is the source for there being several inequivalent strategies to
circumvent the associated problems. In this article, we construct a new
Euclidian Spin Foam Model which is constructed by standard methods from the
Plebanski-Holst path integral with commuting B fields discretised on a 4D
simplicial complex. The resulting model differs from the current ones in
several aspects, one of them being that the closure constraint needs special
care. Only when dropping the closure constraint by hand and only in the large
spin limit can the vertex amplitudes of this model be related to those of the
FK Model but even then the face and edge amplitude differ. Curiously, an ad hoc
non-commutative deformation of the variables leads from our new model
to the Barrett-Crane Model in the case of Barbero-Immirzi parameter goes to
infinity.Comment: 41 pages, 4 figure
The Holst Spin Foam Model via Cubulations
Spin foam models are an attempt for a covariant, or path integral formulation
of canonical loop quantum gravity. The construction of such models usually rely
on the Plebanski formulation of general relativity as a constrained BF theory
and is based on the discretization of the action on a simplicial triangulation,
which may be viewed as an ultraviolet regulator. The triangulation dependence
can be removed by means of group field theory techniques, which allows one to
sum over all triangulations. The main tasks for these models are the correct
quantum implementation of the Plebanski constraints, the existence of a
semiclassical sector implementing additional "Regge-like" constraints arising
from simplicial triangulations, and the definition of the physical inner
product of loop quantum gravity via group field theory. Here we propose a new
approach to tackle these issues stemming directly from the Holst action for
general relativity, which is also a proper starting point for canonical loop
quantum gravity. The discretization is performed by means of a "cubulation" of
the manifold rather than a triangulation. We give a direct interpretation of
the resulting spin foam model as a generating functional for the n-point
functions on the physical Hilbert space at finite regulator. This paper focuses
on ideas and tasks to be performed before the model can be taken seriously.
However, our analysis reveals some interesting features of this model: first,
the structure of its amplitudes differs from the standard spin foam models.
Second, the tetrad n-point functions admit a "Wick-like" structure. Third, the
restriction to simple representations does not automatically occur -- unless
one makes use of the time gauge, just as in the classical theory.Comment: 25 pages, 1 figure; v3: published version. arXiv admin note:
substantial text overlap with arXiv:0911.213
On the Relation between Operator Constraint --, Master Constraint --, Reduced Phase Space --, and Path Integral Quantisation
Path integral formulations for gauge theories must start from the canonical
formulation in order to obtain the correct measure. A possible avenue to derive
it is to start from the reduced phase space formulation. In this article we
review this rather involved procedure in full generality. Moreover, we
demonstrate that the reduced phase space path integral formulation formally
agrees with the Dirac's operator constraint quantisation and, more
specifically, with the Master constraint quantisation for first class
constraints. For first class constraints with non trivial structure functions
the equivalence can only be established by passing to Abelian(ised) constraints
which is always possible locally in phase space. Generically, the correct
configuration space path integral measure deviates from the exponential of the
Lagrangian action. The corrections are especially severe if the theory suffers
from second class secondary constraints. In a companion paper we compute these
corrections for the Holst and Plebanski formulations of GR on which current
spin foam models are based.Comment: 43 page
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