8 research outputs found
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
Canonical path integral measures for Holst and Plebanski gravity. I. Reduced Phase Space Derivation
An important aspect in defining a path integral quantum theory is the
determination of the correct measure. For interacting theories and theories
with constraints, this is non-trivial, and is normally not the heuristic
"Lebesgue measure" usually used. There have been many determinations of a
measure for gravity in the literature, but none for the Palatini or Holst
formulations of gravity. Furthermore, the relations between different resulting
measures for different formulations of gravity are usually not discussed.
In this paper we use the reduced phase technique in order to derive the
path-integral measure for the Palatini and Holst formulation of gravity, which
is different from the Lebesgue measure up to local measure factors which depend
on the spacetime volume element and spatial volume element.
From this path integral for the Holst formulation of GR we can also give a
new derivation of the Plebanski path integral and discover a discrepancy with
the result due to Buffenoir, Henneaux, Noui and Roche (BHNR) whose origin we
resolve. This paper is the first in a series that aims at better understanding
the relation between canonical LQG and the spin foam approach.Comment: 27 pages, minor correction
One vertex spin-foams with the Dipole Cosmology boundary
We find all the spin-foams contributing in the first order of the vertex
expansion to the transition amplitude of the Bianchi-Rovelli-Vidotto Dipole
Cosmology model. Our algorithm is general and provides spin-foams of
arbitrarily given, fixed: boundary and, respectively, a number of internal
vertices. We use the recently introduced Operator Spin-Network Diagrams
framework.Comment: 23 pages, 30 figure
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