1,701 research outputs found
Geometry eigenvalues and scalar product from recoupling theory in loop quantum gravity
We summarize the basics of the loop representation of quantum gravity and describe the main aspects of the formalism, including its latest developments, in a reorganized and consistent form. Recoupling theory, in its graphical tangle-theoretic Temperley-Lieb formulation, provides a powerful calculation tool in this context. We describe its application to the loop representation in detail. Using recoupling theory, we derive general expressions for the spectrum of the quantum area and the quantum volume operators. We compute several volume eigenvalues explicitly. We introduce a scalar product with respect to which area and volume are symmetric operators, and (the trivalent expansions of) the spin network states are orthonormal
Counting surface states in the loop quantum gravity
We adopt the point of view that (Riemannian) classical and (loop-based)
quantum descriptions of geometry are macro- and micro-descriptions in the usual
statistical mechanical sense. This gives rise to the notion of geometrical
entropy, which is defined as the logarithm of the number of different quantum
states which correspond to one and the same classical geometry configuration
(macro-state). We apply this idea to gravitational degrees of freedom induced
on an arbitrarily chosen in space 2-dimensional surface. Considering an
`ensemble' of particularly simple quantum states, we show that the geometrical
entropy corresponding to a macro-state specified by a total area of
the surface is proportional to the area , with being
approximately equal to . The result holds both for case of open
and closed surfaces. We discuss briefly physical motivations for our choice of
the ensemble of quantum states.Comment: This paper is a substantially modified version of the paper `The
Bekenstein bound and non-perturbative quantum gravity'. Although the main
result (i.e. the result of calculation of the number of quantum states that
correspond to one and the same area of 2-d surface) remains unchanged, it is
presented now from a different point of view. The new version contains a
discussion both of the case of open and closed surfaces, and a discussion of
a possibility to generalize the result obtained considering arbitrary surface
quantum states. LaTeX, 21 pages, 6 figures adde
A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements
We study a generalized version of the Hamiltonian constraint operator in
nonperturbative loop quantum gravity. The generalization is based on admitting
arbitrary irreducible SU(2) representations in the regularization of the
operator, in contrast to the original definition where only the fundamental
representation is taken. This leads to a quantization ambiguity and to a family
of operators with the same classical limit. We calculate the action of the
Euclidean part of the generalized Hamiltonian constraint on trivalent states,
using the graphical notation of Temperley-Lieb recoupling theory. We discuss
the relation between this generalization of the Hamiltonian constraint and
crossing symmetry.Comment: 35 pp, 20 eps figures; minor corrections, references added; version
to appear in Class. Quant. Gra
Group Field Theory: An overview
We give a brief overview of the properties of a higher dimensional
generalization of matrix model which arises naturally in the context of a
background independent approach to quantum gravity, the so called group field
theory. We show that this theory leads to a natural proposal for the physical
scalar product of quantum gravity. We also show in which sense this theory
provides a third quantization point of view on quantum gravity.Comment: 10 page
Continuum spin foam model for 3d gravity
An example illustrating a continuum spin foam framework is presented. This
covariant framework induces the kinematics of canonical loop quantization, and
its dynamics is generated by a {\em renormalized} sum over colored polyhedra.
Physically the example corresponds to 3d gravity with cosmological constant.
Starting from a kinematical structure that accommodates local degrees of
freedom and does not involve the choice of any background structure (e. g.
triangulation), the dynamics reduces the field theory to have only global
degrees of freedom. The result is {\em projectively} equivalent to the
Turaev-Viro model.Comment: 12 pages, 3 figure
A spin foam model for pure gauge theory coupled to quantum gravity
We propose a spin foam model for pure gauge fields coupled to Riemannian
quantum gravity in four dimensions. The model is formulated for the
triangulation of a four-manifold which is given merely combinatorially. The
Riemannian Barrett--Crane model provides the gravity sector of our model and
dynamically assigns geometric data to the given combinatorial triangulation.
The gauge theory sector is a lattice gauge theory living on the same
triangulation and obtains from the gravity sector the geometric information
which is required to calculate the Yang--Mills action. The model is designed so
that one obtains a continuum approximation of the gauge theory sector at an
effective level, similarly to the continuum limit of lattice gauge theory, when
the typical length scale of gravity is much smaller than the Yang--Mills scale.Comment: 18 pages, LaTeX, 1 figure, v2: details clarified, references adde
Photons from quantized electric flux representations
The quantum theory of U(1) connections admits a diffeomorphism invariant
representation in which the electric flux through any surface is quantized.
This representation is the analog of the representation of quantum SU(2) theory
used in loop quantum gravity. We investigate the relation between this
representation, in which the basic excitations are `polymer-like', and the Fock
representation, in which the basic excitations are wave-like photons. We show
that normalizable states in the Fock space are associated with `distributional'
states in the quantized electric flux representation. This work is motivated by
the question of how wave-like gravitons in linearised gravity arise from
polymer-like states in non-perturbative loop quantum gravity.Comment: 22 pages, no figure
The Chrono-geometrical Structure of Special and General Relativity: a Re-Visitation of Canonical Geometrodynamics
A modern re-visitation of the consequences of the lack of an intrinsic notion
of instantaneous 3-space in relativistic theories leads to a reformulation of
their kinematical basis emphasizing the role of non-inertial frames centered on
an arbitrary accelerated observer. In special relativity the exigence of
predictability implies the adoption of the 3+1 point of view, which leads to a
well posed initial value problem for field equations in a framework where the
change of the convention of synchronization of distant clocks is realized by
means of a gauge transformation. This point of view is also at the heart of the
canonical approach to metric and tetrad gravity in globally hyperbolic
asymptotically flat space-times, where the use of Shanmugadhasan canonical
transformations allows the separation of the physical degrees of freedom of the
gravitational field (the tidal effects) from the arbitrary gauge variables.
Since a global vision of the equivalence principle implies that only global
non-inertial frames can exist in general relativity, the gauge variables are
naturally interpreted as generalized relativistic inertial effects, which have
to be fixed to get a deterministic evolution in a given non-inertial frame. As
a consequence, in each Einstein's space-time in this class the whole
chrono-geometrical structure, including also the clock synchronization
convention, is dynamically determined and a new approach to the Hole Argument
leads to the conclusion that "gravitational field" and "space-time" are two
faces of the same entity. This view allows to get a classical scenario for the
unification of the four interactions in a scheme suited to the description of
the solar system or our galaxy with a deperametrization to special relativity
and the subsequent possibility to take the non-relativistic limit.Comment: 33 pages, Lectures given at the 42nd Karpacz Winter School of
Theoretical Physics, "Current Mathematical Topics in Gravitation and
Cosmology", Ladek, Poland, 6-11 February 200
Quantum states of elementary three-geometry
We introduce a quantum volume operator in three--dimensional Quantum
Gravity by taking into account a symmetrical coupling scheme of three SU(2)
angular momenta. The spectrum of is discrete and defines a complete set of
eigenvectors which is alternative with respect to the complete sets employed
when the usual binary coupling schemes of angular momenta are considered. Each
of these states, that we call quantum bubbles, represents an interference of
extended configurations which provides a rigorous meaning to the heuristic
notion of quantum tetrahedron. We study the generalized recoupling coefficients
connecting the symmetrical and the binary basis vectors, and provide an
explicit recursive solution for such coefficients by analyzing also its
asymptotic limit.Comment: 15 pages, LaTe
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