8,924 research outputs found
Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction
A Planck-scale minimal observable length appears in many approaches to
quantum gravity. It is sometimes argued that this minimal length might conflict
with Lorentz invariance, because a boosted observer could see the minimal
length further Lorentz contracted. We show that this is not the case within
loop quantum gravity. In loop quantum gravity the minimal length (more
precisely, minimal area) does not appear as a fixed property of geometry, but
rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted
observer can see the same observable spectrum, with the same minimal area. What
changes continuously in the boost transformation is not the value of the
minimal length: it is the probability distribution of seeing one or the other
of the discrete eigenvalues of the area. We discuss several difficulties
associated with boosts and area measurement in quantum gravity. We compute the
transformation of the area operator under a local boost, propose an explicit
expression for the generator of local boosts and give the conditions under
which its action is unitary.Comment: 12 pages, 3 figure
The century of the incomplete revolution: searching for general relativistic quantum field theory
In fundamental physics, this has been the century of quantum mechanics and
general relativity. It has also been the century of the long search for a
conceptual framework capable of embracing the astonishing features of the world
that have been revealed by these two ``first pieces of a conceptual
revolution''. I discuss the general requirements on the mathematics and some
specific developments towards the construction of such a framework. Examples of
covariant constructions of (simple) generally relativistic quantum field
theories have been obtained as topological quantum field theories, in
nonperturbative zero-dimensional string theory and its higher dimensional
generalizations, and as spin foam models. A canonical construction of a general
relativistic quantum field theory is provided by loop quantum gravity.
Remarkably, all these diverse approaches have turn out to be related,
suggesting an intriguing general picture of general relativistic quantum
physics.Comment: To appear in the Journal of Mathematical Physics 2000 Special Issu
Black hole entropy: inside or out?
A trialogue. Ted, Don, and Carlo consider the nature of black hole entropy.
Ted and Carlo support the idea that this entropy measures in some sense ``the
number of black hole microstates that can communicate with the outside world.''
Don is critical of this approach, and discussion ensues, focusing on the
question of whether the first law of black hole thermodynamics can be
understood from a statistical mechanics point of view.Comment: 42 pages, contribution to proceedings of Peyresq
Graphical Evolution of Spin Network States
The evolution of spin network states in loop quantum gravity can be described
by introducing a time variable, defined by the surfaces of constant value of an
auxiliary scalar field. We regulate the Hamiltonian, generating such an
evolution, and evaluate its action both on edges and on vertices of the spin
network states. The analytical computations are carried out completely to yield
a finite, diffeomorphism invariant result. We use techniques from the
recoupling theory of colored graphs with trivalent vertices to evaluate the
graphical part of the Hamiltonian action. We show that the action on edges is
equivalent to a diffeomorphism transformation, while the action on vertices
adds new edges and re-routes the loops through the vertices.Comment: 24 pages, 21 PostScript figures, uses epsfig.sty, Minor corrections
in the final formula in the main body of the paper and in the formula for the
Tetrahedral net in the Appendi
The physical hamiltonian in nonperturbative quantum gravity
A quantum hamiltonian which evolves the gravitational field according to time
as measured by constant surfaces of a scalar field is defined through a
regularization procedure based on the loop representation, and is shown to be
finite and diffeomorphism invariant. The problem of constructing this
hamiltonian is reduced to a combinatorial and algebraic problem which involves
the rearrangements of lines through the vertices of arbitrary graphs. This
procedure also provides a construction of the hamiltonian constraint as a
finite operator on the space of diffeomorphism invariant states as well as a
construction of the operator corresponding to the spatial volume of the
universe.Comment: Latex, 11 pages, no figures, CGPG/93/
Physics with nonperturbative quantum gravity: radiation from a quantum black hole
We study quantum gravitational effects on black hole radiation, using loop
quantum gravity. Bekenstein and Mukhanov have recently considered the
modifications caused by quantum gravity on Hawking's thermal black-hole
radiation. Using a simple ansatz for the eigenstates the area, they have
obtained the intriguing result that the quantum properties of geometry affect
the radiation considerably, yielding a definitely non-thermal spectrum. Here,
we replace the simple ansatz employed by Bekenstein and Mukhanov with the
actual eigenstates of the area, computed using the loop representation of
quantum gravity. We derive the emission spectra, using a classic result in
number theory by Hardy and Ramanujan. Disappointingly, we do not recover the
Bekenstein-Mukhanov spectrum, but --effectively-- a Hawking's thermal spectrum.
The Bekenstein-Mukhanov result is therefore likely to be an artefact of the
naive ansatz, rather than a robust result. The result is an example of concrete
(although somewhat disappointing) application of nonperturbative quantum
gravity.Comment: 4 pages, latex-revtex, no figure
A semiclassical tetrahedron
We construct a macroscopic semiclassical state state for a quantum
tetrahedron. The expectation values of the geometrical operators representing
the volume, areas and dihedral angles are peaked around assigned classical
values, with vanishing relative uncertainties.Comment: 10 pages; v2 revised versio
Holographic Formulation of Quantum Supergravity
We show that supergravity with a cosmological constant can be
expressed as constrained topological field theory based on the supergroup
. The theory is then extended to include timelike boundaries with
finite spatial area. Consistent boundary conditions are found which induce a
boundary theory based on a supersymmetric Chern-Simons theory. The boundary
state space is constructed from states of the boundary supersymmetric
Chern-Simons theory on the punctured two sphere and naturally satisfies the
Bekenstein bound, where area is measured by the area operator of quantum
supergravity.Comment: 30 pages, no figur
Towards the graviton from spinfoams: the 3d toy model
Recently, a proposal has appeared for the extraction of the 2-point function
of linearised quantum gravity, within the spinfoam formalism. This relies on
the use of a boundary state, which introduces a semi-classical flat geometry on
the boundary. In this paper, we investigate this proposal considering a toy
model in the (Riemannian) 3d case, where the semi-classical limit is better
understood. We show that in this limit the propagation kernel of the model is
the one for the harmonic oscillator. This is at the origin of the expected 1/L
behaviour of the 2-point function. Furthermore, we numerically study the short
scales regime, where deviations from this behaviour occur.Comment: 8 pages, 2 figures; v3 revised versio
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
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