595 research outputs found
A Lorentzian Gromov-Hausdoff notion of distance
This paper is the first of three in which I study the moduli space of
isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I
introduce a notion of Gromov-Hausdorff distance which makes this moduli space
into a metric space. Further properties of this metric space are studied in the
next papers. The importance of the work can be situated in fields such as
cosmology, quantum gravity and - for the mathematicians - global Lorentzian
geometry.Comment: 20 pages, 0 figures, submitted to Classical and quantum gravity,
seriously improved presentatio
Short-distance regularity of Green's function and UV divergences in entanglement entropy
Reformulating our recent result (arXiv:1007.1246 [hep-th]) in coordinate
space we point out that no matter how regular is short-distance behavior of
Green's function the entanglement entropy in the corresponding quantum field
theory is always UV divergent. In particular, we discuss a recent example by
Padmanabhan (arXiv:1007.5066 [gr-qc]) of a regular Green's function and show
that provided this function arises in a field theory the entanglement entropy
in this theory is UV divergent and calculate the leading divergent term.Comment: LaTeX, 6 page
Discrete Time from Quantum Physics
't Hooft has recently developed a discretisation of (2+1) gravity which has a
multiple-valued Hamiltonian and which therefore admits quantum time evolution
only in discrete steps. In this paper, we describe several models in the
continuum with single-valued equations of motion in classical physics, but with
multiple-valued Hamiltonians. Their time displacements in quantum theory are
therefore obliged to be discrete. Classical models on smooth spatial manifolds
are also constructed with the property that spatial displacements can be
implemented only in discrete steps in quantum theory. All these models show
that quantization can profoundly affect classical topology.Comment: 21 pages with 2 figures, SU-4240-579 (figures corrected in this
version
The Semiclassical Limit of Loop Quantum Cosmology
The continuum and semiclassical limits of isotropic, spatially flat loop
quantum cosmology are discussed, with an emphasis on the role played by the
Barbero-Immirzi parameter \gamma in controlling space-time discreteness. In
this way, standard quantum cosmology is shown to be the simultaneous limit
\gamma \to 0, j \to \infty of loop quantum cosmology. Here, j is a label of the
volume eigenvalues, and the simultaneous limit is technically the same as the
classical limit \hbar \to 0, l \to \infty of angular momentum in quantum
mechanics. Possible lessons for semiclassical states at the dynamical level in
the full theory of quantum geometry are mentioned.Comment: 10 page
Remarks on Thermal Strings outside Black Holes
We discuss the semiclassical approximation to the level density of (super)
strings propagating in non-compact coset manifolds . We show that the WKB
ansatz agrees with heuristic red-shift arguments with respect to the ``exact"
sigma model metric, up to some deviations from minimal coupling, parametrized
by the dilaton. This approximation is used to study thermal ensembles of free
strings in black holes, with the ``brick wall" regularization of `t Hooft. In
two dimensions the entropy diverges logarithmically with the horizon thickness,
and a local Hagedorn transition occurs in higher dimensional models. We also
observe that supersymmetry improves the regularity of strings at the horizon.Comment: 13 pages, PUPT-94-147
The graviton vacuum as a distributional state in kinematic Loop Quantum Gravity
The quantum behaviour of weak gravitational fields admits an adequate, albeit
approximate, description by those graviton states in which the expectation
values and fluctuations of the linearised gravitational field are small. Such
states must approximate corresponding states in full quantum gravity. We
analyse the nature of this approximation for the graviton vacuum state in the
context of kinematical Loop Quantum Gravity (LQG) wherein the constraints are
ignored. We identify the graviton vacuum state with kinematically
non-normalizable, distributional states in LQG by demanding that relations
between linearised operator actions on the former are mirrored by those of
their non-linear counterparts on the latter. We define a semi- norm on the
space of kinematical distributions and show that the identification is
approximate upto distributions which are small in this semi-norm. We argue that
our candidate states are annihilated by the linearised constraints (expressed
as operators in the full theory) to leading order in the parameter
characterising the approximation. This suggests the possibility, in a scheme
such as ours, of solving the full constraints order by order in this parameter.
The main drawback of our considerations is that they depend on certain
auxilliary constructions which, though mathematically well defined, do not
arise from physical insight. Our work is an attempt to implement an earlier
proposal of Iwasaki and Rovelli.Comment: 44 pages, no figure
The Value of Singularities
We point out that spacetime singularities play a useful role in gravitational
theories by eliminating unphysical solutions. In particular, we argue that any
modification of general relativity which is completely nonsingular cannot have
a stable ground state. This argument applies both to classical extensions of
general relativity, and to candidate quantum theories of gravity.Comment: 5 pages, no figures; a few clarifying comments adde
Entanglement entropy in gauge theories and the holographic principle for electric strings
We consider quantum entanglement between gauge fields in some region of space
A and its complement B. It is argued that the Hilbert space of physical states
of gauge theories cannot be decomposed into a direct product of Hilbert spaces
of states localized in A and B. The reason is that elementary excitations in
gauge theories - electric strings - are associated with closed loops rather
than points in space, and there are closed loops which belong both to A and B.
Direct product structure and hence the reduction procedure with respect to the
fields in B can only be defined if the Hilbert space of physical states is
extended by including the states of electric strings which can open on the
boundary of A. The positions of string endpoints on this boundary are the
additional degrees of freedom which also contribute to the entanglement
entropy. We explicitly demonstrate this for the three-dimensional Z2 lattice
gauge theory both numerically and using a simple trial ground state wave
function. The entanglement entropy appears to be saturated almost completely by
the entropy of string endpoints, thus reminding of a ``holographic principle''
in quantum gravity and AdS/CFT correspondence.Comment: 6 pages RevTeX, 5 figure
On alternative approaches to Lorentz violation in loop quantum gravity inspired models
Recent claims point out that possible violations of Lorentz symmetry
appearing in some semiclassical models of extended matter dynamics motivated by
loop quantum gravity can be removed by a different choice of canonically
conjugated variables. In this note we show that such alternative is
inconsistent with the choice of variables in the underlying quantum theory
together with the semiclassical approximation, as long as the correspondence
principle is maintained. A consistent choice will violate standard Lorentz
invariance. Thus, to preserve a relativity principle in this framework, the
linear realization of Lorentz symmetry should be extended or superseded.Comment: 4 pages, revtex4, no figures, references adde
Event horizon - Magnifying glass for Planck length physics
An attempt is made to describe the `thermodynamics' of semiclassical
spacetime without specifying the detailed `molecular structure' of the quantum
spacetime, using the known properties of blackholes. I give detailed arguments,
essentially based on the behaviour of quantum systems near the event horizon,
which suggest that event horizon acts as a magnifying glass to probe Planck
length physics even in those contexts in which the spacetime curvature is
arbitrarily low. The quantum state describing a blackhole, in any microscopic
description of spacetime, has to possess certain universal form of density of
states which can be ascertained from general considerations. Since a blackhole
can be formed from the collapse of any physical system with a low energy
Hamiltonian H, it is suggested that when such a system collapses to form a
blackhole, it should be described by a modified Hamiltonian of the form
where .I also show
that it is possible to construct several physical systems which have the
blackhole density of states and hence will be indistinguishable from a
blackhole as far as thermodynamic interactions are concerned. In particular,
blackholes can be thought of as one-particle excitations of a class of {\it
nonlocal} field theories with the thermodynamics of blackholes arising
essentially from the asymptotic form of the dispersion relation satisfied by
these excitations. These field theoretic models have correlation functions with
a universal short distance behaviour, which translates into the generic
behaviour of semiclassical blackholes. Several implications of this paradigm
are discussed
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