210 research outputs found
Global Existence of Solutions of the Semiclassical Einstein Equation for Cosmological Spacetimes
We study the solutions of the semiclassical Einstein equation in flat
cosmological spacetimes driven by a massive conformally coupled scalar field.
In particular, we show that it is possible to give initial conditions at finite
time to get a state for the quantum field which gives finite expectation values
for the stress-energy tensor. Furthermore, it is possible to control this
expectation value by means of a global estimate on regular cosmological
spacetimes. The obtained estimates permit to write a theorem about the
existence and uniqueness of the local solutions encompassing both the spacetime
metric and the matter field simultaneously. Finally, we show that one can
always extend local solutions up to a point where the scale factor becomes
singular or the Hubble function reaches a critical value ,
which both correspond to a divergence of the scalar curvature, namely a
spacetime singularity.Comment: 20 pages; corrected reference
Influence of quantum matter fluctuations on geodesic deviation
We study the passive influence of quantum matter fluctuations on the
expansion parameter of a congruence of timelike geodesics in a semiclassical
regime. In particular, we show that, the perturbations of this parameter can be
considered to be elements of the algebra of matter fields at all perturbative
order. Hence, once a quantum state for matter is chosen, it is possible to
explicitly evaluate the behavior of geometric fluctuations. After introducing
the formalism necessary to treat similar problems, in the last part of the
paper, we estimate the approximated probability of having a geodesic collapse
in a flat spacetime due to those fluctuations.Comment: 21 pages, published version, J. Phys. A: Math. Theor. 47 (2014)
37520
Interplay of Boltzmann equation and continuity equation for accelerated electrons in solar flares
During solar flares a large amount of electrons are accelerated within the
plasma present in the solar atmosphere. Accurate measurements of the motion of
these electrons start becoming available from the analysis of hard X-ray
imaging-spectroscopy observations. In this paper, we discuss the linearized
perturbations of the Boltzmann kinetic equation describing an ensemble of
electrons accelerated by the energy release occurring during solar flares.
Either in the limit of high energy or at vanishing background temperature such
an equation reduces to a continuity equation equipped with an extra force of
stochastic nature. This stochastic force is actually described by the well
known energy loss rate due to Coulomb collision with ambient particles, but, in
order to match the collision kernel in the linearized Boltzmann equation it
needs to be treated in a very specific manner. In the second part of the paper
the derived continuity equation is solved with some hyperbolic techniques, and
the obtained solution is written in a form suitable to be compared with data
gathered by hard X-ray imaging-spectroscopy telescopes. Finally, a first
validation of the model with NASA Reuven Ramaty High Energy Solar Spectroscopic
Imager spectrometer measurements is provided.Comment: submitted to SIAM/ASA Journal on Uncertainty Quantificatio
Holography and Conformal Symmetry near black hole horizons
We show here how it is possible to build a QFT on the horizon of a Schwarzschild-like spacetime. That theory, found by restricting bulk quantum elds on the horizon, is equivalent to QFT on the bulk. That fact is called Holography. Moreover the hidden conformal symmetry (SL(2;R)) found for the bulk theory becomes manifest on the horizon in terms of some of its dieomorphisms. Then the extension of group of the generator of that symmetry to the Virasoro algebra is discussed
State independence for tunneling processes through black hole horizons and Hawking radiation
Tunneling processes through black hole horizons have recently been
investigated in the framework of WKB theory discovering interesting interplay
with the Hawking radiation. In this paper we instead adopt the point of view
proper of QFT in curved spacetime, namely, we use a suitable scaling limit
technique to obtain the leading order of the correlation function related with
some tunneling process through a Killing horizon. The computation is done for
certain large class of reference quantum states for scalar fields. In the limit
of sharp localization either on the external side or on opposite sides of the
horizon, the quantum correlation functions appear to have thermal nature, where
in both cases the characteristic temperature is the Hawking one. Our approach
is valid for every stationary charged rotating non extremal black hole,
however, since the computation is completely local, it covers the case of a
Killing horizon which just temporarily exists in some finite region too. These
results give a strong support to the idea that the Hawking radiation, which is
detected at future infinity and needs some global structures to be defined, is
actually related to a local phenomenon taking place even for local geometric
structures (local Killing horizons) existing just for a while.Comment: 19 pages, one figure, some comments added, minor errors corrected,
accepted for publication in Communications in Mathematical Physic
Hadamard States From Light-like Hypersurfaces
This book provides a rather self-contained survey of the construction of
Hadamard states for scalar field theories in a large class of notable
spacetimes, possessing a (conformal) light-like boundary. The first two
sections focus on explaining a few introductory aspects of this topic and on
providing the relevant geometric background material. The notions of
asymptotically flat spacetimes and of expanding universes with a cosmological
horizon are analysed in detail, devoting special attention to the
characterization of asymptotic symmetries. In the central part of the book, the
quantization of a real scalar field theory on such class of backgrounds is
discussed within the framework of algebraic quantum field theory. Subsequently
it is explained how it is possible to encode the information of the observables
of the theory in a second, ancillary counterpart, which is built directly on
the conformal (null) boundary. This procedure, dubbed bulk-to-boundary
correspondence, has the net advantage of allowing the identification of a
distinguished state for the theory on the boundary, which admits a counterpart
in the bulk spacetime which is automatically of Hadamard form. In the last part
of the book, some applications of these states are discussed, in particular the
construction of the algebra of Wick polynomials. This book is aimed mainly, but
not exclusively, at a readership with interest in the mathematical formulation
of quantum field theory on curved backgrounds.Comment: 106 pages, 2 figures, to appear in SpringerBriefs in Mathematical
Physics, references adde
The Casimir effect from the point of view of algebraic quantum field theory
We consider a region of Minkowski spacetime bounded either by one or by two
parallel, infinitely extended plates orthogonal to a spatial direction and a
real Klein-Gordon field satisfying Dirichlet boundary conditions. We quantize
these two systems within the algebraic approach to quantum field theory using
the so-called functional formalism. As a first step we construct a suitable
unital *-algebra of observables whose generating functionals are characterized
by a labelling space which is at the same time optimal and separating and
fulfils the F-locality property. Subsequently we give a definition for these
systems of Hadamard states and we investigate explicit examples. In the case of
a single plate, it turns out that one can build algebraic states via a
pull-back of those on the whole Minkowski spacetime, moreover inheriting from
them the Hadamard property. When we consider instead two plates, algebraic
states can be put in correspondence with those on flat spacetime via the
so-called method of images, which we translate to the algebraic setting. For a
massless scalar field we show that this procedure works perfectly for a large
class of quasi-free states including the Poincar\'e vacuum and KMS states.
Eventually Wick polynomials are introduced. Contrary to the Minkowski case, the
extended algebras, built in globally hyperbolic subregions can be collected in
a global counterpart only after a suitable deformation which is expressed
locally in terms of a *-isomorphism. As a last step, we construct explicitly
the two-point function and the regularized energy density, showing, moreover,
that the outcome is consistent with the standard results of the Casimir effect.Comment: 45 pages, section 2 improved, typos correcte
Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime
The discovery of the radiation properties of black holes prompted the search
for a natural candidate quantum ground state for a massless scalar field theory
on Schwarzschild spacetime, here considered in the Eddington-Finkelstein
representation. Among the several available proposals in the literature, an
important physical role is played by the so-called Unruh state which is
supposed to be appropriate to capture the physics of a black hole formed by
spherically symmetric collapsing matter. Within this respect, we shall consider
a massless Klein-Gordon field and we shall rigorously and globally construct
such state, that is on the algebra of Weyl observables localised in the union
of the static external region, the future event horizon and the non-static
black hole region. Eventually, out of a careful use of microlocal techniques,
we prove that the built state fulfils, where defined, the so-called Hadamard
condition; hence, it is perturbatively stable, in other words realizing the
natural candidate with which one could study purely quantum phenomena such as
the role of the back reaction of Hawking's radiation. From a geometrical point
of view, we shall make a profitable use of a bulk-to-boundary reconstruction
technique which carefully exploits the Killing horizon structure as well as the
conformal asymptotic behaviour of the underlying background. From an analytical
point of view, our tools will range from Hormander's theorem on propagation of
singularities, results on the role of passive states, and a detailed use of the
recently discovered peeling behaviour of the solutions of the wave equation in
Schwarzschild spacetime.Comment: 63 pages, 3 figure
On Quantum Spacetime and the horizon problem
In the special case of a spherically symmetric solution of Einstein equations
coupled to a scalar massless field, we examine the consequences on the exact
solution imposed by a semiclassical treatment of gravitational interaction when
the scalar field is quantized. In agreement with the work of Doplicher,
Fredenhagen and Roberts (DFR), imposing the principle of gravitational
stability against localization of events, we find that the region where an
event is localized, or where initial conditions can be assigned, has a minimal
extension, of the order of the Planck length. This conclusion, though limited
to the case of spherical symmetry, is more general than that of DFR, since it
does not require the use of the notion of energy through the Heisenberg
Principle, nor of any approximation as the linearized Einstein equations.
We shall then describe the influence of this minimal length scale in a
cosmological model, namely a simple universe filled with radiation, which is
effectively described by a conformally coupled scalar field in a conformal KMS
state. Solving the backreaction, a power law inflation scenario appears close
to the initial singularity. Furthermore, the initial singularity becomes light
like and thus the standard horizon problem is avoided in this simple model.
This indication goes in the same direction as those drawn at a heuristic level
from a full use of the principle of gravitational stability against
localization of events, which point to a background dependence of the effective
Planck length, through which a-causal effects may be transmitted.Comment: 26 pages. v3: several discussions and clarifications added, misprints
correcte
Local causal structures, Hadamard states and the principle of local covariance in quantum field theory
In the framework of the algebraic formulation, we discuss and analyse some
new features of the local structure of a real scalar quantum field theory in a
strongly causal spacetime. In particular we use the properties of the
exponential map to set up a local version of a bulk-to-boundary correspondence.
The bulk is a suitable subset of a geodesic neighbourhood of any but fixed
point p of the underlying background, while the boundary is a part of the
future light cone having p as its own tip. In this regime, we provide a novel
notion for the extended *-algebra of Wick polynomials on the said cone and, on
the one hand, we prove that it contains the information of the bulk counterpart
via an injective *-homomorphism while, on the other hand, we associate to it a
distinguished state whose pull-back in the bulk is of Hadamard form. The main
advantage of this point of view arises if one uses the universal properties of
the exponential map and of the light cone in order to show that, for any two
given backgrounds M and M' and for any two subsets of geodesic neighbourhoods
of two arbitrary points, it is possible to engineer the above procedure such
that the boundary extended algebras are related via a restriction homomorphism.
This allows for the pull-back of boundary states in both spacetimes and, thus,
to set up a machinery which permits the comparison of expectation values of
local field observables in M and M'.Comment: 42 pages, xy package is used, typos corrected, clarifications adde
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