345 research outputs found
Scattering of massive Dirac fields on the Schwarzschild black hole spacetime
With a generally covariant equation of Dirac fields outside a black hole, we
develop a scattering theory for massive Dirac fields. The existence of modified
wave operators at infinity is shown by implementing a time-dependent
logarithmic phase shift from the free dynamics to offset a long-range mass
term. The phase shift we obtain is a matrix operator due to the existence of
both positive and negative energy wave components.Comment: LaTex, 17 page
The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes
Quantum fields propagating on a curved spacetime are investigated in terms of
microlocal analysis. We discuss a condition on the wave front set for the
corresponding n-point distributions, called ``microlocal spectrum condition''
(SC). On Minkowski space, this condition is satisfied as a consequence of
the usual spectrum condition. Based on Radzikowski's determination of the wave
front set of the two-point function of a free scalar field, satisfying the
Hadamard condition in the Kay and Wald sense, we construct in the second part
of this paper all Wick polynomials including the energy-momentum tensor for
this field as operator valued distributions on the manifold and prove that they
satisfy our microlocal spectrum condition.Comment: 21 pages, AMS-LaTeX, 2 figures appended as Postscript file
Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes
In this paper, we provide an elementary, unified treatment of two distinct
blue-shift instabilities for the scalar wave equation on a fixed Kerr black
hole background: the celebrated blue-shift at the Cauchy horizon (familiar from
the strong cosmic censorship conjecture) and the time-reversed red-shift at the
event horizon (relevant in classical scattering theory).
Our first theorem concerns the latter and constructs solutions to the wave
equation on Kerr spacetimes such that the radiation field along the future
event horizon vanishes and the radiation field along future null infinity
decays at an arbitrarily fast polynomial rate, yet, the local energy of the
solution is infinite near any point on the future event horizon. Our second
theorem constructs solutions to the wave equation on rotating Kerr spacetimes
such that the radiation field along the past event horizon (extended into the
black hole) vanishes and the radiation field along past null infinity decays at
an arbitrarily fast polynomial rate, yet, the local energy of the solution is
infinite near any point on the Cauchy horizon.
The results make essential use of the scattering theory developed in [M.
Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, A scattering theory for the
wave equation on Kerr black hole exteriors, preprint (2014) available at
\url{http://arxiv.org/abs/1412.8379}] and exploit directly the time-translation
invariance of the scattering map and the non-triviality of the transmission
map.Comment: 26 pages, 12 figure
Local covariant quantum field theory over spectral geometries
A framework which combines ideas from Connes' noncommutative geometry, or
spectral geometry, with recent ideas on generally covariant quantum field
theory, is proposed in the present work. A certain type of spectral geometries
modelling (possibly noncommutative) globally hyperbolic spacetimes is
introduced in terms of so-called globally hyperbolic spectral triples. The
concept is further generalized to a category of globally hyperbolic spectral
geometries whose morphisms describe the generalization of isometric embeddings.
Then a local generally covariant quantum field theory is introduced as a
covariant functor between such a category of globally hyperbolic spectral
geometries and the category of involutive algebras (or *-algebras). Thus, a
local covariant quantum field theory over spectral geometries assigns quantum
fields not just to a single noncommutative geometry (or noncommutative
spacetime), but simultaneously to ``all'' spectral geometries, while respecting
the covariance principle demanding that quantum field theories over isomorphic
spectral geometries should also be isomorphic. It is suggested that in a
quantum theory of gravity a particular class of globally hyperbolic spectral
geometries is selected through a dynamical coupling of geometry and matter
compatible with the covariance principle.Comment: 21 pages, 2 figure
Algebraic approach to quantum field theory on non-globally-hyperbolic spacetimes
The mathematical formalism for linear quantum field theory on curved
spacetime depends in an essential way on the assumption of global
hyperbolicity. Physically, what lie at the foundation of any formalism for
quantization in curved spacetime are the canonical commutation relations,
imposed on the field operators evaluated at a global Cauchy surface. In the
algebraic formulation of linear quantum field theory, the canonical commutation
relations are restated in terms of a well-defined symplectic structure on the
space of smooth solutions, and the local field algebra is constructed as the
Weyl algebra associated to this symplectic vector space. When spacetime is not
globally hyperbolic, e.g. when it contains naked singularities or closed
timelike curves, a global Cauchy surface does not exist, and there is no
obvious way to formulate the canonical commutation relations, hence no obvious
way to construct the field algebra. In a paper submitted elsewhere, we report
on a generalization of the algebraic framework for quantum field theory to
arbitrary topological spaces which do not necessarily have a spacetime metric
defined on them at the outset. Taking this generalization as a starting point,
in this paper we give a prescription for constructing the field algebra of a
(massless or massive) Klein-Gordon field on an arbitrary background spacetime.
When spacetime is globally hyperbolic, the theory defined by our construction
coincides with the ordinary Klein-Gordon field theory on aComment: 21 pages, UCSBTH-92-4
Relativistic Lee Model on Riemannian Manifolds
We study the relativistic Lee model on static Riemannian manifolds. The model
is constructed nonperturbatively through its resolvent, which is based on the
so-called principal operator and the heat kernel techniques. It is shown that
making the principal operator well-defined dictates how to renormalize the
parameters of the model. The renormalization of the parameters are the same in
the light front coordinates as in the instant form. Moreover, the
renormalization of the model on Riemannian manifolds agrees with the flat case.
The asymptotic behavior of the renormalized principal operator in the large
number of bosons limit implies that the ground state energy is positive. In 2+1
dimensions, the model requires only a mass renormalization. We obtain rigorous
bounds on the ground state energy for the n-particle sector of 2+1 dimensional
model.Comment: 23 pages, added a new section, corrected typos and slightly different
titl
Scale Invariance in disordered systems: the example of the Random Field Ising Model
We show by numerical simulations that the correlation function of the random
field Ising model (RFIM) in the critical region in three dimensions has very
strong fluctuations and that in a finite volume the correlation length is not
self-averaging. This is due to the formation of a bound state in the underlying
field theory. We argue that this non perturbative phenomenon is not particular
to the RFIM in 3-d. It is generic for disordered systems in two dimensions and
may also happen in other three dimensional disordered systems
Quantization of Dirac fields in static spacetime
On a static spacetime, the solutions of the Dirac equation are generated by a
time-independent Hamiltonian. We study this Hamiltonian and characterize the
split into positive and negative energy. We use it to find explicit expressions
for advanced and retarded fundamental solutions and for the propagator.
Finally, we use a fermion Fock space based on the positive/negative energy
split to define a Dirac quantum field operator whose commutator is the
propagator.Comment: LaTex2e, 17 page
Distributional Modes for Scalar Field Quantization
We propose a mode-sum formalism for the quantization of the scalar field
based on distributional modes, which are naturally associated with a slight
modification of the standard plane-wave modes. We show that this formalism
leads to the standard Rindler temperature result, and that these modes can be
canonically defined on any Cauchy surface.Comment: 15 pages, RevTe
Recovering the mass and the charge of a Reissner-Nordstr\"om black hole by an inverse scattering experiment
In this paper, we study inverse scattering of massless Dirac fields that
propagate in the exterior region of a Reissner-Nordstr\"om black hole. Using a
stationary approach we determine precisely the leading terms of the high-energy
asymptotic expansion of the scattering matrix that, in turn, permit us to
recover uniquely the mass of the black hole and its charge up to a sign
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