25 research outputs found
Supersymmetric Field-Theoretic Models on a Supermanifold
We propose the extension of some structural aspects that have successfully
been applied in the development of the theory of quantum fields propagating on
a general spacetime manifold so as to include superfield models on a
supermanifold. We only deal with the limited class of supermanifolds which
admit the existence of a smooth body manifold structure. Our considerations are
based on the Catenacci-Reina-Teofillatto-Bryant approach to supermanifolds. In
particular, we show that the class of supermanifolds constructed by
Bonora-Pasti-Tonin satisfies the criterions which guarantee that a
supermanifold admits a Hausdorff body manifold. This construction is the
closest to the physicist's intuitive view of superspace as a manifold with some
anticommuting coordinates, where the odd sector is topologically trivial. The
paper also contains a new construction of superdistributions and useful results
on the wavefront set of such objects. Moreover, a generalization of the
spectral condition is formulated using the notion of the wavefront set of
superdistributions, which is equivalent to the requirement that all of the
component fields satisfy, on the body manifold, a microlocal spectral condition
proposed by Brunetti-Fredenhagen-K\"ohler.Comment: Final version to appear in J.Math.Phy
Hadamard states from null infinity
Free field theories on a four dimensional, globally hyperbolic spacetime,
whose dynamics is ruled by a Green hyperbolic partial differential operator,
can be quantized following the algebraic approach. It consists of a two-step
procedure: In the first part one identifies the observables of the underlying
physical system collecting them in a *-algebra which encodes their relational
and structural properties. In the second step one must identify a quantum
state, that is a positive, normalized linear functional on the *-algebra out of
which one recovers the interpretation proper of quantum mechanical theories via
the so-called Gelfand-Naimark-Segal theorem. In between the plethora of
possible states, only few of them are considered physically acceptable and they
are all characterized by the so-called Hadamard condition, a constraint on the
singular structure of the associated two-point function. Goal of this paper is
to outline a construction scheme for these states which can be applied whenever
the underlying background possesses a null (conformal) boundary. We discuss in
particular the examples of a real, massless conformally coupled scalar field
and of linearized gravity on a globally hyperbolic and asymptotically flat
spacetime.Comment: 23 pages, submitted to the Proceedings of the conference "Quantum
Mathematical Physics", held in Regensburg from the 29th of September to the
02nd of October 201
Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime
We prove that the singularity structure of all n-point distributions of a
state of a generalised real free scalar field in curved spacetime can be
estimated if the two-point distribution is of Hadamard form. In particular this
applies to the real free scalar field and the result has applications in
perturbative quantum field theory, showing that the class of all Hadamard
states is the state space of interest. In our proof we assume that the field is
a generalised free field, i.e. that it satisies scalar (c-number) commutation
relations, but it need not satisfy an equation of motion. The same argument
also works for anti-commutation relations and it can be generalised to
vector-valued fields. To indicate the strengths and limitations of our
assumption we also prove the analogues of a theorem by Borchers and Zimmermann
on the self-adjointness of field operators and of a very weak form of the
Jost-Schroer theorem. The original proofs of these results in the Wightman
framework make use of analytic continuation arguments. In our case no
analyticity is assumed, but to some extent the scalar commutation relations can
take its place.Comment: 18 page
Focusing and the Holographic Hypothesis
The ``screen mapping" introduced by Susskind to implement 't Hooft's
holographic hypothesis is studied. For a single screen time, there are an
infinite number of images of a black hole event horizon, almost all of which
have smaller area on the screen than the horizon area. This is consistent with
the focusing equation because of the existence of focal points. However, the
{\it boundary} of the past (or future) of the screen obeys the area theorem,
and so always gives an expanding map to the screen, as required by the
holographic hypothesis. These considerations are illustrated with several
axisymmetric static black hole spacetimes.Comment: 8 pages, plain latex, 5 figures included using psfi
Hawking Spectrum and High Frequency Dispersion
We study the spectrum of created particles in two-dimensional black hole
geometries for a linear, hermitian scalar field satisfying a Lorentz
non-invariant field equation with higher spatial derivative terms that are
suppressed by powers of a fundamental momentum scale . The preferred frame
is the ``free-fall frame" of the black hole. This model is a variation of
Unruh's sonic black hole analogy. We find that there are two qualitatively
different types of particle production in this model: a thermal Hawking flux
generated by ``mode conversion" at the black hole horizon, and a non-thermal
spectrum generated via scattering off the background into negative free-fall
frequency modes. This second process has nothing to do with black holes and
does not occur for the ordinary wave equation because such modes do not
propagate outside the horizon with positive Killing frequency. The horizon
component of the radiation is astonishingly close to a perfect thermal
spectrum: for the smoothest metric studied, with Hawking temperature
, agreement is of order at frequency
, and agreement to order persists out to
where the thermal number flux is ). The flux
from scattering dominates at large and becomes many orders of
magnitude larger than the horizon component for metrics with a ``kink", i.e. a
region of high curvature localized on a static worldline outside the horizon.
This non-thermal flux amounts to roughly 10\% of the total luminosity for the
kinkier metrics considered. The flux exhibits oscillations as a function of
frequency which can be explained by interference between the various
contributions to the flux.Comment: 32 pages, plain latex, 16 figures included using psfi
Black Hole Evaporation in the Presence of a Short Distance Cutoff
A derivation of the Hawking effect is given which avoids reference to field
modes above some cutoff frequency in the free-fall frame
of the black hole. To avoid reference to arbitrarily high frequencies, it is
necessary to impose a boundary condition on the quantum field in a timelike
region near the horizon, rather than on a (spacelike) Cauchy surface either
outside the horizon or at early times before the horizon forms. Due to the
nature of the horizon as an infinite redshift surface, the correct boundary
condition at late times outside the horizon cannot be deduced, within the
confines of a theory that applies only below the cutoff, from initial
conditions prior to the formation of the hole. A boundary condition is
formulated which leads to the Hawking effect in a cutoff theory. It is argued
that it is possible the boundary condition is {\it not} satisfied, so that the
spectrum of black hole radiation may be significantly different from that
predicted by Hawking, even without the back-reaction near the horizon becoming
of order unity relative to the curvature.Comment: 35 pages, plain LaTeX, UMDGR93-32, NSF-ITP-93-2
Quantum charges and spacetime topology: The emergence of new superselection sectors
In which is developed a new form of superselection sectors of topological
origin. By that it is meant a new investigation that includes several
extensions of the traditional framework of Doplicher, Haag and Roberts in local
quantum theories. At first we generalize the notion of representations of nets
of C*-algebras, then we provide a brand new view on selection criteria by
adopting one with a strong topological flavour. We prove that it is coherent
with the older point of view, hence a clue to a genuine extension. In this
light, we extend Roberts' cohomological analysis to the case where 1--cocycles
bear non trivial unitary representations of the fundamental group of the
spacetime, equivalently of its Cauchy surface in case of global hyperbolicity.
A crucial tool is a notion of group von Neumann algebras generated by the
1-cocycles evaluated on loops over fixed regions. One proves that these group
von Neumann algebras are localized at the bounded region where loops start and
end and to be factorial of finite type I. All that amounts to a new invariant,
in a topological sense, which can be defined as the dimension of the factor. We
prove that any 1-cocycle can be factorized into a part that contains only the
charge content and another where only the topological information is stored.
This second part resembles much what in literature are known as geometric
phases. Indeed, by the very geometrical origin of the 1-cocycles that we
discuss in the paper, they are essential tools in the theory of net bundles,
and the topological part is related to their holonomy content. At the end we
prove the existence of net representations
Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory
On the basis of a thorough discussion of the Batalin-Vilkovisky formalism for
classical field theory presented in our previous publication, we construct in
this paper the Batalin-Vilkovisky complex in perturbatively renormalized
quantum field theory. The crucial technical ingredient is a proof that the
renormalized time-ordered product is equivalent to the pointwise product of
classical field theory. The renormalized Batalin-Vilkovisky algebra is then the
classical algebra but written in terms of the time-ordered product, together
with an operator which replaces the ill defined graded Laplacian of the
unrenormalized theory. We identify it with the anomaly term of the anomalous
Master Ward Identity of Brennecke and D\"utsch. Contrary to other approaches we
do not refer to the path integral formalism and do not need to use
regularizations in intermediate steps.Comment: 34 page
On the Reeh-Schlieder Property in Curved Spacetime
We attempt to prove the existence of Reeh-Schlieder states on curved
spacetimes in the framework of locally covariant quantum field theory using the
idea of spacetime deformation and assuming the existence of a Reeh-Schlieder
state on a diffeomorphic (but not isometric) spacetime. We find that physically
interesting states with a weak form of the Reeh-Schlieder property always exist
and indicate their usefulness. Algebraic states satisfying the full
Reeh-Schlieder property also exist, but are not guaranteed to be of physical
interest.Comment: 13 pages, 2 figure
Axiomatic quantum field theory in curved spacetime
The usual formulations of quantum field theory in Minkowski spacetime make
crucial use of features--such as Poincare invariance and the existence of a
preferred vacuum state--that are very special to Minkowski spacetime. In order
to generalize the formulation of quantum field theory to arbitrary globally
hyperbolic curved spacetimes, it is essential that the theory be formulated in
an entirely local and covariant manner, without assuming the presence of a
preferred state. We propose a new framework for quantum field theory, in which
the existence of an Operator Product Expansion (OPE) is elevated to a
fundamental status, and, in essence, all of the properties of the quantum field
theory are determined by its OPE. We provide general axioms for the OPE
coefficients of a quantum field theory. These include a local and covariance
assumption (implying that the quantum field theory is locally and covariantly
constructed from the spacetime metric), a microlocal spectrum condition, an
"associativity" condition, and the requirement that the coefficient of the
identity in the OPE of the product of a field with its adjoint have positive
scaling degree. We prove curved spacetime versions of the spin-statistics
theorem and the PCT theorem. Some potentially significant further implications
of our new viewpoint on quantum field theory are discussed.Comment: Latex, 44 pages, 2 figure