1,286 research outputs found
Spacelike localization of long-range fields in a model of asymptotic electrodynamics
A previously proposed algebra of asymptotic fields in quantum electrodynamics
is formulated as a net of algebras localized in regions which in general have
unbounded spacelike extension. Electromagnetic fields may be localized in
`symmetrical spacelike cones', but there are strong indications this is not
possible in the present model for charged fields, which have tails extending in
all space directions. Nevertheless, products of appropriately `dressed' fermion
fields (with compensating charges) yield bi-localized observables.Comment: 29 pages, accepted for publication in Annales Henri Poincar\'
Quantum backreaction (Casimir) effect I. What are admissible idealizations?
Casimir effect, in a broad interpretation which we adopt here, consists in a
backreaction of a quantum system to adiabatically changing external conditions.
Although the system is usually taken to be a quantum field, we show that this
restriction rather blurs than helps to clarify the statement of the problem. We
discuss the problem from the point of view of algebraic structure of quantum
theory, which is most appropriate in this context. The system in question may
be any quantum system, among others both finite as infinite dimensional
canonical systems are allowed. A simple finite-dimensional model is discussed.
We identify precisely the source of difficulties and infinities in most of
traditional treatments of the problem for infinite dimensional systems (such as
quantum fields), which is incompatibility of algebras of observables or their
representations. We formulate conditions on model idealizations which are
acceptable for the discussion of the adiabatic backreaction problem. In the
case of quantum field models in that class we find that the normal ordered
energy density is a well defined distribution, yielding global energy in the
limit of a unit test function. Although we see the "zero point" expressions as
inappropriate, we show how they can arise in the quantum field theory context
as a result of uncontrollable manipulations.Comment: 40 pages, AMS-LaTeX; to appear in Ann. H. Poincar
Asymptotic algebra for charged particles and radiation
A C*-algebra of asymptotic fields which properly describes the infrared
structure in quantum electrodynamics is proposed. The algebra is generated by
the null asymptotic of electromagnetic field and the time asymptotic of charged
matter fields which incorporate the corresponding Coulomb fields. As a
consequence Gauss' law is satisfied in the algebraic setting. Within this
algebra the observables can be identified by the principle of gauge invariance.
A class of representations of the asymptotic algebra is constructed which
resembles the Kulish-Faddeev treatment of electrically charged asymptotic
fields.Comment: 28 pages, LaTeX; minor corrections, a reference adde
Infrared problem and spatially local observables in electrodynamics
An algebra previously proposed as an asymptotic field structure in
electrodynamics is considered in respect of localization properties of fields.
Fields are 'spatially local' -- localized in regions resulting as unions of two
intersecting (solid) lightcones: a future- and a past-lightcone. This
localization remains in concord with the usual idealizations connected with the
scattering theory. Fields thus localized naturally include infrared
characteristics normally placed at spacelike infinity and form a structure
respecting Gauss law. When applied to the description of the radiation of an
external classical current the model is free of 'infrared catastrophe'.Comment: 30 pages; accepted for publication in Ann. Henri Poincare; a few
minor correction
Quantum backreaction (Casimir) effect. II. Scalar and electromagnetic fields
Casimir effect in most general terms may be understood as a backreaction of a
quantum system causing an adiabatic change of the external conditions under
which it is placed. This paper is the second installment of a work scrutinizing
this effect with the use of algebraic methods in quantum theory. The general
scheme worked out in the first part is applied here to the discussion of
particular models. We consider models of the quantum scalar field subject to
external interaction with ``softened'' Dirichlet or Neumann boundary conditions
on two parallel planes. We show that the case of electromagnetic field with
softened perfect conductor conditions on the planes may be reduced to the other
two. The ``softening'' is implemented on the level of the dynamics, and is not
imposed ad hoc, as is usual in most treatments, on the level of observables. We
calculate formulas for the backreaction energy in these models. We find that
the common belief that for electromagnetic field the backreaction force tends
to the strict Casimir formula in the limit of ``removed cutoff'' is not
confirmed by our strict analysis. The formula is model dependent and the
Casimir value is merely a term in the asymptotic expansion of the formula in
inverse powers of the distance of the planes. Typical behaviour of the energy
for large separation of the plates in the class of models considered is a
quadratic fall-of. Depending on the details of the ``softening'' of the
boundary conditions the backreaction force may become repulsive for large
separations.Comment: 50 pages, AMS-LaTeX; to appear in Ann. H. Poincar
Infrared limit in external field scattering
Scattering of electrons/positrons by external classical electromagnetic wave
packet is considered in infrared limit. In this limit the scattering operator
exists and produces physical effects, although the scattering cross-section is
trivial.Comment: 12 pages; published version; minor corrections; comments adde
Global vs local Casimir effect
This paper continues the investigation of the Casimir effect with the use of
the algebraic formulation of quantum field theory in the initial value setting.
Basing on earlier papers by one of us (AH) we approximate the Dirichlet and
Neumann boundary conditions by simple interaction models whose nonlocality in
physical space is under strict control, but which at the same time are
admissible from the point of view of algebraic restrictions imposed on models
in the context of Casimir backreaction. The geometrical setting is that of the
original parallel plates. By scaling our models and taking appropriate limit we
approach the sharp boundary conditions in the limit. The global force is
analyzed in that limit. One finds in Neumann case that although the sharp
boundary interaction is recovered in the norm resolvent sense for each model
considered, the total force per area depends substantially on its choice and
diverges in the sharp boundary conditions limit. On the other hand the local
energy density outside the interaction region, which in the limit includes any
compact set outside the strict position of the plates, has a universal limit
corresponding to sharp conditions. This is what one should expect in general,
and the lack of this discrepancy in Dirichlet case is rather accidental. Our
discussion pins down its precise origin: the difference in the order in which
scaling limit and integration over the whole space is carried out.Comment: 32 pages, accepted for publication in Ann. H. Poincar
Massless asymptotic fields and Haag-Ruelle theory
We revisit the problem of the existence of asymptotic massless boson fields
in quantum field theory. The well-known construction of such fields by Buchholz
[2], [4] is based on locality and the existence of vacuum vector, at least in
regions spacelike to spacelike cones. Our analysis does not depend on these
assumptions and supplies a more general framework for fields only very weakly
decaying in spacelike directions. In this setting the existence of appropriate
null asymptotes of fields is linked with their spectral properties in the
neighborhood of the lightcone. The main technical tool is one of the results of
a recent analysis by one of us [11], which allows application of the null
asymptotic limit separately to creation/annihilation parts of a wide class of
non-local fields. In vacuum representation the scheme allows application of the
methods of the Haag-Ruelle theory closely analogous to those of the massive
case. In local case this Haag-Ruelle procedure may be combined with the
Buchholz method, which leads to significant simplification.Comment: 39 pages; to appear in Lett. Math. Phy
Single Jump Processes and Strict Local Martingales
Many results in stochastic analysis and mathematical finance involve local
martingales. However, specific examples of strict local martingales are rare
and analytically often rather unhandy. We study local martingales that follow a
given deterministic function up to a random time at which they jump
and stay constant afterwards. The (local) martingale properties of these single
jump local martingales are characterised in terms of conditions on the input
parameters. This classification allows an easy construction of strict local
martingales, uniformly integrable martingales that are not in , etc. As an
application, we provide a construction of a (uniformly integrable) martingale
and a bounded (deterministic) integrand such that the stochastic
integral is a strict local martingale.Comment: 21 pages; forthcoming in 'Stochastic Processes and their
Applications
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