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 $\gamma$ 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 $H^1$, etc. As an application, we provide a construction of a (uniformly integrable) martingale $M$ and a bounded (deterministic) integrand $H$ such that the stochastic integral $H\bullet M$ is a strict local martingale.Comment: 21 pages; forthcoming in 'Stochastic Processes and their Applications