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Andrzej Herdegen

Publications and source records attributed to Andrzej Herdegen.

At least 19 recordsLinked to original sources
arXiv2024

Long-range effects in asymptotic fields and angular momentum of classical field electrodynamics

Asymptotic properties of classical field electrodynamics are considered. Special attention is paid to the long-range structure of the electromagnetic field. It is shown that conserved Poincare quantities may be expressed in terms of the asymptotic fields. Long-range variables are shown to be responsible for an angular momentum contribution which mixes Coulomb and infrared free field characteristics; otherwise angular momentum and energy-momentum separate into electromagnetic and matter fields contributions.

math-ph↗
arXiv2024

Infrared structure beyond locality in electrodynamics

The infrared problems of quantum electrodynamics, in contrast to ultraviolet difficulties which are of technical nature, are related to fundamental, conceptual physical questions, such as: what is a charged particle, is the particle interpretation of the electromagnetic field complete, does a vacuum state exist, or what is the quantum status of long range degrees of freedom. On the calculational level, the standard local formulations of quantum field theory have achieved procedures to deal with infinities related to long range correlations. However, the answers to the conceptual questions formulated above, based on the locality paradigm, do not seem to be fully convincing, which is confirmed by the fact that no canonical picture did emerge. This contribution briefly characterizes perspectives which open with an admission of nonlocal variables residing in infinity, or at the boundary of spacetime after compactification. Recently, this line of investigation gains popularity.

math-ph↗
arXiv2024

Undressing the electron

The extended algebra of the free electromagnetic fields, including infrared singular fields, and the almost radial gauge, both introduced earlier, are postulated for the construction of the quantum electrodynamics in a Hilbert space (no indefinite metric). Both the Dirac and electromagnetic fields are constructed up to the first order (based on the incoming fields) as operators in the Hilbert space, and shown to have physically well interpretable asymptotic behavior in far past and spacelike separations. The Dirac field tends in far past to the free incoming field, carrying its own Coulomb field, but with no 'soft photon dressing'. The spacelike asymptotic limit of the electromagnetic field yields a conserved operator field, which is a sum of contributions of the incoming Coulomb field, and of the low energy limit of the incoming free electromagnetic field. This should agree with the operator field similarly constructed with the use of outgoing fields, which then relates these past and future characteristics. Higher orders are expected not to change this picture, but their construction needs a treatment of the UV question, which has not been undertaken and remains a problem for further investigation.

math-ph↗
arXiv2024

There is no 'velocity kick' memory in electrodynamics

The memory effect in electrodynamics, as discovered in 1981 by Staruszkiewicz, and also analysed later, consists of adiabatic shift of the position of a test particle. The proposed 'velocity kick' memory effect, supposedly discovered recently, is in contradiction to these findings. We show that the 'velocity kick' memory is an artefact resulting from an unjustified interchange of limits. This example is a warning against drawing uncritical conclusions for spacetime fields, from their asymptotic behavior.

physics.class-ph↗
arXiv2022

Remarks on mathematical structure of the Staruszkiewicz theory

The purpose of this note is threefold: (i) to recall (with some points made more explicit) the mathematical Weyl algebra model formulation, given before, of the Staruszkiewicz theory of quantum Coulomb field; (ii) to add some new elements to the discussion of the representation of the Lorentz group within this model; (iii) to comment on some statements on the structure of the theory which appeared recently.

math-ph↗
arXiv2022

Almost radial gauge

An almost radial gauge $A^\mathrm{ar}$ of the electromagnetic potential is constructed for which $x\cdot A^\mathrm{ar}(x)$ vanishes arbitrarily fast in timelike directions. This potential is in the class introduced by Dirac with the purpose of forming gauge-invariant quantities in quantum electrodynamics. In quantum case the construction of smeared operators $A^\mathrm{ar}(K)$ is enabled by a natural extension of the free electromagnetic field algebra introduced earlier (represented in a Hilbert space). The space of possible smearing functions $K$ includes vector fields with the asymptotic spacetime behavior typical for scattered currents (the conservation condition in the whole spacetime need not be assumed). This construction is motivated by a possible application to the infrared problem in QED.

math-ph↗
arXiv2020

Infrared problem vs gauge choice: scattering of classical Dirac field

We consider the Dirac equation for the classical spinor field placed in an external, time-dependent electromagnetic field of the form typical for scattering settings: $F=F^\mathrm{ret}+F^\mathrm{in}=F^\mathrm{adv}+F^\mathrm{out}$, where the current producing $F^{\mathrm{ret}/\mathrm{adv}}$ has past and future asymptotes homogeneous of degree $-3$, and the free fields $F^{\mathrm{in}/\mathrm{out}}$ are radiation fields produced by currents with similar asymptotic behavior. We show the existence of the electromagnetic gauges in which the particle has 'in' and 'out' asymptotic states approaching free field states, with no long-time corrections of the free dynamics. Using a special Cauchy foliation of the spacetime we show in this context the existence and asymptotic completeness of the wave operators. Moreover, we define a special 'evolution picture' in which the free evolution operator has well-defined limits for $t\to\pm\infty$, thus the scattering wave operators do not need the free evolution counteraction.

math-ph↗
arXiv2019

Electromagnetic gauge choice for scattering of Schrödinger particle

We consider a Schrödinger particle placed in an external electromagnetic field of the form typical for scattering settings in the field theory: $F=F^\mathrm{ret}+F^\mathrm{in}=F^\mathrm{adv}+F^\mathrm{out}$, where the current producing $F^{\mathrm{ret}/\mathrm{adv}}$ has the past and future asymptotes homogeneous of degree $-3$, and the free fields $F^{\mathrm{in}/\mathrm{out}}$ are radiation fields produced by currents with similar asymptotic behavior. We show that with appropriate choice of electromagnetic gauge the particle has 'in' and 'out' states reached with no further modification of the asymptotic dynamics. We use a special quantum-mechanical evolution 'picture' in which the free evolution operator has well-defined limits for $t\to\pm\infty$, thus the scattering wave operators do not need the free evolution counteraction. The existence of wave operators in this setting is established, but the proof of asymptotic completeness is not complete: more precise characterization of the asymptotic behavior of the particle for $|\mathbf{x}|=|t|$ would be needed.

math-ph↗
arXiv2017

Asymptotic structure of electrodynamics revisited

We point out that recently published analyses of null and timelike infinity and long-range structures in electrodynamics to large extent rediscover results present in literature. At the same time, some of the conclusions these recent works put forward may prove controversial. In view of these facts we find it desirable to revisit the analysis taken up more than two decades ago, starting from earlier works on null infinity by other authors.

hep-th↗
arXiv2015

Generalized uncertainty relations

The standard uncertainty relations (UR) in quantum mechanics are typically used for unbounded operators (like the canonical pair). This implies the need for the control of the domain problems. On the other hand, the use of (possibly bounded) functions of basic observables usually leads to more complex and less readily interpretable relations. Also, UR may turn trivial for certain states if the commutator of observables is not proportional to a positive operator. In this letter we consider a generalization of standard UR resulting from the use of two, instead of one, vector states. The possibility to link these states to each other in various ways adds additional flexibility to UR, which may compensate some of the above mentioned drawbacks. We discuss applications of the general scheme, leading not only to technical improvements, but also to interesting new insight.

math-ph↗
arXiv2014

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.

math-ph↗
arXiv2014

On energy-momentum transfer of quantum fields

We prove the following theorem on bounded operators in quantum field theory: if $\|[B,B^*(x)]\|\leq \mathrm{const} D(x)$, then $\|B^k_\pm(ν)G(P^0)\|^2\leq\mathrm{const}\int D(x-y)d|ν|(x)d|ν|(y)$, where $D(x)$ is a function weakly decaying in spacelike directions, $B^k_\pm$ are creation/annihilation parts of an appropriate time derivative of $B$, $G$ is any positive, bounded, non-increasing function in $L^2(\mathbb{R})$, and $ν$ is any finite complex Borel measure; creation/annihilation operators may be also replaced by $B^k_t$ with $\check{B^k_t}(p)=|p|^k\check{B}(p)$. We also use the notion of energy-momentum scaling degree of $B$ with respect to a submanifold (Steinmann-type, but in momentum space, and applied to the norm of an operator). These two tools are applied to the analysis of singularities of $\check{B}(p)G(P^0)$. We prove, among others, the following statement (modulo some more specific assumptions): outside $p=0$ the only allowed contributions to this functional which are concentrated on a submanifold (including the trivial one -- a single point) are Dirac measures on hypersurfaces (if the decay of $D$ is not to slow).

math-ph↗
arXiv2012

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.

hep-th↗
arXiv2011

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.

hep-th↗
arXiv2010

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.

hep-th↗
arXiv2009

Flat spacetime in a capsule

We propose a parallel introduction to Galilean and Einsteinian relativity based on the causal structure and inertial motions. Galilean and Poincare transformations, as objects secondary to the geometrical structure, are left aside.

physics.class-ph↗
arXiv2007

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'.

hep-th↗