Resummation of the divergent perturbation series for a hydrogen atom in an electric field

Jentschura, Ulrich D.

doi:10.1103/physreva.64.013403

Cited by 43 publications

(51 citation statements)

References 76 publications

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“…[207][208][209]. These contours are just straight lines tilted in the complex plane so that the singularities of the Padé approximants are appropriately encircled when Jordan's Lemma is applied, to give the pole contributions that finally yield numerical approximations to the imaginary part of the Borel sum.…”

Section: Borel-padé Methodsmentioning

confidence: 99%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance, and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

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Section: Borel-padé Methodsmentioning

confidence: 99%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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“…In the current case, the interpretation (i) is relevant. The resonances have complex resonance energies; the waves are evanescent (exponentially damped) just like the diffracted wave under total reflection, or a wave in a waveguide below the minimum frequency for the TE 1 0 mode necessary for propagation, and the resonance energies are complex just as in the case of a resonance energy of the Stark effect [40]. Resonances are damped for propagation forward in time, antiresonances for propagation backward in time, in accordance with the Feynman prescription.…”

Section: Tachyonic Dirac Equationmentioning

confidence: 74%

Tachyonic field theory and neutrino mass running

Jentschura

2012

Open Physics

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Abstract:In this paper three things are done. (i) We investigate the analogues of Cerenkov radiation for the decay of a superluminal neutrino and calculate the Cerenkov angles for the emission of a photon through a W loop, and for a collinear electron-positron pair, assuming the tachyonic dispersion relation for the superluminal neutrino. The decay rate of a freely propagating neutrino is found to depend on the shape of the assumed dispersion relation, and is found to decrease with decreasing tachyonic mass of the neutrino.(ii) We discuss a few properties of the tachyonic Dirac equation (symmetries and plane-wave solutions), which may be relevant for the description of superluminal neutrinos seen by the OPERA experiment, and discuss the calculation of the tachyonic propagator. (iii) In the absence of a commonly accepted tachyonic field theory, and in view of an apparent "running" of the observed neutrino mass with the energy, we write down a model Lagrangian, which describes a Yukawa-type interaction of a neutrino coupling to a scalar background field via a scalar-minus-pseudoscalar interaction. This constitutes an extension of the standard model. If the interaction is strong, then it leads to a substantial renormalization-group "running" of the neutrino mass and could potentially explain the experimental observations. PACS

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“…If we take into account that the hyperbolic cotangent is odd according to coth(−z) = − coth(z), we see from (27) and (34) or from (28) and (35) that the poles and residues of F (z) have to satisfy for all k ∈ N the (anti-)symmetry relations…”

Section: (±)mentioning

confidence: 99%

“…If we now insert the poles (27) and (28) and the corresponding residues (34) and (35), respectively, into (51), we obtain the following partial-fraction decomposition:…”

Section: (±)mentioning

confidence: 99%

QED effective action revisited

Gies¹,

Valluri²,

Lamm³

et al. 2002

Can. J. Phys.

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The derivation of a convergent series representation for the quantum electrodynamic effective action obtained by two of us (S.R.V. and D.R.L.) in [Can. J. Phys. 71, 389 (1993)] is reexamined. We present more details of our original derivation. Moreover, we discuss the relation of the electric-magnetic duality to the integral representation for the effective action, and we consider the application of nonlinear convergence acceleration techniques which permit the efficient and reliable numerical evaluation of the quantum correction to the Maxwell Lagrangian.

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Resummation of the divergent perturbation series for a hydrogen atom in an electric field

Cited by 43 publications

References 76 publications

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

Tachyonic field theory and neutrino mass running

QED effective action revisited

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