Instantons and Nucleon Magnetism

Forkel, Hilmar

doi:10.1142/9789812811653_0006

Cited by 6 publications

(9 citation statements)

References 13 publications

(29 reference statements)

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“…An explicit infrared cutoff destroys the gauge invariance of the theory. For a survey of field-theoretic aspects related to instantons, one may consult the following review articles [25][26][27][28][29] (the list is necessarily incomplete). The nonlinear σ model (see e.g.…”

Section: Discussionmentioning

confidence: 99%

Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations

2004

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In this second part of the treatment of instantons in quantum mechanics, the focus is on specific calculations related to a number of quantum mechanical potentials with degenerate minima. We calculate the leading multi-instanton constributions to the partition function, using the formalism introduced in the first part of the treatise [J. Zinn-Justin and U. D. Jentschura, e-print quantph/0501136]. The following potentials are considered: (i) asymmetric potentials with degenerate minima, (ii) the periodic cosine potential, (iii) anharmonic oscillators with radial symmetry, and (iv) a specific potential which bears an analogy with the Fokker-Planck equation. The latter potential has the peculiar property that the perturbation series for the ground-state energy vanishes to all orders and is thus formally convergent (the ground-state energy, however, is nonzero and positive). For the potentials (ii), (iii), and (iv), we calculate the perturbative B-function as well as the instanton Afunction to fourth order in g. We also consider the double-well potential in detail, and present some higher-order analytic as well as numerical calculations to verify explicitly the related conjectures up to the order of three instantons. Strategies analogous to those outlined here could result in new conjectures for problems where our present understanding is more limited.

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Section: Discussionmentioning

confidence: 99%

Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations

2004

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“…The generalized perturbative expansions, which are also known as resurgent expansions, and their connection to perturbation series in quantum chromodynamics has been explored in other works (e.g., Refs. [182][183][184][185][186][187]). The so-called Lipatov asymptotic form, which describes factorially divergent contributions to perturbation series due to the explosion of the number of Feynman diagrams in higher orders, has been discussed in Ref.…”

Section: Quick Startmentioning

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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“…In [28], it has been stressed that the formulation in terms of the Euclidean action follows naturally from an analytic continuation of the usual definition of the path integral to imaginary time.…”

Section: Path Integrals and Spectra Of Hamiltoniansmentioning

confidence: 99%

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

2004

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We consider specific quantum mechanical model problems for which perturbation theory fails to explain physical properties like the eigenvalue spectrum even qualitatively, even if the asymptotic perturbation series is augmented by resummation prescriptions to "cure" the divergence in large orders of perturbation theory. Generalizations of perturbation theory are necessary which include instanton configurations, characterized by nonanalytic factors exp(−a/g) where a is a constant and g is the coupling. In the case of one-dimensional quantum mechanical potentials with two or more degenerate minima, the energy levels may be represented as an infinite sum of terms each of which involves a certain power of a nonanalytic factor and represents itself an infinite divergent series. We attempt to provide a unified representation of related derivations previously found scattered in the literature. For the considered quantum mechanical problems, we discuss the derivation of the instanton contributions from a semi-classical calculation of the corresponding partition function in the path integral formalism. We also explain the relation with the corresponding WKB expansion of the solutions of the Schrödinger equation, or alternatively of the Fredholm determinant det(H − E) (and some explicit calculations that verify this correspondence). We finally recall how these conjectures naturally emerge from a leading-order summation of multi-instanton contributions to the path integral representation of the partition function. The same strategy could result in new conjectures for problems where our present understanding is more limited.

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Instantons and Nucleon Magnetism

Cited by 6 publications

References 13 publications

Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations

Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations

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

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

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