Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions
By J. Zinn-Justin, U. Jentschura. 2004
190 citations recorded by the source corpus.
Abstract 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 non-analytic 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 non-analytic 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 Schrodinger 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.
Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Physical and Theoretical Chemistry
View Paper: https://www.semanticscholar.org/paper/bfb49d71f1971b38c38bcc780386c535b9cd2f7d
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In Atomic and Molecular Physics, and Optics
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Citations in the corpus
- Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings (2012, background): “eed the content of (2.119). This phenomenon was noticed in the literature on renormalons in gauge theories (see for example [73] for a clear presentation), as well as in the study of instantons in QM [126], and is called in those contexts the cancellation of non-perturbative ambiguities. 2.5 Non-per”
- Resurgence theory, ghost-instantons, and analytic continuation of path integrals (2013, unknown): “We used the Zinn Justin/Jentschura approach [36, 37], in which one expands the exact WKB quantization condition in a perturbative series in the coupling constant, and then inverts to express the energy in terms of the level index.”
- Nonperturbative effects and nonperturbative definitions in matrix models and topological strings (2008, background): “We show that the 1/N series can be resummed in such a way that the ambiguities coming from the Borel resummation cancel against nonperturbative instanton effects, providing a nice illustration of the cancellation mechanism much discussed in renormalon physics [25, 26] and in some quantum-mechanical ”
- Multi-instantons and exact results II: Specific cases, higher-order effects, and numerical calculations (2004, unknown): “Many of the instanton calcul tions rely on generalizations of the ideas and methods introduced in chapter 4 (which is again part of the first paper [1] of this series).”