Multi-instantons and exact results IV: Path integral formalism

U. Jentschura, Jean Zinn-Justin

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

This is the fourth paper in a series devoted to the large-order properties of anharmonic oscillators. We attempt to draw a connection of anharmonic oscillators to field theory, by investigating the partition function in the path integral representation around both the Gaussian saddle point, which determines the perturbative expansion of the eigenvalues, as well as the nontrivial instanton saddle point. The value of the classical action at the saddle point is the instanton action which determines the large-order properties of perturbation theory by a dispersion relation. In order to treat the perturbations about the instanton, one has to take into account the continuous symmetries broken by the instanton solution because they lead to zero-modes of the fluctuation operator of the instanton configuration. The problem is solved by changing variables in the path integral, taking the instanton parameters as integration variables (collective coordinates). The functional determinant (Faddeev-Popov determinant) of the change of variables implies nontrivial modifications of the one-loop and higher-loop corrections about the instanton configuration. These are evaluated and compared to exact WKB calculations. A specific cancellation mechanism for the first perturbation about the instanton, which has been conjectured for the sextic oscillator based on a nonperturbative generalized Bohr-Sommerfeld quantization condition, is verified by an analytic Feynman diagram calculation.

Original languageEnglish
Pages (from-to)2186-2242
Number of pages57
JournalAnnals of Physics
Volume326
Issue number8
DOIs
Publication statusPublished - Aug 2011

Fingerprint

instantons
formalism
saddle points
oscillators
determinants
perturbation
Feynman diagrams
configurations
cancellation
partitions
broken symmetry
eigenvalues
perturbation theory
operators
expansion

Keywords

  • Asymptotic problems and properties
  • General properties of perturbation theory
  • Summation of perturbation theory

ASJC Scopus subject areas

  • Physics and Astronomy(all)

Cite this

Multi-instantons and exact results IV : Path integral formalism. / Jentschura, U.; Zinn-Justin, Jean.

In: Annals of Physics, Vol. 326, No. 8, 08.2011, p. 2186-2242.

Research output: Contribution to journalArticle

Jentschura, U. ; Zinn-Justin, Jean. / Multi-instantons and exact results IV : Path integral formalism. In: Annals of Physics. 2011 ; Vol. 326, No. 8. pp. 2186-2242.
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