### 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 language | English |
---|---|

Pages (from-to) | 2186-2242 |

Number of pages | 57 |

Journal | Annals of Physics |

Volume | 326 |

Issue number | 8 |

DOIs | |

Publication status | Published - Aug 2011 |

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### Keywords

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

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Annals of Physics*,

*326*(8), 2186-2242. https://doi.org/10.1016/j.aop.2011.04.002

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

Research output: Contribution to journal › Article

*Annals of Physics*, vol. 326, no. 8, pp. 2186-2242. https://doi.org/10.1016/j.aop.2011.04.002

}

TY - JOUR

T1 - Multi-instantons and exact results IV

T2 - Path integral formalism

AU - Jentschura, U.

AU - Zinn-Justin, Jean

PY - 2011/8

Y1 - 2011/8

N2 - 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.

AB - 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.

KW - Asymptotic problems and properties

KW - General properties of perturbation theory

KW - Summation of perturbation theory

UR - http://www.scopus.com/inward/record.url?scp=79957992898&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957992898&partnerID=8YFLogxK

U2 - 10.1016/j.aop.2011.04.002

DO - 10.1016/j.aop.2011.04.002

M3 - Article

VL - 326

SP - 2186

EP - 2242

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

IS - 8

ER -