Multi-instantons and exact results IV: Path integral formalism

Journal Article · Mon Aug 15 00:00:00 EDT 2011 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2011.04.002· OSTI ID:21583312

Zinn-Justin, Jean ^[1]

---

1. CEA, IRFU and Institut de Physique Theorique, Centre de Saclay, F-91191 Gif-Sur-Yvette (France)

Highlights: > O(N) anharmonic oscillators are considered in the path integral formalism. > Higher-order corrections to instantons are obtained using Feynman diagram calculations. > Generalized Bender-Wu formulas are confirmed. > The O(N) functional determinant describing the zero modes leads to higher-order terms. > Surprising cancellations are observed for the sextic oscillator. - 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.

• OSTI ID: 21583312
• Journal Information: Annals of Physics (New York), Vol. 326, Issue 8; Other Information: DOI: 10.1016/j.aop.2011.04.002; PII: S0003-4916(11)00047-9; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANHARMONIC OSCILLATORS
ASYMPTOTIC SOLUTIONS
CORRECTIONS
DISPERSION RELATIONS
DISTURBANCES
EIGENVALUES
FEYNMAN DIAGRAM
FIELD THEORIES
FLUCTUATIONS
INSTANTONS
OSCILLATORS
PARTITION FUNCTIONS
PATH INTEGRALS
PERTURBATION THEORY
QUANTIZATION
SYMMETRY BREAKING
WKB APPROXIMATION
APPROXIMATIONS
CALCULATION METHODS
DIAGRAMS
ELECTRONIC EQUIPMENT
EQUIPMENT
FUNCTIONS
INFORMATION
INTEGRALS
MATHEMATICAL SOLUTIONS
QUASI PARTICLES
VARIATIONS