Multi-instantons and exact results IV: Path integral formalism Ulrich D. Jentschura a,⇑ , Jean Zinn-Justin b a Department of Physics, Missouri University of Science and Technology, Rolla, MO 65409-0640, USA b CEA, IRFU and Institut de Physique Théorique, Centre de Saclay, F-91191 Gif-Sur-Yvette, France article info Article history: Received 10 March 2011 Accepted 5 April 2011 Available online 8 April 2011 Keywords: General properties of perturbation theory Asymptotic problems and properties Summation of perturbation theory abstract This is the fourth paper in a series devoted to the large-order prop- erties of anharmonic oscillators. We attempt to draw a connection of anharmonic oscillators to field theory, by investigating the par- tition function in the path integral representation around both the Gaussian saddle point, which determines the perturbative expan- sion 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 deter- minant (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 nonpertur- bative generalized Bohr–Sommerfeld quantization condition, is verified by an analytic Feynman diagram calculation. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Anharmonic oscillators continue to intrigue the minds of theoretical physicists because they represent one of the most interesting, nontrivial but still solvable (in terms of generalized nonanalytic 0003-4916/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2011.04.002 ⇑ Corresponding author. Tel.: +1 573 3416221; fax: +1 573 3414715. E-mail address: ulj@mst.edu (U.D. Jentschura). Annals of Physics 326 (2011) 2186–2242 Contents lists available at ScienceDirect Annals of Physics journal homepage: www.elsevier.com/locate/aop