The Power of Perturbation Theory

Abstract

We study quantum mechanical systems with a discrete spectrum. We show that the asymptotic series associated to certain paths of steepest-descent (Lefschetz thimbles) are Borel resummable to the full result. Using a geometrical approach based on the Picard-Lefschetz theory we characterize the conditions under which perturbative expansions lead to exact results. Even when such conditions are not met, we explain how to define a different perturbative expansion that reproduces the full answer without the need of transseries, i.e. non-perturbative effects, such as real (or complex) instantons. Applications to several quantum mechanical systems are presented.

v1: 42 pages, 8 figures; v2: 43 pages, 9 figures, minor improvements and references added, matches JHEP published version; v3: minor corrections and references added

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• Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb{C}{P}^{N-1}$ models
• $\lambda {\varphi }^4$ Theory I: The Symmetric Phase Beyond NNNNNNNNLO
• Delayed Deconfinement and the Hawking-Page Transition
• Functional methods for false vacuum decay in real time
• Probabilistic definition of the perturbative theoretical uncertainty from missing higher orders
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• $\mathcal{P}\mathcal{T}$-symmetric $-g{\phi }^4$ theory
• Critical Points at Infinity, Non-Gaussian Saddles, and Bions
• Instantons from Perturbation Theory
• Nonlinear trident in the high-energy limit: Nonlocality, Coulomb field and resummations
• Looking through the QCD Conformal Window with Perturbation Theory
• Resummation and renormalons in a general Quantum Field Theory
• Renormalization Scheme Dependence, RG Flow and Borel Summability in ${\varphi }^4$ Theories in $d<4$
• Gravitational Decoupling and Picard-Lefschetz
• Exploring High Multiplicity Amplitudes in Quantum Mechanics
• A Deal with the Devil: From Divergent Perturbation Theory to an Exponentially-Convergent Self-Consistent Expansion
• Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories
• $1/\epsilon$ problem in resurgence
• Resonances and PT symmetry in quantum curves
• On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues
• High order perturbation theory for difference equations and Borel summability of quantum mirror curves
• An Introduction to Motivic Feynman Integrals
• Transseries for causal diffusive systems
• Exploring High Multiplicity Amplitudes: The QM Analogue of the Spontaneously Broken Case
• The Breakdown of Resummed Perturbation Theory at High Energies
• Exact instanton transseries for quantum mechanics
• Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models
• Resurgence of one-point functions in a matrix model for 2D type IIA superstrings
• All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability
• Resurgence and $1/N$ Expansion in Integrable Field Theories
• Two-point functions at arbitrary genus and its resurgence structure in a matrix model for 2D type IIA superstrings
• Constructive expansion for quartic vector fields theories. I. Low dimensions
• Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation
• The hand-made tail: Non-perturbative tails from multifield inflation
• Statistical mechanics of a cold tracer in a hot bath
• Instantons and transseries of the Mathieu potential deformed by a $\mathcal{P}\mathcal{T}$-symmetry parameter
• From Divergent Series to Geometry: Resurgence of the Quantum Metric