- The paper introduces a framework that uses Borel summation to extract non-perturbative insights from divergent asymptotic series.
- The paper employs trans-series and alien calculus to connect perturbative expansions with underlying physical phenomena in quantum theories.
- The paper highlights practical implications for computing multi-instanton effects and decoding the Stokes phenomenon in theoretical physics.
An Examination of Resurgence, Trans-Series, and Alien Calculus
The paper "An Introduction to Resurgence, Trans-Series and Alien Calculus" by Daniele Dorigoni provides a comprehensive exploration into the theory of resurgence, focusing on its application within physics, particularly by contextualizing it within asymptotic series and non-perturbative phenomena. The paper aims to elucidate how resurgence theory, through tools such as trans-series expansions and the alien calculus, can enable the extraction of non-perturbative physical insights from perturbation theory.
The premise of resurgence theory is based on understanding asymptotic expansions, which often arise in perturbative calculations in quantum mechanics and quantum field theories (QFT). Perturbation theory typically yields formal power series whose coefficients grow factorially, rendering the series asymptotic with zero radius of convergence. The resurgence framework, particularly through Borel summation, provides a mechanism to formally "resum" these divergent series into meaningful analytic functions, highlighting connections to deeper non-perturbative phenomena.
The paper introduces key constructs in resurgence theory:
- Borel Transform: Mapping a formal power series into a series with potentially better convergence properties.
- Directional Laplace Transform: Used to analytically continue the Borel transform back to a resummed power series.
A central concept in this framework is the notion of resurgent functions having a controlled singularity structure characterized by "simple" singularities on their Borel plane. This singular behavior can map back to non-perturbative corrections in the original theories. Dorigoni introduces the Borel sum, lateral Borel sums, and the Stokes phenomenon, showing how the discontinuities along specific Stokes lines (directions in the Borel plane) are linked to physical phenomena via alien derivatives.
Alien Calculus—one of the core tools introduced—is crucial for understanding the structure of singularities and their connection via the Stokes automorphism. This provides a systematic approach to account for non-perturbative effects lacking in the original asymptotic expansion. The "bridge equation" formalizes the connection between the non-perturbative alien calculus and the standard differential calculus, offering practical computational pathways for deriving holomorphic data.
The modular framework of resurgence is particularly adept at illuminating non-perturbative effects hidden in the perturbative sector. This inherently involves the notion of trans-series, offering an extension beyond simple perturbative series to include non-perturbative orders characterized by exponential terms and potentially logarithmic contributions. This formalism naturally encloses descriptions of multi-instanton effects and infrared renormalon contributions in QFT.
The paper emphasizes practical implications, particularly the ability of resurgence and trans-series expansions to understand phenomena like the Stokes phenomenon and the reality of physical observables in asymptotic expansions. For example, in quantum mechanical toy models as well as conformal field theories, resurgence provides a dual approach to formulate the intricate dance between divergent perturbative series, their asymptotic approximations, and the overarching resurgent trans-series capturing the full physical content.
Looking forward, the paper speculates on broader implications of resurgence theory in path integrals, suggesting possible connections with ideas like the Lefschetz thimble decomposition in quantum field theories, reinforcing the structure of trans-series as a fundamental insight into non-perturbative dynamics capable of reshaping our understanding of asymptotic analysis in physics.
This paper serves not only as a primer on resurgence theory with its mathematical constructs but also as an intriguing exposition for experienced researchers interested in the resurgence phenomenon's potential applicability to a range of complex systems in theoretical physics. It opens avenues for further exploration into the algebraic and analytic properties of resurgence and trans-series with an eye on their practical applications and to gaining insight into non-perturbative domains traditionally obscured within the framework of formal perturbation expansions.