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This paper is a set of lecture notes that reviews a class of strong positivity properties found in functions that appear in quantum field theory. The authors focus on two related families: completely monotone functions, which obey an infinite list of sign constraints on all derivatives, and Stieltjes functions, which satisfy even stronger analytic conditions. These concepts are classical in analysis, but the notes collect recent evidence that they show up in many physical building blocks such as scalar Feynman integrals (in the Euclidean region) and certain amplitudes on the Coulomb branch of N=4 supersymmetric Yang–Mills theory.
The notes explain what these properties mean in plain terms and why they are useful. A completely monotone function is one whose derivatives alternate sign in a regular way; this is equivalent to the function being a Laplace transform of a positive measure. Stieltjes functions admit a related spectral representation and have tightly constrained behaviour in the complex plane. The text uses familiar examples to motivate the ideas, including the Gamma function and classical uniqueness theorems, and it summarizes rigorous theorems about analyticity and representations of these functions.
The authors survey where these positivity structures come from in physics. In scattering amplitudes they often trace back to unitarity (probability conservation) and analyticity (causality and locality). For Feynman integrals the same patterns can follow from the structure of their parametric representations and the Symanzik polynomials that appear there. The notes also discuss a geometric angle: when an observable can be written as a “dual volume” or as a Laplace-type integral over a positive domain, complete monotonicity follows automatically.
The lecture notes collect several applications that have appeared recently. Those include constraints on the analytic S-matrix (the function that encodes scattering), implications for numerical bootstrap methods that try to reconstruct physics from limited data, and links to positive geometry, where amplitudes can sometimes be seen as volume-like quantities. The notes present evidence for a close relation between completely monotone / Stieltjes structure and geometric volume interpretations, and they give examples and plots (for instance, the cusp anomalous dimension is expected to be completely monotone on a certain interval).
Important caveats are discussed. The appearance of these positivity properties is not universal: the underlying reasons can differ case by case, and an observable must admit the appropriate Laplace-type or dual-volume representation for the conclusions to apply. The notes are a review and survey of proofs, examples, and evidence rather than a single new theorem that covers all amplitudes. They also include technical appendices on related topics such as the moment problem and Tauberian theorems, which explain limitations of reconstructing functions from partial data.
These notes are an extended set of lectures given at the “Positive Geometry in Scattering Amplitudes and Cosmological Correlators” workshop held at the International Centre for Theoretical Sciences, Bengaluru, in February 2025. They aim to provide a pedagogical bridge between mathematical facts about completely monotone and Stieltjes functions and their emerging role as organizing principles in modern studies of scattering amplitudes.