  We investigate positivity properties in quantum field theory (QFT). We find
that planar Feynman integrals in QFT, as well as many related quantities,
satisfy an infinite number of positivity conditions: the functions, as well as
all their signed derivatives, are non-negative in a specified kinematic region.
Such functions are known as completely monotonic (CM) in the mathematics
literature. A powerful way to certify complete monotonicity is via integral
representations. We thus show that it applies to non-planar integrals
possessing a Euclidean region, to cosmological correlators, as well as to
certain stringy integrals. Motivated by Positive Geometry, we investigate
positivity properties in planar maximally supersymmetric Yang-Mills theory. We
present evidence, based on known analytic multi-loop results, that the CM
property extends to several physical quantities in this theory. This includes
the (suitably normalized) finite remainder function of the six-particle
maximally-helicity violating (MHV) amplitude, four-point scattering amplitudes
on the Coulomb branch, four-point correlation functions, as well as the
angle-dependent cusp anomalous dimension. Our findings are however not limited
to supersymmetric theories. It is shown that the CM property holds for the QCD
and QED cusp anomalous dimensions, to three and four loops, respectively. We
comment on open questions, and on possible numerical applications of complete
monotonicity.
