  We discuss the general method for obtaining full positivity bounds on
multi-field effective field theories (EFTs). While the leading order forward
positivity bounds are commonly derived from the elastic scattering of two
(superposed) external states, we show that for a generic EFT containing 3 or
more low-energy modes, this approach only gives incomplete bounds. We then
identify the allowed parameter space as the dual to a spectrahedron,
constructed from crossing symmetries of the amplitude, and show that finding
the optimal bounds for a given number of modes is equivalent to a geometric
problem: finding the extremal rays of a spectrahedron. We show how this is done
analytically for simple cases, and numerically formulated as semidefinite
programming (SDP) problems for more complicated cases. We demonstrate this
approach with a number of well-motivated examples in particle physics and
cosmology, including EFTs of scalars, vectors, fermions and gravitons. In all
these cases, we find that the SDP approach leads to results that either improve
the previous ones or are completely new. We also find that the SDP approach is
numerically much more efficient.
