  We discuss the general method for obtaining full positivity bounds on
multi-field effective field theories (EFTs), from analyticity and unitarity
requirements on the UV theory. 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.
