  We propose a new non-perturbative method for studying UV complete unitary
quantum field theories (QFTs) with a mass gap in general number of spacetime
dimensions. The method relies on unitarity formulated as positive
semi-definiteness of the matrix of inner products between asymptotic states (in
and out) and states created by the action of local operators on the vacuum. The
corresponding matrix elements involve scattering amplitudes, form factors and
spectral densities of local operators. We test this method in two-dimensional
QFTs by setting up a linear optimization problem that gives a lower bound on
the central charge of the UV CFT associated to a QFT with a given mass spectrum
of stable particles (and couplings between them). Some of our numerical bounds
are saturated by known form factors in integrable theories like the
sine-Gordon, E8 and O(N) models.
