  The wavefunction in quantum field theory is an invaluable tool for tackling a
variety of problems, including probing the interior of Minkowski spacetime and
modelling boundary observables in de Sitter spacetime. Here we study the
analytic structure of wavefunction coefficients in Minkowski as a function of
their kinematics. We introduce an off-shell wavefunction in terms of amputated
time-ordered correlation functions and show that it is analytic in the complex
energy plane except for possible singularities on the negative real axis. These
singularities are determined to all loop orders by a simple energy-conservation
condition. We confirm this picture by developing a Landau analysis of
wavefunction loop integrals and corroborate our findings with several explicit
calculations in scalar field theories. This analytic structure allows us to
derive new UV/IR sum rules for the wavefunction that fix the coefficients in
its low-energy expansion in terms of integrals of discontinuities in the
corresponding UV-completion. In contrast to the analogous sum rules for
scattering amplitudes, the wavefunction sum rules can also constrain
total-derivative interactions. We explicitly verify these new relations at
one-loop order in simple UV models of a light and a heavy scalar. Our results,
which apply to both Lorentz invariant and boost-breaking theories, pave the way
towards deriving wavefunction positivity bounds in flat and cosmological
spacetimes.
