The positivity of energy in quantum field theory is traditionally but not altogether clearly encoded in the possibility of ‘Wick rotation’ – the analytic continuation of various observable quantities to ‘imaginary time’. My talk, an account of joint work with Maxim Kontsevich, will describe a somewhat different point of view on this. I shall motivate the definition of a specific infinite-dimensional domain of complex-valued metrics on a smooth manifold which is a complexification of the usual space of real Riemannian metrics, but has the Lorentzian metrics on its boundary. The positivity of energy is then encoded in the fact that the theory is the boundary value of a holomorphic theory defined for space-time manifolds with complex metrics belonging to the domain. Questions of the unitarity of field theory in a curved space-time are thereby related to familiar phenomena in representation theory.