Unitarity of interacting fields in curved spacetime

On globally hyperbolic spacetimes, each foliation by spacelike hypersurfaces corresponds to a Hamiltonian description of field theory, and unitarity follows formally from the Hermiticity of the Hamiltonian. For a renormalizable theory, unitarity at each order in perturbation theory follows from the corresponding Hermiticity of each term in the time-ordered product of interaction Hamiltonians. For more general spacetimes, one can still use the path integral to obtain a generalized Lehmann-Symanzik-Zimmermann reduction formula for $S$-matrix elements and the corresponding perturbative expansion. Unitarity imposes an infinite set of identities on the scattering amplitudes, which are the generalizations of the flat-spacetime Cutkosky rules. We find these explicitly to $O\left({\lambda }^3\right)$ in a $\lambda {\varphi }^4$ theory, and show how to find the relations to any order. For globally hyperbolic spacetimes the unitarity identities are satisfied [at least to $O\left({\lambda }^3\right)$] because of a single property of the configuration-space propagator that reflects the causal structure of the spacetime.