Local and covariant flow relations for OPE coefficients in Lorentzian spacetimes

As their spacetime points approach coincidence, the n-point functions of local quantum field theories can be approximated to arbitrary precision by their operator product expansions (OPEs). The coefficients of OPEs are c-number distributions which contain important information about both the quantum fields and the physical states. Under variations of the interaction parameters, Hollands et al. have shown the OPE coefficients of renormalizable Euclidean QFTs satisfy "flow equations". These Euclidean flow equations have been proven to hold order-by-order in perturbation theory, but remain mathematically well defined under very general assumptions for any value of the interaction parameters. The flow equations, therefore, potentially provide a non-perturbative approach to obtaining OPE coefficients. However, there exist serious obstacles to deriving flow relations for OPE coefficients on Lorentzian spacetimes in a manner compatible with locality and covariance. In this talk, I describe these issues and our resolutions to them for a solvable toy model: Klein-Gordon theory with the mass viewed as an "interaction parameter". Our approach to obtaining local and covariant flow relations for the Klein-Gordon OPE coefficients on Lorentzian spacetimes generalizes to interacting QFTs.