Local and covariant flow relations for OPE coefficients in curved spacetime.

Although pointwise products of quantum field observables are not well defined, products of field observables evaluated at distinct spacetime points are known to satisfy asymptotic relations called operator product expansion (OPEs) in the coincidence limit. Hollands and Wald have argued much of the key information about a quantum field theory (QFT) is contained in the coefficients of its OPEs. For interacting λφ⁴-theory in flat Euclidean spacetime, Holland and Hollands have derived non-perturbative "flow relations" which govern how OPE coefficients vary under changes in the interaction parameter, λ. However, serious obstacles arise if one attempts to generalize their result to curved Lorentzian spacetimes in a local and covariant manner. In this talk, I will describe these issues and sketch our resolutions for a "toy model" which is Klein-Gordon theory in curved Lorentzian spacetime with the mass parameter, m², viewed as an interaction parameter. The strategies I describe for ensuring locality and covariance of this toy model's flow relations are expected to be applicable more generally to QFTs with nonlinear interactions in curved Lorentzian spacetimes.