3:30–4:30 pm
Please Join us:
Mark G. Klehfoth’s PhD Thesis Defense
Local and Covariant Flow Relations for OPE Coefficients in Lorentzian Spacetimes
The short-distance "ultraviolet" divergences produced by pointwise products of quantum fields have posed an intractable obstacle to formulating interacting quantum field theories outside of perturbation theory. In the coincidence limit, the behavior of products of quantum fields is characterized by their operator product expansion (OPE). For self-interacting quantum field theories in flat Euclidean space, Holland and Hollands have shown the OPE coefficients satisfy "flow equations" with respect to the interaction parameter(s). These flow equations were proven to hold order-by-order in perturbation theory, but they are well defined non-perturbatively and thus provide a possible route towards a non-perturbative construction of the interacting theory.
In this thesis, we generalize the Euclidean flow equations to Lorentzian spacetimes for the solvable "toy model" of massive Klein-Gordon scalar field theory, with the mass viewed as the "self-interaction parameter". Because the flow relations contain a spacetime integral with a cutoff, several modifications must be made in order to obtain local and covariant flow relations in Lorentzian spacetimes: In Minkowski spacetime, the cutoff breaks Lorentz invariance. Nevertheless, we show how Lorentz covariance can be restored to the flow relations by introducing "counterterms" that cancel the non-covariant dependence on the cutoff. In curved spacetimes, the integral in the flow relations depends non-locally on the spacetime metric. However, we show how this difficulty may be overcome by suitably replacing the metric with a local polynomial approximation. We thereby obtain local and covariant flow relations for the Klein-Gordon OPE coefficients in curved Lorentzian spacetimes. We then develop an algorithm for constructing local and covariant flow relations beyond this toy model based on the "associativity" properties of the OPE coefficients. We apply our algorithm to the flow relations of an interacting scalar field theory with a quartic potential.
Committee Members:
Robert M. Wald (Chair)
Daniel E. Holz
Stephan S. Meyer
Dam T. Son