Stefan Hollands, Robert M. Wald
On the Renormalization Group in Curved Spacetime
Preprint series: ESI preprints

MSC:

81T17 Renormalization group methods

81T20 Quantum field theory on curved space backgrounds

PACS: 11.10.Hi,04.62.+v
Abstract: We define the renormalization group flow for a renormalizable
interacting quantum field in curved spacetime via its behavior under
scaling of the spacetime metric, $\g \rightarrow \lambda^2 \g$. We
consider explicitly the case of a scalar field, $\varphi$, with a
self-interaction of the form $\kappa \varphi^4$, although our results
should generalize straightforwardly to other renormalizable
theories. We construct the interacting field---as well as its Wick
powers and their time-ordered-products---as formal power series in the
algebra generated by the Wick powers and time-ordered-products of the
free field, and we determine the changes in the interacting field
observables resulting from changes in the renormalization
prescription. Our main result is the proof that, for any fixed
renormalization prescription, the interacting field algebra for the
spacetime $(M, \lambda^2 \g)$ with coupling parameters $p$ is
isomorphic to the interacting field algebra for the spacetime $(M,
\g)$ but with different values, $p(\lambda)$, of the coupling
parameters. The map $p \to p(\lambda)$ yields the renormalization
group flow. The notion of essential and inessential coupling
parameters is defined, and we define the notion of a fixed point as a
point, $p$, in the parameter space for which there is no change in essential
parameters under renormalization group flow.