On the Renormalization Group in Curved Spacetime

TL;DR

This paper develops a framework for understanding the renormalization group flow of quantum fields in curved spacetime by analyzing how the algebra of interacting fields changes under metric scaling, identifying fixed points in the parameter space.

Contribution

It introduces a method to define the renormalization group flow in curved spacetime using metric scaling and characterizes fixed points and essential couplings in this context.

Findings

Proves the isomorphism of interacting field algebras under metric scaling with adjusted parameters.

Defines essential and inessential coupling parameters in curved spacetime.

Establishes the concept of fixed points where essential parameters remain unchanged.

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, $\g \to \lambda^2 \g$. We consider explicitly the case of a scalar field, $\phi$, with a self-interaction of the form $\kappa \phi^4$, 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.