# Connection between the renormalization groups of Stückelberg-Petermann and Wilson

December 27, 2010

The Stueckelberg-Petermann renormalization group is the group of finite
renormalizations of the S-matrix in the framework of causal perturbation
theory. The renormalization group in the sense of Wilson relies usually on a
functional integral formalism, it describes the dependence of the theory on a
UV-cutoff $\Lambda$; a widespread procedure is to construct the theory by
solving Polchinski's flow equation for the effective potential.
To clarify the connection between these different approaches we proceed as
follows: in the framework of causal perturbation theory we introduce an
UV-cutoff $\Lambda$, define an effective potential $V_\Lambda$, prove a
pertinent flow equation and compare with the corresponding terms in the
functional integral formalism. The flow of $V_\Lambda$ is a version of Wilson's
renormalization group. The restriction of these operators to local interactions
can be approximated by a subfamily of the Stueckelberg-Petermann
renormalization group.

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