Contents

# Contents

## Idea

In perturbative quantum field theory one way to construct an S-matrix $\mathcal{S}$ via ("re"-)normalization of time-ordered products/Feynman amplitudes is to consider a sequence of UV cutoffs $\Lambda$ with corresponding effective S-matrices $\mathcal{S}_\Lambda$ and then form the limit as $\Lambda \to \infty$, which exists if in the course one applies suitable interaction vertex redefinitions $\mathcal{Z}_\Lambda$

$\mathcal{S}(g S_{int} + j A) = \underset{\Lambda \to \infty}{\lim} (\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda)(g S_{int} + j A)$

(See at effective QFT this prop.).

This may be read as saying that as one “removes the UV-cutoff” the original interaction action functional $g S_{int} + j A$ is to be corrected by “counterterm-interactions

$S_{counter, \Lambda} \;\coloneqq\: \left( \mathcal{Z}_\Lambda - id \right) \left( g S_{int} + j A \right) \,.$

## Details

See at effective action around this remark.

## References

The rigorous formulation of ("re"-)normalization via UV cutoff and counterterms in causal perturbation theory/perturbative QFT is due to

reviewed in