nLab
counterterm

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

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

𝒮(gS int+jA)=limΛ(𝒮 Λ𝒵 Λ)(gS int+jA) \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 gS int+jAg S_{int} + j A is to be corrected by “counterterm-interactions

S counter,Λ(𝒵 Λid)(gS int+jA). 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

See also

Last revised on August 1, 2018 at 08:17:22. See the history of this page for a list of all contributions to it.