nLab
renormalization condition

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, ("re"-)normalization is an extension of time-ordered products, uniquely defined away from coinciding spacetime arguments, to the locus of coinciding spacetime arguments. Before the extension these time-ordered products may generally enjoy further properties beyond the causal factorization that characterizes them as time-ordered products. Not all these properties necessarily need to be preserved by a choice of extension of distributions to coinciding spacetime arguments. A renormalization condition is the condition that some such properties ought to be preserved by the choice of ("re-")normalization.

The main theorem of perturbative renormalization holds true if at least the renormalization conditions “field independence” and “translation equivariance” are imposed. Depending on how many other renormalization conditions are imposed, the resulting Stückelberg-Petermann renormalization group becomes smaller. (Duetsch 18, remark 3.102)

Besides “field indepence” and “translation invariance” a key renormalization condition for gauge theories is the quantum master equation or quivalently master Ward identity. If this renormalization condition cannot be satisfied, one says that the corresponding perturbative QFT does not exist due to a “gauge anomaly”. (Dütsch 18, section 4.2)

Definition

For the moment see at S-matrix this definition.

Examples

For the moment see at S-matrix this prop.

References

Last revised on February 20, 2018 at 00:36:19. See the history of this page for a list of all contributions to it.