# nLab local quantum field theory

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## Surveys, textbooks and lecture notes

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• Axiomatizations

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• Tools

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• Structural phenomena

• Types of quantum field thories

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• examples

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# Contents

## Idea

A central aspect of quantum field theory usually desired or demanded or expected of fundamental QFTs is that it is local in the rough sense that

In some effective quantum field theories locality may be violated, if fundamental local processes are averaged out to a single non-local macroscopic process, but for fundamental physics of the observable universe it is thought to hold.

### Formalization in quantum field theory

Locality is formalized in the two main axiomatizations of quantum field theory as follows.

• In AQFT the algebras of observables are required to form a local net meaning that

1. there is one such algebra assigned to each suitable subset of spacetime, compatible with inclusion of such subsets,

2. that under these inclusions algebras associated to spacelike separated regions commute with each other.

Often in the literature the term “local quantum field theory” is meant to refer specifically to these AQFT axioms (some authors use the terms synonymously, dating from a time when this was the only axiomatization of quantum field theory considered.)

• In FQFT locality is encoded in the functor-property of the functor on the category of cobordisms: being a functor means that the assignment to a cobordism $\Sigma$ is obtained by composing the assignments to any decomposition of $\Sigma$ into small cobordisms. In particular, in extended quantum field theory (now also sometimes called “fully localized” QFT) this is n-functorial meaning that this gluing condition holds in all dimensions and in all directions.

### Formalization in pre-quantum field theory

There are also properties of locality in prequantum field theory.

A Lagrangian density is called a local Lagrangian if it “depends only on finitely many derivatives of the fields” at any point of spacetime, which formally means that it is a horizontal differential form on the jet bundle of the field bundle. Local Lagrangian densities are expected to yield local quantum field theories under quantization. This is the case at least in perturbative quantum field theory formalized via causal perturbation theory/perturbative AQFT, see at S-matrix the section Quantum observables and retarded products

## References

Steps towards a local version of BV-formalism are undertaken in

Last revised on September 20, 2017 at 02:37:01. See the history of this page for a list of all contributions to it.