nLab n-point function



Algebraic Quantum Field Theory

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



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT

Measure and probability theory



In relativistic quantum field theory an observable that evaluates the time-ordered products of the point-evaluation observables of the basic fields at nn points of spacetime in some state on a star-algebra is called an nn-point function, typically denoted

:Φ(x 1)Φ(x 2)Φ(x n): \left\langle : \mathbf{\Phi}(x_1) \mathbf{\Phi}(x_2) \cdots \mathbf{\Phi}(x_n) : \right\rangle

For details on what this means see at geometry of physics – perturbative quantum field theory chapter 7. Observables and chapter 15. Interacting quantum fields

Specifically the 2-point functions are also known as Feynman propagators, see Section 9. Propagators.

In Euclidean field theory (say under Wick rotation) nn-point functions are also called correlators, but in fact both terms are often used interchangeably.

Traditionally nn-point functions are thought of as distributions of several variables. In relativistic field theory these have singularities on the “relative light cones”, hence whenever two points x ix_i are lightlike.

On the other hand, in Euclidean field theories the nn-point functions/correlators are distributions with singularities only on the fat diagonal, hence when at least two of their arguments coincide. This means that in Euclidean field theory nn-point functions/correlators restrict to smooth (non-singular) differential forms on configuration spaces of points. For more on this perspective see at correlators as differential forms on configuration spaces of points.


Discussion in AdS-CFT:

  • Adam Bzowski, Paul McFadden, Kostas Skenderis, A handbook of holographic 4-point functions [arXiv:2207.02872]

Last revised on December 10, 2022 at 04:23:00. See the history of this page for a list of all contributions to it.