nLab n-point function

Idea

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

$$\left\langle : \mathbf{\Phi}(x_1) \mathbf{\Phi}(x_2) \cdots \mathbf{\Phi}(x_n) : \right\rangle$$

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) $n$-point functions are also called correlators, but in fact both terms are often used interchangeably.

Traditionally $n$-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 $x_i$ are lightlike.

On the other hand, in Euclidean field theories the $n$-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 $n$-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.

Related concepts

• Feynman amplitude

References

Discussion in AdS-CFT:

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