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

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

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*.

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.