nLab
linear chord diagram

Contents

Contents

Idea

A linear chord diagram or arc diagram or rooted chord diagram is a trivalent finite undirected graph with an embedded oriented line and all vertices on that line.

Equivalemtly this is just an n-tuple equipped with a partition into pairs.

The following shows a generic example of a linear chord diagram:

The graphics on the right shows all linear chord diagrams with precisely four vertices.

Closing up the line of a linear chord diagram to a circle and remembering the ordering of vertices only op to cyclic permutation, it becomes a round chord diagram, usually just called a chord diagram. Conversely, a linear chord diagram is equivalently a round chord diagram with one of its vertices singled out.

Applications

Wick’s theorem

The combinatorics of contractions in Wick's theorem is governed by linear chord diagrams:

Let {Z i}\{Z_i\} be a set of quantum fields/random variables which are free fields/multivariate normally distributed with

Z iZ j=k ij. \big\langle Z_i Z_j \big\rangle = k_{i j} \,.

Then Wick's theorem says that the expectation value of the product of nn \in \mathbb{N} of these fields/random variables is the sum over linear chord diagrams with nn vertices of the product over the edges e ie je_i \to e_j of the given chord diagram of the factors k ijk_{i j}.

For example, for n=4n = 4, Wick's theorem says this:

References

Last revised on December 2, 2019 at 13:34:06. See the history of this page for a list of all contributions to it.