Correlation Functions

Abstract

In this chapter, our focus is on providing a comprehensive and mathematically rigorous treatment of the correlation diagrams and their associated correlation functions. From the previous chapter, recall that a correlation diagram for N (where N is tacitly assumed to be at least 3) indistinguishable Fermions graphically exhibits the potencies of their mutual correlations. In purely mathematical terms, such a diagram is an undirected, loopless multi-graph on N vertices. Here, the term multi-graph simply means a graph in which there may be multiple (albeit, finitely many) edges between a vertex-pair. In what follows, we regard correlation diagram and multi-graph as equivalent terms. Let $\Gamma$ be a multi-graph on N vertices with some chosen labeling of its vertices by numbers $1, 2, \ldots , N$. Then, to $\Gamma$ corresponds a product of the terms $(z_{i} - z_{j})^{p_{ij}}$, denoted by $\mu (\Gamma )$, in which $z_{1}, \ldots , z_{N}$ are indeterminates and for $1 \le i < j \le N$, the nonnegative integer $p_{ij}$ is the number of edges between the vertices labeled i and j (in $\Gamma$). In the classical theory of invariants, $\mu (\Gamma )$ is called the graph-monomial of $\Gamma$. Note that since our N Fermions are indistinguishable, we must consider each of the possible choices of vertex-labelings, for the correlation diagram under consideration, on an equal footing. Two multi-graphs $\Gamma _{1}$ and $\Gamma _{2}$, each with N labeled vertices, are said to be isomorphic provided one can be obtained from the other by a relabeling of its vertices (see Fig. 2.1 for an example of isomorphic multi-graphs).