Feynman Diagram Rules

Every non-vanishing Wick contraction corresponds to a Feynman diagram — a graph that encodes the topology of the contraction. sft-wick automatically constructs these diagrams as FeynmanDiagram objects backed by a networkx.MultiGraph.

Elements of a Diagram

Symbol

Element

Description

●

External point

An observable field operator (e.g. \(\phi_a(x)\)). Rendered as a filled circle.

■

Interaction vertex

A term from \(S_{\mathrm{int}}\) (e.g. \(F_{ijk} \phi_i\phi_j\psi_k\)). Rendered as a filled square.

—

C propagator

Correlation \(C_{ij}(x,x') = \langle\phi_i(x)\, \phi_j(x')\rangle_{S_0}\). Drawn as a solid blue line.

→

R propagator

Response \(R_{ij}(x,x') = \langle\phi_i(x)\, \psi_j(x')\rangle_{S_0}\). Drawn as a dashed red arrow (pointing from \(\psi\) to \(\phi\)).

Constructing a Diagram from a Pairing

Given an observable \(\mathcal{O}\) and an expansion of \(S_{\mathrm{int}}^n\), the Wick contraction produces a set of pairings. Each pairing maps to a diagram as follows:

  1. Create nodes: one external node per observable operator and one vertex node per interaction vertex instance.

  2. Create edges: for each contracted pair \((i,j)\), draw a propagator edge between the nodes that own operators i and j.

The class method from_pairing() performs this construction automatically.

Reading Off Expressions

To reconstruct the algebraic expression from a diagram:

  • Each vertex \(v\) contributes:

    • A coupling factor (e.g. \(F_{ijk}\))

    • Integration(s) over internal spatial variable(s) (\(\int \mathrm{d}y\,\ldots\))

    • Summation(s) over internal component indices (\(\sum_{i=1}^N \ldots\))

  • Each propagator edge contributes a two-point function: \(C_{ij}(x,x')\) or \(R_{ij}(x,x')\).

  • The overall prefactor for an order-n diagram is \((-1)^n / n!\) times the multinomial coefficient from the vertex combination.

Topological Properties

Loop count:

\[L = E - V + C\]

where \(E\) is the number of edges (propagators), \(V\) the number of nodes (external + vertex), and \(C\) the number of connected components. Accessed via the n_loops property.

Connectivity:

A diagram is connected if every node can be reached from every other node. Disconnected diagrams factorise into independent sub-diagrams. Checked via is_connected.

Self-loops (tadpoles):

When a propagator connects a node to itself (e.g. \(C(x,x)\) at a vertex), it forms a tadpole. These are rendered as small loops in the diagram.

Rendering

Use DiagramRenderer (or the convenience method draw_diagrams()) to visualise diagrams:

result = compute_moment(obs, action, order=1)

# Quick visualisation
result.draw_diagrams(order=1)

# Manual rendering with custom layout
from sft_wick import DiagramRenderer
renderer = DiagramRenderer(figsize=(8, 6))
for d_info in result.diagrams_by_order[1]:
    fd = d_info.to_feynman_diagram()
    renderer.draw(fd, title=fd.summary())

See Feynman Diagrams for the full usage guide.