# Wick four-point pairing diagram: source and reuse

Original diagram by OpenAI Codex for QFT.org, 2026. The diagram and its editable source are licensed under [Creative Commons Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/). Credit the creator and QFT.org, link the license, and indicate your changes when adapting the diagram. This license statement does not cover the consulted books or article.

- [Editable TikZ source](./wick-contractions.tex)
- [SVG diagram](./wick-contractions.svg)
- [Explanation and equivalent pairing table](https://qft.org/foundations/correlators-sources-effective-actions/wick-theorem-free-gaussian-factorization/)

The upper panel shows all three complete pairings of four centered Gaussian scalar fields. Each term has a plus sign. The lower panel uses the selected charge-preserving free Dirac vacuum and the field order psi, bar-psi, psi, bar-psi. Only two complete pairings survive; putting the exchanged pair into the declared psi/bar-psi order produces the relative minus. Curved arcs are contraction notation, not particle trajectories or interaction lines. The two-point functions are raw time-ordered correlators, with no extra factor of i. Other states require their own covariance and allowed-pairing data.

The drawing is schematic. Pairings, signs, state restrictions and formulas were preserved in the September 2026 typography revision; small parity labels and qualifications were enlarged. No third-party figure was copied or traced.

## Consulted sources

- Schwartz, Matthew D. *Quantum Field Theory and the Standard Model*. Cambridge University Press, 2014, § 7.A, pp. 100–103, and § 14.3.2, p. 263. [DOI](https://doi.org/10.1017/9781139540940). These passages supply the normal-ordered operator identity and Gaussian four-point differentiation. The creation/annihilation labels in § 7.A are opposite to the page's labels; the actual operators and contraction are translated accordingly.
- Weinberg, Steven. *The Quantum Theory of Fields, Volume I: Foundations*. Cambridge University Press, 1995; first edition, 2005 paperback and 2012 printing consulted, § 6.1, pp. 261–262 and 267–268. [DOI](https://doi.org/10.1017/CBO9781139644167). The graded rearrangement gives the exchange sign. Weinberg's raw contraction notation must not be identified by name with the page's delta-normalized inverse kernel.
- Wick, Gian-Carlo. “The Evaluation of the Collision Matrix.” *Physical Review* 80, no. 2 (1950): 268–272, § II, pp. 269–270, and Theorems 1–2, pp. 270–271. [DOI](https://doi.org/10.1103/PhysRev.80.268). The graded chronological and normal-product rules, including the reversed fermion contraction, determine the signs.

## Regeneration

Compile in a temporary directory with LaTeX/TikZ and Latin Modern fonts, then convert the DVI to SVG:

```sh
latex -interaction=nonstopmode -halt-on-error wick-contractions.tex
dvisvgm --no-fonts --exact-bbox -o wick-contractions.svg wick-contractions.dvi
```

The website revision used TeX Live 2025 and dvisvgm 3.4.3. Its SVG additionally canonicalizes generated font-glyph identifiers and duplicate glyph paths. Repeating the source compilation reproduced the canonical output; comparisons with both raw outputs at several raster widths found identical pixels. Raw SVG bytes can differ because of generated identifiers even when the drawing is unchanged.