An open, community-editable reference for two-particle quantities in quantum field theory
Definitions, conventions, diagrammatics, and the technical derivations that papers usually leave out — with a focus on strongly correlated electron systems.
📖 Read it online · Symbols & notations · Contribute
Two-particle quantities — vertices, four-point functions, susceptibilities — sit at the heart of the theory of strongly correlated electrons, from the Bethe–Salpeter and parquet equations to modern diagrammatic extensions of dynamical mean-field theory. Unfortunately, there is a vast body of literature on them, and different communities use different (and often mutually incompatible) conventions and notations — sometimes even from paper to paper. This is a perennial source of confusion, and the main motivation for this project.
TwoParticleQFT is a documentation site that aims to:
- lay out the fundamental definitions of two-particle quantities from a single, self-consistent set of conventions;
- present the diagrammatic frameworks (parquet theory, single-boson exchange) built on top of them;
- work through the technical derivations that are usually omitted from papers "for the sake of brevity"; and
- explicitly cross-reference competing conventions in the literature (e.g. the "Vienna" vs. "Munich" conventions, and the differing channel labelings) so results can be translated between them.
The rendered site lives at https://nepomukritz.github.io/TwoParticleQFT/.
The content is written in MyST Markdown and organized into the sections below (see myst.yml for the full table of contents).
| Page | Focus | |
|---|---|---|
| Symbols and notations | Lookup tables for every object used on the site, each cross-listed against its common synonyms in the literature. | |
| Definitions | Starting point | The imaginary-time action, the bare propagator, and the antisymmetrized quartic interaction in Hugenholtz notation. |
| Basic definitions | Propagator, self-energy, four-point correlation function, and the four-point vertex, with the index-ordering conventions used throughout. | |
| Two-particle channels | Two-particle reducibility and the three channels (pp, ph, ph̄); connectors, channel identity operators, and bubbles. | |
| Frequency parametrizations | Choosing the independent frequency/momentum arguments; channel-native parametrizations that diagonalize bubble contractions. | |
| Spin parametrizations | SU(2) spin structure; the magnetic/density and singlet/triplet bases. | |
| Diagrammatic frameworks | Parquet theory | Parquet decomposition, Bethe–Salpeter and Schwinger–Dyson equations, and the parquet approximation. |
| Single boson exchange | The SBE decomposition, Hedin vertices, the screened interaction, and the SBE approximation (with GW as a limit). | |
| Advanced topics | Keldysh formalism | Two-particle QFT on the real-frequency Keldysh contour: index structure, the Keldysh rotation, and component Dyson equations. |
| w2dynamics conventions | The frequency conventions and definitions of the w2dynamics impurity solver — one- and two-particle Green's functions, channel parametrizations, and crossing symmetries — and how they relate to the Vienna/Munich conventions. |
- 🧭 One consistent convention, with explicit translation notes to the "Vienna" and "Munich" conventions, to the various channel labelings used across communities, and to the conventions of specific codes (e.g. the w2dynamics impurity solver).
- ✏️ Hugenholtz (antisymmetrized-vertex) diagrammatics used from the ground up to unify direct and exchange scattering.
- 🌡️ Both formalisms: the imaginary-time Matsubara formalism and the real-frequency Keldysh formalism.
- 🧮 Worked examples (e.g. the Hubbard model) and collapsible step-by-step derivations of results that are normally quoted without proof.
- 🖼️ Diagrams and figures for propagators, vertices, bubbles, and the parquet/SDE relations.
Note
The site is under active construction — several pages carry visible "to do" notes marking derivations still to be filled in or expanded. See the wishlist for the current priorities.
The site is a Jupyter Book 2 / MyST project. You only need the jupyter-book CLI; a recent Node.js is bundled/bootstrapped for you either way.
Option A — Python (pip):
pip install jupyter-book # provides the v2 (MyST) CLI, ≥ 2.1.0Option B — Node (npm), matching the CI:
npm install -g jupyter-bookThen, from the repository root:
# Live preview with hot reload
jupyter-book start
# Or produce a static HTML build in _build/html
jupyter-book build --htmlThe deployment to GitHub Pages is fully automated: every push to main triggers the deploy.yml workflow, which builds the HTML and publishes it to Pages.
TwoParticleQFT/
├── myst.yml # MyST project config + table of contents
├── src/
│ ├── intro.md # Landing page
│ ├── symbols_and_notations.md
│ ├── starting_point.md
│ ├── basic_definitions.md
│ ├── two-particle-channels.md
│ ├── frequency_parametrizations.md
│ ├── spin_parametrizations.md
│ ├── parquet_theory.md
│ ├── single_boson_exchange.md
│ ├── keldysh_formalism.md
│ ├── w2dynamics.md
│ ├── diagrams/ # Feynman diagrams (SVG/PNG)
│ └── images/ # Title image, logo
├── Vertex_conventions.pdf # Supplementary notes on vertex conventions
├── .github/workflows/ # GitHub Pages deployment
└── LICENSE
This site is editable by anyone, and contributions are very welcome! Whether you have found a typo or a sign error, want to clarify a convention, or would like to add a whole new section, please open an issue or a pull request.
Good places to start:
- the wishlist of planned sections and improvements;
- the "to do" admonitions scattered throughout the individual pages, marking derivations and references still to be added.
To preview your changes, edit the relevant Markdown file under src/ and run jupyter-book start (see above).
Released under the MIT License. © 2025 Nepomuk Ritz.
If this reference is useful for your work, a link back to the site is appreciated:
N. Ritz, TwoParticleQFT — Open review of definitions, conventions and equations for two-particle quantum field theory. https://nepomukritz.github.io/TwoParticleQFT/