From Feynman Diagrams to the GW Approximation

A visual guide to many-body perturbation theory for condensed-matter physics
From equations to diagrams — and from diagrams back to equations

Author: Dmitry Skachkov

Content

• Introduction
• How to Use This Tutorial
• Part I — Feynman Diagrams
• Part II — The Dyson Equation
• Part III — Self-Energy
• Part IV — The GW Approximation
• A Final Perspective

Introduction

Feynman diagrams were originally introduced in the context of quantum electrodynamics as a graphical language for representing the propagation, interaction, creation, and annihilation of elementary particles. Instead of manipulating increasingly complicated terms in perturbation theory only as algebraic expressions, one could represent propagators by lines, interactions by vertices, and entire classes of physical processes by compact graphical structures.

Over time, this diagrammatic language became an essential tool far beyond high-energy physics. In condensed-matter and many-body theory, Feynman diagrams are used to represent electron propagation, exchange, polarization, screening, self-energy corrections, and collective many-particle effects. Diagrammatic expansions also provide a natural language for constructing and understanding approximations such as the GW approximation.

However, the transition from particle-physics diagrams to condensed-matter diagrams is not always straightforward.

In high-energy physics, diagrams are often introduced through scattering processes: particles enter, interact at vertices, transform or exchange other particles, and leave the interaction region. In condensed-matter physics, the same graphical language is adapted to a different type of problem. Here we are usually interested not in isolated scattering events, but in particles propagating through an interacting many-body environment.

A solid line may represent a bare or dressed fermionic propagator. A wavy interaction line may represent either the bare Coulomb interaction or a dynamically screened interaction. Closed electron-hole structures describe polarization processes. Insertions into propagator lines represent self-energy corrections. Repeated interaction processes generate infinite diagrammatic series that can be reorganized through the Dyson equation.

The underlying mathematics is well established, but the visual presentation of these diagrams is much less uniform. Different papers and textbooks often use different line styles, arrow conventions, colors, orientations, and graphical layouts. As a result, students may learn the equations but still find it difficult to understand how a mathematical expression is translated into a diagram—or how to draw a clear and internally consistent diagram of their own.

This tutorial takes a practical approach.

We introduce a consistent graphical language for the condensed-matter many-body problem considered here. The goal is not only to recognize standard diagrams, but also to build a two-way translation between equations and diagrams: to construct diagrams systematically from mathematical expressions, and to read diagrams back into explicit many-body equations. Once the graphical conventions are fixed, this reading is not ambiguous: the lines, arrows, vertices, and internal variables determine the corresponding mathematical expression.

The tutorial follows four connected steps.

Part I — Feynman Diagrams

We begin with the basic graphical elements of many-body perturbation theory: electron propagators, interaction lines, vertices, polarization bubbles, and self-energy structures. We establish a consistent set of drawing conventions that allow a two-way translation between mathematical expressions and diagrams. These rules show how to construct diagrams from many-body equations and how to read diagrams back into explicit formulas. Once the conventions are fixed, this translation is unambiguous: propagator lines, interaction lines, arrows, vertices, and internal variables determine the corresponding mathematical expression.

Part II — The Dyson Equation

We then show how repeated interaction corrections generate an infinite perturbative series and how this series can be written compactly through the Dyson equation,

$$G = G_0 + G_0 \Sigma G.$$

This introduces one of the central objects of many-body theory: the self-energy.

Part III — Self-Energy

The self-energy describes how interactions with the surrounding many-body environment modify single-particle propagation. Its real part shifts and renormalizes excitation energies, while its imaginary part, after analytic continuation to real frequencies, is connected with finite quasiparticle lifetimes. We introduce the physical meaning of the self-energy, quasiparticle renormalization, and the connection with the spectral function.

Part IV — The GW Approximation

Finally, we bring the pieces together. Electronic polarization modifies the bare Coulomb interaction $v$ and produces the screened interaction $W$. The GW approximation then approximates the self-energy in the compact form

$$\Sigma^{GW} = iGW.$$

We show how this expression is related to its Feynman diagram, how screening emerges from polarization processes, and how the coupled diagrammatic structure of $G$, $P$, $W$, and $\Sigma$ forms one of the central frameworks of modern electronic-structure theory.

The purpose of this tutorial is therefore broader than learning how to draw diagrams. By the end of the tutorial, the reader should be able to read a many-body diagram, relate it to its mathematical expression, understand the physical process it represents, and see how the visual language of Feynman diagrams leads naturally to the GW approximation.

How to Use This Tutorial

This tutorial is organized in two layers.

The main pages contain the essential concepts needed to understand Feynman diagrams, the Dyson equation, self-energy, and the GW approximation. They focus on the physical meaning of the equations, the interpretation of the diagrams, and the logical connections between the main quantities.

Reading only the main pages provides a compact introduction to the subject. After completing them, the reader should be able to:

• identify the main propagator and interaction lines in Feynman diagrams;
• understand how the Dyson equation connects $G_0$, $\Sigma$, and $G$;
• explain how the real and imaginary parts of the self-energy affect excitation energies and lifetimes;
• understand the physical meaning of screening;
• recognize the central relations of the GW approximation;
• follow the general workflow from a mean-field reference to quasiparticle energies.

This core material is intended for students encountering many-body Green’s functions for the first time, as well as researchers who need a concise conceptual introduction before using GW software or reading more specialized literature.

Each main page also links to an Advanced Topics section. These sections provide a more detailed treatment of the mathematical and diagrammatic structure, including operator notation, different representations, irreducible diagrams, analytic properties, vertex corrections, self-consistency, and practical computational approximations.

The advanced material is most useful for readers who plan to:

• derive or manipulate many-body equations;
• interpret detailed Feynman-diagram expansions;
• develop or modify Green’s-function and GW codes;
• understand the approximations used in research software;
• study quasiparticle lifetimes, spectral functions, vertex corrections, or methods beyond standard GW;
• continue toward the Bethe–Salpeter equation, dynamical mean-field theory, or other many-body approaches.

The two levels can be used independently. A first-time reader may complete the main sequence and return to the advanced sections later. A reader with prior experience may use the main pages as a structured overview and open the advanced sections whenever a more formal explanation is needed.

The recommended first path is

$$ \text{Feynman diagrams} \longrightarrow \text{Dyson equation} \longrightarrow \text{Self-energy} \longrightarrow GW.$$

The advanced pages then provide a second, deeper pass through the same concepts.

A Final Perspective

The many-body interactions described in this tutorial can lead to complicated integral equations involving numerous internal variables, intermediate states, and repeated interaction processes. Written only in algebraic form, even a relatively simple approximation can quickly become difficult to visualize and interpret.

One of the remarkable achievements of Feynman’s diagrammatic language is that this complexity can be transformed into geometry. Propagation becomes a line, interaction becomes a connection between lines, internal variables become vertices to be integrated over, and an entire many-body expansion becomes a sequence of graphical structures that can be read, analyzed, and compared.

This visual language did much more than make complicated equations easier to illustrate. It provided a new way of thinking about interacting quantum systems. Processes that are deeply hidden inside multidimensional integrals become visible as topology, connectivity, and structure.

The diagrams used in condensed-matter and many-body physics today have evolved far beyond their original applications, but the central idea remains extraordinarily powerful: a complex quantum theory can sometimes become clearer when its mathematical structure is given a visual form.

This tutorial is built around that idea. We hope that learning not only to recognize Feynman diagrams, but also to construct them from equations and read equations back from them, will help make the language of many-body theory more transparent and intuitive.

Readers who would like to explore Green’s function theory in greater depth, apply it to solving the Kohn–Sham equation, and earn a certificate of completion are invited to continue with our full course, Green's Functions in Quantum Many-Body Theory (to be published soon).

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© 2026 Delta Science Institute
This tutorial is freely available as an open educational resource.
Unless otherwise stated, the tutorial materials are licensed under the
Creative Commons Attribution 4.0 International License (CC BY 4.0).
Preferred citation: Skachkov, D. (2026). From Feynman Diagrams to the GW Approximation. Delta Science Institute. Available at: https://www.dsedu.org/tutorials/FeynmanGW