Lecture Notes on Quantum Many-Body Theory: A Pedagogical Introduction

Abstract

In these notes, we present a rigorous and self-contained introduction to the fundamental concepts and methods of quantum many-body theory. The text is designed to provide a solid theoretical foundation for the study of interacting quantum systems, combining clarity with mathematical precision. Core topics are developed systematically, with detailed derivations and comprehensive proofs that aim to make the material accessible to graduate students and beginning PhD students. Special attention is given to formal consistency and pedagogical structure, so as to guide the reader through both the conceptual and technical aspects of the subject. This work is intended as a reliable starting point for further exploration and research in modern quantum many-body physics.

Figures (61)

• Figure 1: The 14 3-dimensional Bravais lattices, grouped into seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Each lattice is defined by its symmetry, the lengths and angles of its primitive vectors, and can occur in primitive (P), body-centered (I), face-centered (F), or base-centered (C) forms. This classification encompasses all possible periodic arrangements of points in three-dimensional space consistent with crystallographic symmetry.
• Figure 2: Integration region corresponding to \ref{['eq: primaregioneintegrazioneordinesecondointerazioneserieDyson']}, where $0 \leq \tau" \leq \tau' \leq \tau$. This domain reflects the time ordering $\tau" \leq \tau'$ required by the Dyson series at second order. The integration is performed first over $\tau"$ (from $0$ to $\tau'$), then over $\tau'$ (from $0$ to $\tau$), and matches the natural ordering of operators in $\hat{T}_D \lbrace \hat{\mathcal{H}}_I^{(0)}(\tau') \hat{\mathcal{H}}_I^{(0)}(\tau") \rbrace$.
• Figure 3: Integration region corresponding to \ref{['eq: secondaregioneintegrazioneordinesecondointerazioneserieDyson']}, where $0 \leq \tau" \leq \tau$ and $\tau" \leq \tau' \leq \tau$. This is an equivalent rewriting of the domain in Figure \ref{['fig: primaregioneintegrazioneHtauprimoHtausecondo']}, preserving the time ordering $\tau" \leq \tau'$. The integration is performed first over $\tau"$, and for each fixed $\tau"$, over $\tau'$ from $\tau"$ to $\tau$.
• Figure 4: Integration region corresponding to \ref{['eq: terzaregioneintegrazioneordinesecondointerazioneserieDyson']}, where $0 \leq \tau' \leq \tau" \leq \tau$. This region corresponds to the opposite time ordering, $\tau' \leq \tau"$, and appears in the second-order Dyson term when operators are explicitly reordered by $\hat{T}_D$. The integration is carried out first over $\tau'$, and then over $\tau"$ from $\tau'$ to $\tau$.
• Figure 5: Schematic representation of a basic interaction in Feynman diagrammatics. In particular, bosonic lines are represented as wavy curves.

Theorems & Definitions (168)

• Example 1: Spin-$\frac{1}{2}$ particles
• Theorem 1: Construction of coherent states
• proof
• Remark 1: Notation
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof