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Some Basic Tools of QFT

A. V. Ivanov, M. A. Russkikh

Abstract

This work has a methodological nature and is a set of lecture notes for undergraduate students. It is devoted to the study of the basic tools of quantum field theory on the example of the simplest cubic "toy" model. We introduce such concepts as the functional integral, the generating functions, the background field method and the Feynman diagram technique, and also consider relations among them. The model under consideration allows us to perform all the calculations explicitly, which significantly increases the visibility and clarity of the notes. We also provide neat proofs that are usually omitted in standard educational publications.

Some Basic Tools of QFT

Abstract

This work has a methodological nature and is a set of lecture notes for undergraduate students. It is devoted to the study of the basic tools of quantum field theory on the example of the simplest cubic "toy" model. We introduce such concepts as the functional integral, the generating functions, the background field method and the Feynman diagram technique, and also consider relations among them. The model under consideration allows us to perform all the calculations explicitly, which significantly increases the visibility and clarity of the notes. We also provide neat proofs that are usually omitted in standard educational publications.

Paper Structure

This paper contains 9 sections, 7 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Functional integral
  4. Method of calculation
  5. Cubic functional
  6. Asymptotic expansion
  7. Diagrams
  8. Mathematical point of view
  9. Exponential decomposition

Key Result

Lemma 2.1

Let $x,y\in V$, and $F(x)$ be a polynomial of a finite degree, then we have

Figures (2)

  • Figure 1: The function $I(\alpha_k)$ for $\alpha_k\in[-2.1,2.1]$.
  • Figure 2: Star-triangle relation.

Theorems & Definitions (15)

  • Definition 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 5 more