Introduction

 

Quantum field theory is the relativistic generalization of (non-relativistic) quantum mechanics. It is a very huge subject which physics graduate students normally study for two or three semesters after finishing graduate-level quantum mechanics in their first year of study. Nevertheless, as Fundamental Physics Prize laureate Prof. Nima Arkani-Hamed did in his third year as a physics undergraduate, many students successfully learn quantum field theory without the knowledge of graduate level quantum mechanics; along with the very basics of special relativity and Cauchys residue theorem, undergraduate quantum mechanics is more or less sufficient as a prerequisite.

 

In these articles, we will teach you quantum field theory assuming you have already read my previous articles on physics and math. Since I treated there several topics usually treated in quantum field theory textbook, you may not understand some of our articles here if you only know the very basics of special relativity and undergraduate quantum mechanics without reading my previous articles on physics and math.

 

I will add more sections and articles later.

 

Preliminaries

Notations and conventions in our review articles for quantum field theory

Dimension

 

Comments:

Every quantum field theory book uses slightly different notations and conventions. In the first article, we will set ours.

 

Klein-Gordon equation and Noethers theorem

Klein-Gordon equation and Klein-Gordon action

 

Comments:

Klein-Gordon equation is the relativistic generalization of Schrödinger equation.

 

Noethers theorem in field theory

Energy-momentum tensor

 

Comments:

The first article explains the field theory version of Noethers theorem which is the one familiar from classical mechanics. Only minor details are different. The second article treats one of the most important examples of Noethers theorem.

 

A simple derivation of the conserved current in complex Klein-Gordon theory

The variation of a field in terms of the Noether charge

 

Comments:

If an action exhibits a symmetry, one can easily derive the conserved current using Noethers theorem. However, there is an equivalent, easier method, if you cannot readily remember the formula for Noethers theorem. The first article concerns this method. The second article shows how the variation of a field can be expressed by the Noether charge associated with the variation. Readers should be familiar with this from my earlier article Noethers theorem in field theory. However, I approach this from a different perspective. These two articles concern concepts which are very important and often used in quantum field theory, but which are usually left out of textbooks.

 

Complex analysis

Cauchy-Riemann equation

Cauchys integral formula and Cauchys residue theorem

Application of residue theorem

 

Comments:

Cauchys residue theorem plays an important role in quantum field theory. For our purpose, it is needed in our article The Feynman propagator in the section Quantization of the scalar field.

 

Quantization of the scalar field

The scalar field as harmonic oscillators

Complex scalar field

The Feynman propagator of the scalar field

 

Comments:

By building on the ideas in the section Klein-Gordon equation and Noethers theorem we quantize Klein-Gordon field (also known as the scalar field) in this section.

 

The Lorentz group

The Lorentz group and its representations

Spinor representations of the Lorentz group

 

Comments:

The articles here are very beautiful and elegant.

 

The Dirac field

The Dirac equation and gamma matrices

Angular momentum of the Dirac field

Dirac Lagrangian

Solutions of the Dirac equation

Dirac bilinears

Quantization of the Dirac field

The Feynman propagator of the Dirac field

QED Lagrangian

Electron magnetic moment from the Dirac equation

C, P, T transformation for the Dirac field

 

Comments:

Having dealt with the scalar field in earlier sections, we deal with the next simplest field, namely, the Dirac field in this section. In the first reading, you can skip the last article.

 

Scattering

Mandelstam variables

S-matrix

Cross sections

 

Perturbation theory

LSZ reduction formula

The Dyson series and the perturbative expansion of n-point function

Expectation values in quantum field theory, revisited

Wicks theorem

Feynman rules for phi^4 theory I

Feynman rules for phi^4 theory II

Tree diagrams and loop diagrams

Connected diagrams and disconnected diagrams

The effective action

Wick rotation

Euclidean formulation

 

Comments:

Those of the readers who read my earlier article Expectation values in quantum field theory (1) may wonder why the Gaussian function is present in the definition of the formula for the expectation values. You will find the reason in the article about the Expectation values.

 

Quantum Electrodynamics

Quantization of electromagnetic field

 

Comments:

You have to know the Maxwell Lagrangian to understand the article Quantization of electromagnetic field

 

Renormalization

 

Path Integral for Fermions

 

Quantization of Non-Abelian Gauge theory

Faddeev-Popov Ghost

BRST symmetry

Feynman rules for non-Abelian gauge theory

Asymptotic freedom

 

Standard Model

Spontaneous symmetry breaking

Higgs mechanism

Glashow-Weinberg-Salam model

GIM mechanism and CKM matrix

 

Anomalies

 

Group Theory

SU(3) and the Gell-Mann matrices

Tensor methods

Clebsch-Gordan decomposition

Quark model

SU(5) and the Georgi-Glashow model