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Quantum Field Theory from the Ground Up

Modern particle physics posits that the fundamental constituents of the universe are not particles but fields. Instead, particles are merely excitations of these fields. Quantum Field Theory (QFT) is a theoretical framework that combines quantum mechanics and special relativity to study these fields and their interactions. Our current understanding of the universe is based on the Standard Model of particle physics, which is a quantum field theory that describes the electromagnetic, weak, and strong nuclear forces. It is defined by the Standard Model Lagrangian:

We classify particles into two categories: fermions and bosons. Fermions are, in essence, matter particles, while bosons are force carriers. Fermions are spin-1/2 particles, while bosons are spin-0 or spin-1 particles. The spin-0 boson is the Higgs boson. Spin-1/2 particles are described by spinors, spin-0 by scalars, and spin-1 by vectors.

In quantum field theory, all particles are treated as excitations of their respective fields. For example, electrons are excitations of the electron field, while photons are excitations of the electromagnetic field. The Lagrangian density describes the dynamics of these fields and their interactions. Additionally, the fields interact with each other through the exchange of force-carrying particles, which are also excitations of their respective fields.

To study quantum field theory, we need to have a solid understanding of three key concepts:

  1. Quantum Mechanics: In the early 20th century, various physical phenomena could not be explained by classical physics. Quantum mechanics emerged as a new framework to describe the behavior of particles at the atomic and subatomic levels. Understanding quantum mechanics is essential for grasping the principles of quantum field theory because QFT builds upon the foundations of quantum mechanics.

  2. Special Relativity: Special relativity, formulated by Albert Einstein, describes the behavior of objects moving at high speeds, close to the speed of light. Fundamentally, its core postulate leads to the Lorentz transformation, which relates the space and time coordinates of two observers in relative motion. QFT is a relativistic theory, meaning it incorporates the principles of special relativity into its framework.

  3. Classical Field Theory: Classical field theory describes the behavior of classical fields, such as electromagnetic fields, gravitational fields, and scalar fields. Classical field theory ensures that the principle of locality is satisfied, meaning that interactions occur at a single point in space and time. For example, Newtonian gravity does not satisfy locality, as it acts instantaneously at a distance. This is why a field theory of gravity is required (general relativity). Once we attempt to make quantum mechanics relativistic, we realize that we need to use a field-theoretic approach to describe particles and their interactions.

Quantum field theory can be thought of as a method to ensure quantum mechanics adheres to both the principles of special relativity and locality. In modern physics, quantum field theory is mostly used for two purposes:

  1. Particle Physics: QFT is the theoretical framework for understanding the fundamental particles and their interactions. The Standard Model of particle physics is a quantum field theory that describes all known fundamental interactions except for gravity.
  2. Condensed Matter Physics: QFT is also used to describe many-body systems in condensed matter physics. While particle physics focuses on a single particle or a few particles, condensed matter physics deals with systems containing a large number of particles. Instead of describing each particle individually, we use statistics to describe the collective behavior of the system.

In addition to the physics, quantum field theory also has a very rich mathematical structure. This includes concepts from group theory, topology, and differential geometry. We will encounter many of these concepts throughout the course, such as group representations, Lie algebras, fiber bundles, Clifford algebras, and much more. As such, I have included a section purely dedicated to mathematical concepts. This section will be a reference for the mathematical concepts we will encounter throughout the course.

Prerequisites​

The prerequisites for this set of notes are not strict. I will assume that the reader knows calculus, linear algebra, and some basic physics. No prior knowledge of quantum mechanics or special relativity is required.

Here are some resources for the prerequisites. Although they are by no means exhaustive, they should be enough to get you started.