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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:

As one can see, there are a bunch of probably unfamiliar symbols in the equation above. Being very abstract and mathematical, QFT is often considered one of the most challenging subjects in theoretical physics, and it is usually taught at the graduate level. (The other challenging subject being General Relativity. Personally I find QFT more challenging.)

However, QFT is incredibly successful in describing the behavior of elementary particles and their interactions.

This set of notes aims to introduce the basics of quantum field theory from the ground up. The goal is to start from high-school level physics and mathematics and build up all the way to the full theory of quantum field theory, with all of its complexities and subtleties. The main topics covered will be:

  • Quantum mechanics arose in the early 20th century as a response to the failure of classical mechanics to explain the behavior of certain things. It is a fundamental theory in physics that describes the behavior of matter and energy on the smallest scales.

  • Classical field theory is a generalization of classical mechanics that describes fields, which are functions of space and time. It is generally described by the Lagrangian density, which is a function of the fields and their derivatives.

  • Special relativity is a theory that describes the relationship between space and time. Starting from the postulates of special relativity, we can derive the Lorentz transformations, which describe how coordinates change under a change of reference frame.

These are then combined into relativistic field theory, relativistic quantum mechanics, and nonrelativistic quantum field theory, which are then finally combined into quantum field theory.

This will be a very long set of notes, starting from the foundations of all these theories and building up to the full theory of quantum field theory.