(Written:- July, 2024).

The Need for Gauge Field Theories

Introduction

Let’s begin by trying to understand the necessity of studying a field theory approach to quantum mechanics and as the title suggests why we need gauge field theories. We will start off by understanding how gauge theory was formalized, invariance and symmetry and from there we will work our way up to how we can use gauge theory to construct strong and weak interactions (the strong force and the weak force). One thing that needs to be considered while studying field theories , we are trying to understand processes that occur at very small i.e. (quantum-mechanical) scales and very large (relativistic) energies. One might ask why we must study the quantization of fields. Just like the classical formalism approach,~ why can’t we just quantize relativistic particles the way we quantized non relativistic particles? . The answer to that question is simple. When we try to quantize relativistic particles at high energies and shorter distances, the wave equation breaks down. In other words, the Schrodinger equation works for a free particle , but when you try to add relativity to it, it breaks down. We also run into other problems like negative energy solutions, a disorder in first and second order derivatives, negative probability densities etc.~ Therefore, a totally new framework is needed to deal with particles (fixed number of particles or otherwise) at relativistic levels. This is how quantum field theory was formalized and As we have been hearing since high school, particles are just field excitations acting as operators on a 2D Hilbert space. I am assuming that you as the reader are familiar with concepts like principle of least action, four vectors, Lagrangian and Hamiltonian formalism. If not, check out any MIT OCW playlist.

For the full write-up, download the pdf above.