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  • Print publication year: 2020
  • Online publication date: May 2020

Chapter 3 - Quantization of Free Particle Fields

Summary

Introduction

The concept of associating particles with fields originated during the study of various physical phenomena involving electromagnetic radiation. For example, the observations and theoretical explanations of the black body radiation by Planck, the photoelectric effect by Einstein, and the scattering of a photon off an electron by Compton established that electromagnetic radiation can be described in terms of “discrete quanta of energy” called photon, identified as a massless particle of spin 1. Consequently, Maxwell's equations of classical electrodynamics, describing the time evolution of the electric and magnetic fields are interpreted to be the equations of motion of the photon, written in terms of the massless spin 1 electromagnetic field. Later, the quantization of the electromagnetic field was formulated to explain the emission and absorption of radiation in terms of the creation and annihilation of photons during the interaction of the electromagnetic field with the physical systems. The concept of treating photons as quanta of the electromagnetic fields was successful in explaining the physical phenomena induced by the electromagnetic interactions; methods of field quantization were used leading to quantum electrodynamics (QED), the quantum field theory of electromagnetic interactions. The concept was later generalized by Fermi [23, 207] and Yukawa [208, 209] to formulate, respectively, the theory of weak and strong interactions in analogy with the theory of QED.

In order to describe QED, the quantum field theory of electromagnetic fields and their interaction with matter, in terms of the massless spin 1 fields corresponding to photons, the equations of motion of should be fully relativistic. This requires the reformulation of classical equations of motion for the fields to obtain the quantum equations of motion for the fields and find their solutions, in case of free fields as well as fields interacting with matter. This is generally done using perturbation theory for which a relativistically covariant perturbation theory is required.

The path of transition from a classical description of fields to a quantum description of fields, requiring the quantization of fields, their equations of motion, propagation, and interaction with matter involves understanding many new concepts and mathematical methods. For this purpose, the Lagrangian formulation for describing the dynamics of particles and their interaction with the fields is found to be suitable.