Densities

One of the most important concepts in field theory is the shift from particles to fields. Part of this shift is the shift from quantities like energy, momentum, the Lagrangian, etc, to their densities. This is a very important concept in field theory, and it is worth spending some time on it.

Table of Contents

• Table of Contents
• Introduction

Introduction

One way to introduce a field theory from a particle theory is to create a force field. For example, a common field everyone is familiar with is the electric field. The fundamental concept is that a force depends on two particles—their positions and their charges. In this case, we have . On the other hand, if we fix one of the particles, we can think of the force as a function of the other particle's position. This leads to the electric field, defined such that .

We can further generalize a force field to its potential. Unlike the electric field, which is a vector field, its associated electric potential is a scalar field. The electric potential is defined such that . The reason that a single scalar can encapsulate all information in the electric field is that the electric field is conservative. This incurs many constraints on the electric field, which leads it to have a lower amount of information.