From Particles to Fields

In the previous section, we discussed the Lagrangian formalism and how it can be used to describe the dynamics of a system of particles. In this section, we will discuss a fundamental shift in perspective: from particles to fields. We will apply it to an example of the electromagnetic field, from which we will derive the tensor form of Maxwell's equations.

Table of Contents

• Table of Contents
• Introduction
• Generalizing Electromagnetism

Introduction

In classical mechanics, we describe the dynamics of a system by specifying the positions and momenta of particles. For example, the Sun-Earth system can be described by the positions and momenta of the Sun and the Earth. Then, we introduce forces that act on these particles, such as the gravitational force between the Sun and the Earth:

Likewise, for two charges and , the electrostatic force between them is given by Coulomb's law:

In this description, the forces act instantaneously across space. Any interaction between two objects only requires that the objects are there, and that there is empty space between them. This is known as action-at-a-distance.

However, this is not how the universe works. There is a fundamental principle in physics known as locality, which states that interactions between objects can only occur at the same point in space and time. This prevents instantaneous transmission of information across space, as it would violate the principles of special relativity.

To see why action-at-a-distance is problematic, consider two events and that are spacelike separated (i.e., light cannot travel between them). If event affects event , then there must be a causal relationship between them. If causes , then we would hope that always precedes in all reference frames. However, if the events are spacelike separated, there are reference frames in which precedes . This leads to a contradiction, as causality would be violated.

As a step towards resolving this issue, we introduce the concept of field theory. Instead of describing the dynamics of individual particles, we describe the dynamics of fields, which are functions of space and time.

For example, in the gravitational case, instead of describing the force between two particles, we describe the gravitational field at every point in space and time:

The field tells us how a test mass would move at the point at time . Similarly, in the electromagnetic case, we describe the electric and magnetic fields and at every point in space and time.

These field vectors are functions of space and time and describe the forces that act on charged particles. They can be further generalized as potentials, which are scalar and vector fields that describe the forces acting on particles. For instance, the electric and magnetic fields and can be derived from the electric potential and the magnetic vector potential :

As a consequence of locality, fields can evolve even in the absence of particles - think of electromagnetic waves propagating through empty space. Even when and , the electromagnetic fields and can still evolve according to Maxwell's equations.

Generalizing Electromagnetism

In the familiar formulation of electromagnetism, we describe the electric and magnetic fields and in terms of the charge density and the current density . The equations that govern the evolution of the fields are Maxwell's equations. In SI units, they are given by: