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Introduction

Classical field theory is a generalization of classical mechanics that describes fields, which are functions of space and time.

In "regular" classical mechanics, we have particles moving around in space, characterized by their positions and momentums. For example, the Sun-Earth system can be described by the positions and momentums of the Sun and the Earth. Then, we introduce various forces that act on these particles. In the Sun-Earth system, the gravitational force acts on both the Sun and the Earth. It is equal to , i.e. inversely proportional to the square of the distance between the two objects.

Suppose that the Sun mysteriously disappears at some point in time . Using what we know so far, we would say that at exactly , the force would vanish, and the Earth will start moving in a straight line (because there are no external forces left). However, the Sun and Earth have a distance of , and light takes minutes to travel that distance. From special relativity, we know that information cannot travel faster than the speed of light, or else there would be reference frames in which the information is received before it is sent. But in this case, the Earth would start moving in a straight line before the information that the Sun disappeared reaches it. Hence, there is an inherent contradiction, and something is wrong with our current description of the system.

What I just described is known as action-at-a-distance, where the force acts instantaneously across space. Any interaction between two objects only requires that objects are there, and that there is empty space between them. Mutual forces are instantaneously transmitted across space, regardless of the distance between the objects.

This is not how the universe works.

In order to fix this, we introduce the concept of field theory. It comes with the restriction of locality, which states that interactions between objects can only occur at the same point in space and time. For instance, when we apply locality and special relativity to gravity, we get general relativity.

This section is extremely important—despite the fact that it is called "classical" field theory, it is one of the most important parts of physics. Although the reader may already be familiar with classical mechanics, it is crucial to understand the foundations of classical field theory, as it is the basis for quantum field theory. Furthermore, many of the concepts introduced in this section will likely be at a more advanced level than those in an introductory mechanics course.

Scientific pedagogy works this way—we build upon the same foundations, but we go deeper and deeper into the subject matter. For instance, in an introductory course, we might learn about the conservation of energy, that energy is conserved in isolated systems, and that it can be transformed from one form to another. This works well for simple systems, such as a pendulum or a spring, and helps explain various phenomena in daily situations. In a more advanced course, we might learn about the conservation of energy-momentum in relativistic systems with the energy-momentum tensor , which is a more general concept that encompasses both energy and momentum. Then, we might learn about how the conservation of energy arises from a symmetry of the action under time translations, as described by Noether's theorem. To take another example, we might first learn about electromagnetism qualitatively, that opposite charges attract and like charges repel, and that electric fields are created by charges. Then, we might learn about the mathematical formulation of electromagnetism, such as Coulomb's law, Gauss's law, and the Lorentz force law. Finally, we might learn about how electromagnetism is a gauge theory, how conservation of electric charge arises from gauge invariance, and further concepts in quantum electrodynamics (QED). As one can see, the same concepts are revisited, but they are explored in greater depth and with more mathematical rigor.

Overview​

In this set of notes, we will explore classical field theory. We will start by a brief overview of classical mechanics and the Lagrangian formalism, which are the basis of classical field theory. When whifting to fields, we will deal with densities rather than "point" quantities - for example, we will introduce concepts like energy density and the Lagrangian density. The Euler-Langrange equations will be generalized to be able to deal with fields and densities.

A specific "field theory" is defined by its Lagrangian density. Hence, we will explore a few examples of field theories, such as the following, based on the spin of the field:

  • Spin-0 (scalar) - Klein-Gordon Equation:

  • Spin-1/2 (spinor) - Dirac Equation:

  • Spin-1 (vector) - Proca Equation:

  • Spin-1 (vector, massless) - Maxwell's Equations:

  • Spin-2 (tensor) - Einstein's Equations:

We will go through a deep dive into several of these equations. One such example is Maxwell's equations, which describe the electromagnetic field. We will see how electromagnetic theory is an example of a gauge theory. Specifically, it is a gauge theory, which means that the Lagrangian density is invariant under local phase transformations of the field. We will also see how the gauge invariance leads to the conservation of electric charge, and how it is related to the concept of gauge symmetry. This will be extremely important for quantum field theory.