The word "field" is sexy in modern physics partly because it appears in the term quantum field theory, and quantum field theories form the basis for our current understanding of particle physics down to incredibly small length scales. But fields have been useful in physics for well over a century. For example, one of the most successful physical frameworks, that of classical electrodynamics, is a rather old field theory, and is not a quantum field theory.

In this post, I'd like to take some time to explain the term "field" as it is used in physics and many areas of mathematics. In later posts, I hope to specialize to particular kinds of fields, such as vector, spinor, and tensor fields, and focus on the relationships between field theory and group theory. Before we give the general definition of the term "field" as it will be used on my blog, let's look at an example of a field in physics to motivate the more general definition.

Since you are alive and reading this post, you are most probably breathing air, and in fact, there is a bunch of air all around you. Imagine you have a machine that can measure the temperature of this air at any given point in space, and output all of this information in the form of a function that assigns to each point in space, the temperature of the air at that point. We would call this function a scalar field because it assigns a scalar, in this case a real number, to every point in space. The essential property here is that the field is a function that assigns some value to each point in a space.

In the general definition of a field, we would like to generalize the notion of assigning a certain kind of object to each point in some space, and we would like the definition to be general enough that it will describe most of the fields we care to consider in physical models.

The scalar field in the temperature example assigns a scalar to each point in three-dimensional Euclidean space, but in physics, one considers fields defined on all sorts of spaces that aren't necessarily three-dimensional. For most purposes, especially when it comes to field theory, the spaces on which fields in physics are defined are manifolds. Hence, in our general definition of "field," we will require that a field be a function that assigns some value to each point on a manifold.

However, when we say that the field is assigning values to points on a space, we don't want to restrict ourselves to scalar fields, those that assign numbers, such as real or complex numbers, to points in a space. In fact, we might as well keep things completely general and allow the field, at this stage, to assign any mathematical object whatsoever to each point in the space under consideration. This leads to the following general definition for the term "field:"

A field is a function where is a manifold, and is a set. The manifold is called the field's domain, and the set is called the field's target set.

Let's give some examples to gain some intuition for the definition and to examine the extent to which it can accommodate some pretty exotic mathematical objects that arise in physics

Example 1. The temperature field in a room.

This is the example that was already given to motivate the general definition of a field, but let's put it in slightly more precise terms to make contact with the terminology in the definition. In particular, what are the manifold and the target set in this case? Since the temperature function assigns a real number to each point in the room, the manifold must model the space occupied by the room, and the set must model the values that temperature can take on. If we take the room to have length , width , and height , if we call one corner of the room the origin , and if we orient cartesian axes emanating from the origin that correspond to the intersections of the walls, then we can model the space in the room as the manifold . If we take temperature to be measured in Kelvin, then temperature can be modeled as a real number greater than or equal to , so the target set in this case is . This can all be summarized by saying that temperature in a room is a function such that

(1)  

Since the region is a three-dimensional manifold, the temperature function satisfies our general definition of the term field.

Example 2. Velocity field on a lake.

Consider a circular lake with radius . We can label the position of each point on the surface by a point in the Euclidean plane . If we choose coordinate axes such that the center of the lake is at the origin , then the region in the plane corresponding to the surface of the lake consists of the the set of all points whose distance from the origin is less than or equal to . Let's call this set where the "" stands for "disk" since the surface of the lake looks like a disk.

At each point on the lake, the water at that point will be moving with some velocity which is itself a two-dimensional vector or, more mathematically, an element of . The function that specifies the velocity of the water at each point on the lake's surface is called a vector field because it a assigns a vector value to each point in the domain ;

   

Since is two-dimensional manifold (with boundary), this function satisfies our general definition of the term field.

Example 3. Gauge field on Minkowski space.

In classical electrodynamics, the electric and magnetic fields can be written in terms of a single "gauge field" (a term I'll define in a later post) called the vector potential. This field takes a point in four-dimensional Minkowski spacetime , and maps that point to a four-component vector also in Minkowski space.

(2)  

where I have used the shorthand . Assuming that the region over which the vector potential is defined is a very large subset of spacetime, we can just model this region as all of spacetime. Mathematically then, the vector potential is a function

(3)  

The domain of the vector potential is a (pseudo-Riemannian) manifold, so it satisfies our general definition of the term field.

In later posts, we'll discuss more exotic kinds of fields and their significance in physics. In particular, my central focus will be to explain what physicists mean when they say that

"a field transforms in such and such way"

This is a phrase that often causes students confusion, and the concept of field transformation is a hugely important concept in physics because it allows one to discuss symmetry in classical and quantum field theory in a systematic, rigorous way, and symmetry is one of the central guiding properties behind constructing physically relevant theories.