Introduction to Field Theory

Introduction to Field Theory
Image Courtesy of DeltaWorks via Pixabay

Prerequisites Required- Basic Mathematics, Standard Vectors

Today's topic is fields.

Figure 1: Field landscape, courtesy of Matthew Smith via Unsplash

Nope, not that kind of field; mathematical fields. Fields are actually very basic. A field consists of a property that is mapped over space (in one or more dimensions).

A Note on Symmetry and Invariance

The concept of symmetry is very useful in studying and classifying fields. To be symmetric means that if you transform an object, it is indistinguishable from the untransformed version of that same object. To be symmetric with respect to a transformation is to also be invariant to that transformation.

Below are several examples:

Uniform Scalar Fields

Below is an example of a field. The axes, shown in white aren't part of the field, they are just there to help us assign spacial coordinates to parts of the field. This is called a color field because each location in 2D space is assigned a color.

Figure 3: Depiction of uniform scalar field

The field above is an example of a uniform field. It is blue everywhere. Notably, this is also a scalar field because the property color is a directionless quantity. This kind of field is appropriately called a uniform scalar field.