Introduction to Field Theory
Prerequisites Required- Basic Mathematics, Standard Vectors
Today's topic is fields.
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:
Figure 2: Upper Left- A circle is invariant to any rotation. | Upper Center- A square is invariant to 90 degree rotations. | Upper Right- Pentagons are invariant to 72 degree rotations. | Lower Left- This droplet is invariant to a flip about its horizontal center | Lower Center- Solid colors extending out to infinity are invariant to rotation and translation | Lower Right- A square is invariant to two sequential 45 degree rotations, irrespective of the order or direction they are applied.
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.
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.