2.2. Symmetry and Unification Review

summary

Symmetry is a unifying principle of mathematics.

The key concept that permeates all of the seemingly esoteric and abstract domains of mathematics is the idea of symmetry.
We can experience symmetry in our daily lives—in snowflakes, butterflies, the human body, for instance.
What makes something symmetrical? Can something be more symmetrical than something else?

What does it mean to be symmetrical?

Let’s consider two glasses, one rounded and one squared. Which one is more symmetrical? There is in fact a clear answer, but we have to understand what it means to be symmetrical.
If we rotate the round glass about its central vertical axis, any rotation preserves the appearance of the glass. However, with the square glass, only rotations of 90°, 180°, 270°, or 360° will preserve the appearance.
Transformations that preserve the object—its position and shape—are called symmetries.
The round glass has infinitely many symmetries, whereas the square glass only has four rotational symmetries. Thus we can say that the round glass is more symmetrical.

Groups of symmetries are mathematical objects.

Let’s look at all the symmetries of a given object. For the round glass, each symmetry corresponds to a rotation by a particular angle. So the set of all symmetries of this glass is the set of all angles. We think of this abstract set as a geometric object—the circle.
Similarly, the set of all symmetries of the square glass are the four points equally spaced 90° apart—plotting this set manifests the shape of a diamond.
We have arrived at the concept of a group of symmetries. A group is a set of elements with an operation (in our case, the addition of angles) that satisfy some natural properties.
For example, the circle is a group. The set of four rotations {90°, 180°, 270°, 360°} is also a group.
We can apply transformations back-to-back. If we rotate the round glass by 30°, then again by 20° more, the result is a rotation by 50°. Any two rotations give rise to a third one.

Now consider more complex rotations.

Imagine a sphere that can be freely rotated. In the case of a sphere, there are many more possible transformations that preserve its shape and position.
Consider the globe. We can rotate it by any angle about the axis connecting the north and south poles. But of course, we can also choose any other axis of rotation through the center of the globe, and all rotations about this axis are also symmetries.
This symmetry group is known as SO(3), which stands for special orthogonal group—responsible for rotations of a sphere in 3-dimensional Euclidean space.

Symmetries are also paramount to quantum physics.

By the 1960s, physicists were discovering many dozens of elementary particles called hadrons. Wolfgang Pauli joked that physics was turning into botany. They desperately needed an explanation of this rapid proliferation of particles.
The answer came in the form of a classification theme that used group theory. It turns out that these elementary particles like protons and neutrons are actually made up of smaller particles known as quarks.
There is a symmetry group behind this classification of hadrons, known as SU(3) (special unitary group). SU(3) is similar to SO(3), using complex numbers instead of real numbers.
Octets (groups of eight) and decuplets (groups of 10) are “representations” of this group, and quarks are the building blocks of these representations.
The mathematical concept of symmetry groups helps to explain the behavior of elementary particles and the interactions between them.