Quick Intro: Group Theory

A group is an abstract description of how a system can transform.

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Now and again in higher-level Physics you might run into terms like this: SO(3), Lie Group, U(2), etc… These sound like strange and disconnected terms but they are all part of how we categorize systems into groups.

What is a group you might ask? A group is an abstract description of how that system can transform. Specifically, a group consists of a set of elements that create our group space. With this group there is an operator that allows us to combine any two elements into a third element. The operator is usually represented by a plus + symbol so that if A, B and C are group elements, a statement like A + B = C makes sense. You might notice that this looks a lot like simple algebra, and groups do form their own algebra in a sense, but they aren’t limited to the math operators that we learned in primary school.

Since a group is completely abstract, it can have different ‘representations’ that express the group. A cyclic group can be represented by the way we rotate an object, so that if you add two rotations, you might end up rotating the object 360 degrees and returning to the original position. Another example of a group representation would be how we can combine simple twists of string to form complex braids (the group in this case is called the braid group).

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One example of a group is the ways that we can flip a mattress. This helpful article explores group theory in that scenario: https://opinionator.blogs.nytimes.com/2010/05/02/group-think/

While there are infinite representations of groups, we try to reduce each group to it’s simplest form, known as the irreducible representation. This allows us to compare different systems in terms of their groups.

Breaking systems down into their fundamental groups allows us to predict transitions that appear in the group before we can even observe them. Group theory can also be used to understand symmetry, this perspective can not only be applied to the physical sciences, but the study of art.

As far as physics goes, here are a few common groups that come up a lot:

SO(3): The special orthogonal group in three dimensions. This group consists of simple rotations in 3 dimension space around a fixed point.

U(2): The unitary group of degree 2. This group consists of all 2×2 matrices that are unitary.

SU(2): The special unitary group of degree 2. Like the unitary group, but the determinant of each matrix is equal to 1.

There are a few ways of discussing different types of groups. Many groups are characterized by some kind of symmetry, which is defined by a parameter of the system that doesn’t change during a transformation. For example matrices in the special unitary group always have a determinant equal to 1, or the special orthogonal group doesn’t change the length of a given radius as it rotates.

A Lie group is any group that is defined by a continuous transformation. A particle might have discrete transformation between one energy state of another, but rotations can be made continuously.

There are so many different ways of looking at groups. Even if it doesn’t become your field of study, it can make for a very interesting way of looking at different systems. Here are a few sources where you can learn a little bit more about groups:

PDF/BOOK: Introduction to Group Theory for Physicists – Marina von Steinkirch
This book breaks gives us the mathematical formulation of group theory and looks at a number of groups that have special significance for Physics.
http://www.astro.sunysb.edu/steinkirch/books/group.pdf

VIDEO: Group Theory – Robert de Mello Koch
The first in a series of lectures that cover the study of group theory and representations. A great place to start if you have some experience with Quantum Mechanics.
https://www.youtube.com/watch?v=3wNPrSwbtQ8

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