The Beauty of Group Theory
For those of us working in quantum mechanics group theory is an indispensable tool. Group theory has an enormous range of application, however. In essence it is the language of symmetry. While to physicists it is thus obvious as to why it is important for the study of most modern physics topics, one may ask what its relation to visual symmetry might be. I am nearly finished reading Mario Livio’s fantastic book
The Equation That Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry. In it he gives a terrific introduction to groups via its relation to visual symmetry!
So what is a group? Groups are actually related to sets, the fundamental building blocks of modern mathematics. Many people mistakenly assume that the elements (members) of sets are always numbers. In fact anything can be an element of a set. A set simply is a way to categorize items. A group is a set whose elements may be combined by some operation. The elements and the operation have certain properties that may essentially be summarized as
- Any combination of elements by the operation produces another element of the same set. This is called closure. For example, in the set of integers with the operation of addition, adding any two integers produces another integer.
- When combining three or more elements of the set using the operation, it does not matter how the operations are combined. For example, (3+5)+2=3+(5+2)=10. This is called associativity.
- The set contains one element (called the identity element) that, when combined with any other by the operation, leaves that other element unchanged. For example, working with the natural numbers (1,2,3…), if the operation is addition the identity element is 0. If the operation is multiplication the identity element is 1.
- Each element has an inverse that is also a member of the set. When you combine anything with its inverse you get the identity element. If your operation is multiplication, this implies that their product is 1. If it is addition, their sum is zero.
This last point is a crucial one since, from this, we can see that if the operation is multiplication the set of all natural numbers is not a group! Why? Take the number 6 for example. Its inverse is 1/6. But 1/6 is not a natural number! Now, there are ways around this. One way is to employ modular arithmetic, but there is not time or space to explain that here. (Note: if the operation is multiplication, then the set of all natural numbers, that is all positive integers, is still not a group since the inverse of 6, for example, would be -6).
Now if we’re trying to relate groups (which we have just defined) to symmetry, it would be helpful to have some idea of what symmetry is. In physics and mathematics, any time we perform an operation on something and it remains unchanged, our operation represents a symmetry. So, for example, the laws of physics should look the same regardless of how we look at them or where we look at them from (i.e. they should be the same to us as they are to people living in a different galaxy). This is the Principle of Relativity and forms the conceptual foundation of both special and general relativity theory. In nuclear physics we see symmetries in the types of particles that exist – each type seems to have a similar partner. So another way to look at symmetry is as a representation of balance.
How does this apply to visual symmetries though? Take the human body – in fact just about any animal or insect body – for example. It exhibits bilateral symmetry, sometimes referred to as mirror symmetry since one side is (roughly) the mirror image of the other. This objects that have this symmetry remain unchanged under two symmetry transformations – reflections (we’ll call this r) and the identity (meaning doing nothing and we’ll call this I). Does this set of operations constitute a group? Indeed it does! Let’s see why.
Our operation will simply be the combination of transformations (so it’s like addition). For example, we could write I º r where the º symbol just means we applied the I transformation first, then the r. Let’s make sure these transformations and the operation satisfy the four conditions of a group.
- First, let’s check to make sure that any combination gives another element of the group. So if we do nothing (I) and then do nothing (I) we’ve done nothing (I). In other words IºI=I. Now suppose we do nothing (I) then reflect one half of a creature’s body (r). That’s equivalent to just the reflection alone. In other words Iºr=r. So far so good. What happens if we reflect half the body and then reflect that same half again? It ends up back where it started which is just like doing nothing! Thus rºr=I. It seems as if any combination of I and r produces either I or r again which is exactly what we want.
- Can we combine them in any order? Sure! Does it matter whether we first do nothing (I) and then reflect (r) or reflect (r) first and then do nothing (I)? Nope! (Think about it.)
- Is there an identity element? Yep! It’s I (doing nothing).
- Does each element have an inverse? This might seem a bit trickier, but it isn’t. The definition says that combining an operation with its inverse should give the identity element. In our case, taking any element twice in a row does the trick, i.e. rºr=I and IºI=I.
Thus the symmetry transformations of animal and insect bodies form a group! In the process of understanding this we catch a glimpse of the beauty and elegance of mathematics. As Einstein said in one of my favorite quotes, “Pure mathematics is the poetry of logical ideas.”