Symmetry:

I have a habit of throwing around the term symmetry as if it is something that everyone knows. Well, perhaps everyone has an intuitive picture of what symmetry is but I thought it might be useful alert you to some of the various ways the term is employed in modern math and physics. In truth, I needed to take many courses before I really understood what symmetry is. I'm still learning.

Let me attempt to discuss some of what a symmetry is to mathematicians and physicists. Ordinary geometric symmetries are probably familar to you. Given some object there are certain ways you can rotate it and yet it looks the same. The set of all operations that leaves some object the same forms a symmetry group. I'm being a little vague here about what "same" means. To say they form a group means that if you take several operations and combine them you get another such operation. Essentially, a group is a set with an operation which is associative with inverses.

In practice there are many different kinds of groups. Finite groups have a finite number of objects in them. Continuous groups in contrast have an infinite number of objects. A smooth continuous group is a Lie group. From the modern perspective Sophius Lie actually worked with local Lie groups (late 19-th century) which are not even technically groups since you can take to operations and get another which is not in the same class. However, the terminology in physics is less rigid. Mathematicians mean a very specific technical thing when they lable some object a group. The physicist's notion of a group is more flexible. Especially when physicists speak of local symmetry groups, the mathematics of local symmetries are rather intricate and beautiful.

Local symmetry groups demand the introduction of a so-called compensating field in the langauge of minimal coupling in physics. This is what is known as gauge theory, or more precisely Yang-Mills-Utiyama theory. I usually leave off the Utiyama, but given that Utiyama's work was more general perhaps its a bad practice (by the way gauge theory is much more general than Utiyama's theory nowadays...). Anyway, the surprising thing is that during the time this gauge theory was developed in the physics community there was a mathematical theory of fiber bundles and connections developed concurrently and independently. My advisor realized they were the same in the early 1970's, but it was not widely known at the time. By the early 1980's modern gauge theory was lending results to the theory of fiber bundles. Physics was used to calculate topological results. This interplay is part of what makes both math and physics so exiciting in my view.

The Standard Model of particle physics is based on a gauge theory. The gauge group is the tensor product of several different groups. The different groups yield the various forces found within the Standard Model, namely strong, weak and electromagnetic. The complete story involves symmetry breaking and quantum mechanics, it takes a while to absorb all those details... but the basic point is this: modern physical theories are at their base centered on some symmetry principles.

In fact, you may here from time to time people speak of the four forces, this is just a force of habit, a historical homage to Newton. There are not really "forces" as much as there are symmetry principles. What is basic is symmetry and the ideas of quantum mechanics and energy.

That said, physics is just a model, so what is basic may deserve less credit than we are sometimes tempted bestow.

Do you understand what I meant in the last sentence? It took me some time as a physics student to get that through my skull.

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Last Modified: 7-16-08