What is Symmetry

Here is a quick, painless introduction to the kind of symmetry that you probably learned in school. In that lesson, we learn that there are different kinds of symmetry:

• Rotational
• Translational
• Mirror

and so on. What this video (and your teacher) did not explain was what symmetry is in general. If you learned about symmetry in University Math, your idea was expanded a bit to come closer to what symmetry is in general. The first step is to think of symmetry as a set of transformations (rotate, move, flip) rather than a property of whatever it is that is being transformed.

And that's just the start. For example, in physics, you learned the Lorentz Transformation is a symmetry group in the "real world" that leaves the laws of physics unchanged. Newton's laws are not symmetric in this sense but with a bit of tweaking (Special Relativity) they can be fixed.

In physics, the Lorentz transformation (or transformations) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other.

Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating in curved paths, rotational motion with constant angular velocity, etc.). The term "Lorentz transformations" only refers to transformations between inertial frames, usually in the context of special relativity.

Special relativity boils down to "fixing" the laws of physics so that they apply "locally" in non-inertial frames of reference.  I promise: no more about General Relativity.

Lorentz transformations are a special case of Gauge Symmetry, which is a concept used in the "Core Theory" of Quantum Mechanics and gets us finally to the point where we are talking about "real" symmetry in the "real" world. You need it to understand quarks for example.

This can all get pretty abstract and make your head hurt if you follow it all the way to Quantum Mechanics. It's best to stand back and ask what symmetry is. Wilczek describes it briefly as "change without change".  Or perhaps we could say you can change one thing that doesn't matter and everything that does matter is still the same.

In this blog we will be interested in a few kids of symmetry that aren't covered in school and are perhaps just a bit more general than even Gauge Symmetry. These involve transformations between domains I, M and R and within those domains or within sub-domains. For example the relationship between two ideas that are "like" each other (within the M-domain) is seen as a symmetry transformation. Some transformations are "better" than others and the criterion is symmetry. Does the transformation leave the idea the "same" in important aspects. This is similar to Hofstadter's idea of the "Essence" that is preserved in a "good" analogy.

Human beings seem to love all kinds of symmetry for good reason - they help economize on "representation" of "things" in the brain. Wylczek draws an interesting parallel with data compression. The Core Theory of Quantum Mechanics, which more or less represents everything we know about the physical world, can be written down in a few dozen strange looking characters, yet everything in human experience is a special case of it. Wylcze's treatment of this issue inspires this whole blog since he wrote a whole book about it: "A Beautiful Question" which basically points out the astonishing symmetry between our ideas (especially the Core Theory) and the real world. The equations of the Core Theory are incredibly symmetric and the real world seems to "like" symmetry a lot too. The "Beautiful Question" is a response to this situation. You are left with astonishment and wonder: a question rather than a "theory".