Sean Carroll in Cosmic Variance:
This year we give thanks for an idea that is central to our modern understanding of the forces of nature: gauge symmetry. (We’ve previously given thanks for the Standard Model Lagrangian, Hubble’s Law, the Spin-Statistics Theorem, conservation of momentum, effective field theory, andthe error bar.)
When you write a popular book, some of the biggest decisions you are faced with involve choosing which interesting but difficult concepts to tackle, and which to simply put aside. In The Particle at the End of the Universe, I faced this question when it came to the concept of gauge symmetries, and in particular their relationship to the forces of nature. It’s a simple relationship to summarize: the standard four “forces of nature” all arise directly from gauge symmetries. And the Higgs field is interesting because it serves to hide some of those symmetries from us. So in the end, recognizing that it’s a subtle topic and the discussion might prove unsatisfying, I bit the bullet and tried my best to explain why this kind of symmetry leads directly to what we think of as a force. Part of that involved explaining what a “connection” is in this context, which I’m not sure anyone has ever tried before in a popular book. And likely nobody ever will try again! (Corrections welcome in comments.)
Physicists and mathematicians define a “symmetry” as “a transformation we can do to a system that leaves its essential features unchanged.” A circle has a lot of symmetry, as we can rotate it around the middle by any angle, and after the rotation it remains the same circle. We can also reflect it around an axis down the middle. A square, by contrast, has some symmetry, but less — we can reflect it around the middle, or rotate by some number of 90-degree angles, but if we rotated it by an angle that wasn’t a multiple of 90 degrees we wouldn’t get the same square back. A random scribble doesn’t have any symmetry at all; anything we do to it will change its appearance.