I understand the concept of symmetry in geometry, but what do theoretical physicists mean when they speak of symmetry?
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I am seeing a lot of CPT Symmetry discussion here. And that is a great example but I figured I would hop in with another.
I want to discuss U(1)XSU(2)XSU(3) symmetries, because, in my opinion, that is where symmetries in physics really shine. In physics symmetries mean something very similar to in geometry, it is all about how changing one aspect of a system (or a shape in geometry) doesn’t change the overall system. A good example in geometry is a circle, which is rotationally invariant, it has rotational symmetry. You can spin a circle by any amount and it will still look the same.
In quantum physics fundamental particles are described as a waves. With a typical wave it will have ‘global phase shift invariance’ which is a fancy way of saying that if you shift a wave from left to right, or right to left, without changing the amplitude (height) of the wave or the wavelength (distance between peaks) then the particle being described wont change. Just like with the circle we have a symmetry, you can shift the whole wave back and forth all you want and it wont change the particle.
That much is pretty normal, but the fun idea started when they worked out what it would mean for a wave to have ‘*local* phase invariance’ this is where you shift each point on the wave to the left or right by *different amounts* this will completely destroy the wave, it will be all disconnected and a total mess.
But that is where things get really cool, I have no idea who figured this out or how, but they realized that you can add a term to the equation for your wave function that will offset the issues created by any local phase shift. This equation then describes a particle that is invariant to local phase shifts. And it turns out the term you need to add describes how an electrically charged particle self interacts via the electromagnetic force, all electrically charged particles will have this self interaction term, and that shows that *what it means for a particle to be electrically charged is that it has local phase invariance.*
Fundamental forces come from symmetries. What I described above is a simplified version of U(1) symmetry and it is reasonable for the electromagnetic force. SU(2) symmetry is another symmetry that is responsible for the weak force and SU(3) is the one for the strong force. In quantum physics fundamental forces are completely tied to and rise from symmetries.
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