Why are symmetries important in physics? Here’s a heuristic answer. We wish our physical theories to be such that nothing of importance depends on an arbitrary choice, say the choice of a coordinate system. Let me make this a bit more precise: Since trivially, if I place the origin in the centre of my apartment I will measure different velocities, distances, angles, and so on, than I would if I place the origin at the front-end of a train in uniform motion. But the actual physics of the situation should not be affected by such arbitrary decisions. In such (classical) inertial frames it appears that accelerations and masses, and thus forces are independent of the choice.
Let me flesh this out a bit more. In classical physics all ‘physical’ magnitudes should remain invariant under certain coordinate transformations. The choice of where to place the origin should be free, the orientation of the axes should be free, and the coordinate frame should be able to either be stationary (compared to some other system) or to move at a constant velocity. Also one should be free to choose which moment of time to mark as t = 0. From these considerations a lot follows. From translational invariance (by Noether’s theorem) both the conservation of momentum (translational invariance with respect to space) and the conservation of energy (translational invariance with respect to time) follow. From rotational invariance follows the conservation of angular momentum. So far so good.
There’s a caveat, though. It is an empirical discovery that we can do such reasoning from symmetries. Had something like Aristotelian physics been correct, all these symmetries would not have helped us in our physical reasoning. This brings out the following point: what if there are symmetries that we think that fundamental physical theories should obey, but have no empirical means of seeing whether such symmetries in fact lead to correct theories, conservation laws and so on? This, I submit, is the case with some of the more exotic symmetries the physics community has brought forward. Take the case of “supersymmetry”. Supersymmetry is, roughly, the idea that all the particles of the standard model of particle physics have “superpartners”. For fermions such superpartners would be bosons, and vice versa. However, at the moment the only reasons to assume supersymmetry are purely theoretical.
So, should we take supersymmetry seriously as physics? Simply put, I do not know. There is, however, a deep problem within theoretical physics: the ways to test theories at the cutting edge of physics seem to be missing. To see this more clearly, I direct the reader to Sabine Hossenfelder’s wonderful book Lost in Math: How Beauty Leads Physics Astray (2018). In it Hossenfelder, a professional physicist, takes a deep and at times uncomfortable look at her own field. Her verdict: something has gone wrong with theoretical physics – mathematics has blinded (some) practitioners of the field. But do not take my word for it; read the book!
Hossenfelder, Sabine (2018): Lost in Math: How Beauty Leads Physics Astray, New York: Basic Books.