The notes of a lecture on basic group and representation theory I attended last semester begin with a bit of motivation for the argument. They give the following examples for applications in physics:
Unfortunately, we were didn't get to treat any of the examples above. I am really interested in this kind of stuff, though, and I was wondering if there is some book or other reference with a purely group representation theoretical of similar topics in physics. Mainly I am interested in the following examples:
- How is the shape of the Maxwell and Einstein equations dictated by symmetries?
- What information can be found about the hydrogen atom by looking at its symmetries?
- I have been told that spin also arises from some symmetry. How so?
Thank you in advance for any directions you can give me.
Best Answer
I'll begin with Maxwell's Equations. I would say that the shape of Maxwell's equations demand Minkowski geometry where the isometries include Lorentz transformations. Before the advent of relativity, Maxwell's equations were already what we now term "relativistic". So, I'm not sure I agree with your question historically. That said, if we accepted that space and time was prescribed in the Newtonian sense (absolute time, and euclidean stand-alone space) then we have to either discard Maxwell's equations or make bizarre adhoc modifications of classical mechanics. It's an older text, but Resnick is a nice read to see explicitly what the Lorentz transformations are and also to see explicitly how Maxwell's equations are covariant. His proof is explicit at the level of PDEs, which is nice for some students. Others will be more pleased by something like Misner Thorne and Wheeler where the equation is recast in terms of differential forms to break free of the coordinates.
The possible wavefunctions of the hydrogen atom are indicated by it's symmetries. Wavefunctions provide representations of that symmetry group. One interesting idea, if left alone, electrons tend to stay in inequivalent representations of the symmetry group. The Hamiltonian matrix is block-diagonal and no mixing occurs between energy eigen states. If a pertubation occurs then coupling terms appear which join different energy eigen states. From a symmetry perspective, the departure from the perfect symmetry opens new interactions before not possible. I recommend Greiner's text on Lie groups and representation theory in physics. This is certainly not a math book, but it helps answer some of the questions you raise. I found it more readable than other books with similar goals. For gory details on spectroscopy and so forth the classic by Hammermesch still finds a home on the bookshelf of many aspiring theorists who wish to master rep-theory for Atomic and Molecular physics. On the math side of things, the text Symmetry Groups and their Applications by Willard Miller is worth a look.
Spin is a particular number which is associated with matter. It labels how the particle interacts with the magnetic field. Mathematically, it has do with the double-cover of the Lorentz group and the concept of a "spin-bundle" has considerable generalization over the usage of the term in standard physics. Typical examples, electrons have spin 1/2 whereas photons have spin 1. The graviton has spin 2. Spin, like mass or charge of a particle is an intrinsic property. It's part of what defines the identity of a given elementary particle. In short, spin labels the type of spinorial representation of the Lorentz group.
To see how physicists think about it perhaps look at the text by Wu Ki Tung. There is a three-volume set by Cornwell, I suspect there is a reasonable deep explanation in those, but sadly I only own a poor copy of one at the present. Bleeker's Gauge Theory and Variational Principles discusses spin and Lorentz groups, and it's a Dover now. To see a detailed construction of spinors from other algebra there is a text by Naber which appears quite accessible if you have some time. Personally, I haven't worked it out, but it might be of interest. See Geometry of Minkowski Spacetime