Hermann Weyl

Hermann Weyl (1885-1955)

Herman Weyl was born at a German town near Hamburg and studied at Goettingen. His Advisor was David Hilbert. He spent time together with Einstein in Zurich, before returning to Geottingen to become Hilbert's successor in 1930. He then came to the Institute for Advanced Study in Princeton and stayed there until 1951. He spent his rerirement years in Zurich until he died in 1955.

It is not clear how much interaction he had with Einstein while in Princeton, but he learned relativity from Einstein while they were together in Zurich before they immigrated to the United States. Hermann Weyl was the first one to come up with the idea of gravitation as a gauge field like electromagnetic field.

Eugene Wigner, a long-time Princetonian, told me he did not like Weyl personally. I did not ask and he did not explain his reasons for disliking Weyl. Like Einstein, he was an isolated person in Princeton.

This does not prevent from learning lessons from Hermann Weyl. If you like to be regarded as a respected theoretical physicist, you must have the following three books on your bookshelf. I choose to show my Weyl collection using this webpage.
Since I have these three books, I am qualified to talk about Weyl's mathemtical approach to physics.
In his book on Classical Groups, Weyl introduced the word "Symplectic Group," but he did not give too many physical examples. Guilemmin and Sternberg made a laudable effort to teach physicists this group. Arnold attempted to reformulate classical mechanics within the framework of the symplectic group.
However, most of us do not know we are doing symplectic stuff while doing physics. Eugene Wigner once asked me whether I could explain what symplectic group is, but I could not at that time, even though we both were intensely interested in the Lorentz groups. Here are the reasons.
1. The Lorentz group applicable to two space and one time dimensions, or O(2,1) is isomophic to the symplectic group Sp(2), the group of two-by-two matrices with real elements and with unit determinants.
2. However, the Lorentz group applicable to the four-dimensional Minkowskian sapce (with one time-like and three space-like dimensions) is not a symplectic group. I was confused on this issue and published one misleading paper.
3. The symplectic group Sp(4) is isomorphic to the deSitter group O(3,2) with three space-like and two time-like dimensions, as P.A.M Dirac observed in 1963. This group later became the basic language for two-mode squeezed states.
While working with Wigner in his late years (1985-90), I was not able to explain these to him. When he was younger, Wigner disliked Weyl and was never interested what Weyl said in his books, even though they were neighbors in Princeton from 1933 to 1951.
We cannot do physics without two-by-two matrices, and the simplest two-by-two matrix contains four real elements. For convenience, we can set its determinant to be one. Then this matrix contains three independent parameters. We are using matrices of this kind everyday. Did you know you are doing the symplectic group Sp(2)?
In Optics, this two-by-two matrix is called the [ABCD] matrix. It serves primarily as a beam transfer matrix. In other branches of physics it serves as the one-dimenensional scattering matrix. All four elements are real, and its determinant is one. Thus it has three independent parameters.
This matrix can be brought to one of the three equi-diagonal matrices by a rotation. They then have two independent parameters, and take three different forms depending on their traces.
1. You can do this mathematical operation without knowing any physics. Indeed, this operation is called the construction of representation according to conjugate classes.
2. Did you know that this fundamental classification in mathematics corresponds to Wigner's classification of internal space-time symmetries? On this subject, my recent publication include
  - One anlytic form for four branches of the ABCD matrix,
    J. Mod. Opt. [57], 1251-1259 (2010). -- ArXiv
  - ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics
    J. Opt. Soc. Am. A [26], 3049-2054 (2009). -- ArXiv
    It is a pleasure to tell you I wrote these papers with Sibel Baskal.
3. Why in Optics journals? Those Wigner matrices are more useful in optical sciences. They allow us to handle optical periodic systems, including laser cavities, lense optics, multi-layer optics, and many others.
4. Click here for my other publications in optics. They all have to applications for the Lorentz group in optical sciences.
So far, I have been talking about the symplectic group Sp(2). You would agree that this group is very rich in physics. Why simplectic, not simplectic? It is possible that Weyl chose this word to tell that this group is neither simple nor complicated.
It is a simple matter if we deal only with the rotation matrix R, and the same is true for the boost or squeeze matrix S. If we use both in our algebra, we end up with the question of Wigner rotations. This problem is still a head ache for most of you. I also went through this problem. Here is the paper I published (in JoP B) on this problem with Sibel Baskal. Not many people know the Wigner rotation is a exercise problem for the symplectic group.
As you know by now, I believe in the physics of harmony. I should be able to put both R and S matrices into one matrix. You will note that you can recover these two matrices from the following expression. If lambda = 0, this matrix becomes R, and it becomes S if theta = 0.
The question is how I was able to derive this expression from R and S. This has to do with the so-called Bargmann decomposition. I have been doing this since 1986, but was not able to identify this aspect of Sp(2) until recently. Indeed, the symplectic group put many strange words into one package.
On this issue, I gave a talk at the 10th International Conference on Quantum Optics and Quantum Information (May 2010). Here is the Arxived article to be published in the proceedings.

This conference was organized by Prof. Sergei Kilin of Belarus who started this series in his home base of Minsk. I attended a number of his earlier conferences. I am indeed thankful for him for giving me opportunities for developing my ideas. It is a pleasure to show you my phto with him taken in Dubna (Russia 2000). We both were attending the 22nd International Colloquium on Group Theoretical Methods in Physics.
This conference was held in Kiev, rich in history and culture. It was my second visit to Kiev. I was there in 2006 and 2010. I went there with my camera and took photos.

Click here for some of the Kiev Photos.

Click here for his home page.