Wigner's Last Lunch with Kim

This lunch took place at a seafood restaurant near Princeton in 1991.

In 33 AD, there was a much more elegant meeting somewhere in Jerusalem. We do not have a photo of this dinner meeting, but Leonardo da Vinci painted this scene:

Athough I was a graduate student at Princeton (1958-61), Wigner was not my thesis advisor. My advisor was Sam Treiman. Yet, I am known as Wigner's student among my physics colleagues. How did this happen? Click here for a story.
The issue is Wigner's 1939 paper on the inhomogeneous Lorentz group. Wigner in this paper formulated his "little groups" which dictate the internal space-time symmetries of particles in the Lorentz-covariant world. Here he concludes that the symmetry for a massive particle is like the three-dimensional rotation group. This symmetry leads to the concept of spin.
For massless particles, Wigner observed the symmetry group is isomorphic to the two-dimensional Euclidean group or E(2), consisting of one rotational and two translational degrees of freedom. It is easy to associate the rotational degree to the helicity. What about the translational degree of freedom? This question was not completely addressed until 1990, 51 years after 1939.
Thus, this is an interesting topic in history of physics. In 1939, Wigner stated that the symmetry is isomorphic to E(2) and wrote down the matrix of the form
This is the "ugliest" matrix in physics. Even these days, physicists give up if the matrix is not unitary. I know this from my own experience. Since 1973, my papers were based on non-unitary squeeze matrices. Did you know that Lorentz boosts are squeeze transformations? As soon as I start telling this, my colleagues run away from me.

Weinberg talking to Wigner (1957) (top), my interpretation of Weinberg's interpretation of the E(2) symmetry.

It was Steven Weinberg who started tackling the above "ugly" matrix in one of his 1964 papers. In 1979, I studied Weinberg's papers with my younger colleagues, namely D. Han and D. Son, and came up with this figure.
This two-dimensional figure tells about rotations and translations in the E(2) plane, and the translations as gauge transformations. As for the issue of gauge transformations, there are several other authors who said the translations correspond to gauge transformations, and they were quoted in my papers. Most certainly, I am not the first one to observe this aspect of Wigner's E(2)-like symmetry.

After seeing this, I thought the person who would appreciate this most was Eugene Wigner who wrote the above "ugly" matrix in his 1939 paper. In 1985, I went to Princeton to tell him about the translation-like variables in his E(2)-like group. He became very happy, but asked why there have to be two gauge components ( x and y coordinates).

I then had to work hard to make Wigner happy. I pulled out Wigner's 1953 paper with Inonu on group contractions, telling that a spherical surface can become flat if the radius becomes large. Thus, we can consider a tangential plane for the E(2) symmetry.

The E(2) symmetry comes from the plane tangential at the north pole, while the cylindrical symmetry comes from the cylinder tangential at the equatorial belt (left figure). Click here for the 1987 paper by Kim and Wigner.
The four-by-four Lorentz transformation matrix produces a geometry which deforms a sphere into an egg and a pancake, which eventually become a cylinder and a plane respectively. Click here for the 1990 paper by Kim and Wigner.

Indeed, both translation-like variables correspond to one up-down translation on the cylindrical surface. Thus, Wigner's E(2)-like symmetry has one rotational degree and one translational degree of freedom. This translational degree of freedom corresponds to the gauge degree of freedom.
During the period of five years (1985-90), I went to Princeton frequently to seek guidance from Eugene Wigner. I was like a graduate student working on his thesis research under Wigner's guidance. When I was a "real graduate student" (1958-61), I was afraid of him.
These days, I am frequently introduced as Wigner's youngest student, while my thesis advisor was Sam Treiman when I got my degree in 1961. I become angry when this introduction is based on my photos with Wigner. I will be very happy if I am so introduced based on the two papers quoted above.
The above story is based on the following papers.
1. E. Wigner, Ann. Math. 40 149 (1939).
2. E. Inonu and E. P. Wigner, Proc. Natl. Acad. Sci. (U.S.) 39, 510 (1953).
  Click here for further information.
3. S. Weinberg, Phys. Rev. 133, B1318 (1964).
4. S. Weinberg, Phys. Rev. 134, B882 (1964).
5. S. Weinberg, Phys. Rev. 135, B1049 (1964).
6. A. Janner and T. Jenssen, Physica 53, 1 (1971); ibid. 60, 292 (1972).
7. K. Kuperzstych, Nuovo Cimento B 31, 1 (1976).
8. D. Han, Y. S. Kim, and D. Son, Phys. Rev. D 25, 461 (1982).
9. D. Han, Y. S. Kim, and D. Son, Phys. Rev. D 31, 328 (1985).
10. D. Han, Y. S. Kim, and D. Son, J. Math. Phys. 27, 2228 (1986).
11. Y. S. Kim and E. P. Wigner, J. Math. Phys. 28, 1175 (1987).
12. Y. S. Kim and E. P. Wigner, J. Math. Phys. 31, 55 (1990).