Why am I regarded as Wigner's student?Before Newton, physicists tried to find physical laws by looking at the sky. People still do. Do you know how many stars there are in the sky? These days, there are the same number number of physics papers to read. Thus, some physicists try to find something new from those papers.
- 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
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).
While studying Weinberg's 1964 paper on massless particles, I and my colleagues came up with the following interpretation of the photon spin and gauge transformations.
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.
- E. Wigner, Ann. Math. 40 149 (1939).
- E. Inonu and E. P. Wigner, Proc. Natl. Acad. Sci. (U.S.)
39, 510 (1953).
Click here for further information.
- S. Weinberg,
Phys. Rev. 133, B1318 (1964).
- S. Weinberg,
Phys. Rev. 134, B882 (1964).
- S. Weinberg,
Phys. Rev. 135, B1049 (1964).
- A. Janner and T. Jenssen, Physica 53, 1 (1971); ibid. 60, 292 (1972).
- K. Kuperzstych, Nuovo Cimento B 31, 1 (1976).
- D. Han, Y. S. Kim, and D. Son, Phys. Rev. D 25, 461 (1982).
- D. Han, Y. S. Kim, and D. Son, Phys. Rev. D 31, 328 (1985).
- D. Han, Y. S. Kim, and D. Son,
J. Math. Phys. 27, 2228 (1986).
- Y. S. Kim and E. P. Wigner, J. Math. Phys. 28, 1175 (1987).
- Y. S. Kim and E. P. Wigner, J. Math. Phys. 31, 55 (1990).
- This page was constructed by Y.S. Kim (2014.4.3).
| Eugene Wigner and his wife at |
their home in Princeton (1991).
copyright@2014 by Y. S. Kim, unless otherwise specified. The portrait of Copernicus is from a postcard I purchased from the Jagiellonian University book store. The map of the Vistula Lagoon and the portrait of Galileo Galilei are from the public domain.