Why am I regarded as Wigner's student?

Eugene Wigner and his wife at their home in Princeton (1991).

Among them, one of the most difficult papers is Wigner's 1939 paper dealing with the internal space-time symmetries. He said the internal symmetries are isomorphic to O(3) and E(2) for massive and massless particles respectively. The physics is transparent for O(3)-like symmetry for a massive particle. If it is at rest its spin can rotate in the three-dimensional space.
• 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). 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.

  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.