Personal Background

I came to the United States from Korea in 1954 after high-school graduation, and became a freshman at the Carnegie Institute of Technology (now called Carnegie-Mellon University). In 1958, I became a graduate student at Princeton University. After receiving my PhD degree from Princeton in 1961, I stayed there for one more year as a post-doctoral fellow. Since 1962, I have been on the faculty of the University of Maryland at College Park. The United States has been very nice to me.

With General Maxwell Taylor (1954), the commander of U.S. Forces in Korea. I was in my high-school uniform. There were 330,000 US combat troops under Taylor's command. Later, Gen. Taylor served as the chairman of the Joint Chiefs for the Kennedy administration and designed Kennedy's Cuban invasion plan.
I spent my high-school years in Korea from 1948 to 1954, covering the Korean War period (1950-53). There were 360 boys in my entering class, but only 250 were able to graduate. This tells how destructive the war was on Korea's educational system. Yet, my class produced three Harvard PhDs, two Princeton PhDs, and one MIT PhD. I contend that my high-school class was the No. 1 class in the world.

You may click here for my Korean background.

In Korea, I had a Hallicrafters short-wave radio and was able to listen to the world. From Japanese broadcasts, I picked up many music programs. I used to listen to the "Voice of America" English programs directly from California, and heard clearly what was happening in the United States after the Supreme Court ordered the school integration in May of 1954, three months before coming to the United States. If you see my Carnegie Tech photo given on this page, you will see a black box on my left. It was another Hallicrafter radio which I had while I was at Carnegie.

In addition, I was born in a Christian family. My mother used to say I was able to recite verses of the Gospel of Matthew before I learned how to speak. This was of course an exaggeration, but I discovered easily that the super-constitution of the United States is the Gospel of Matthew. I have many Jewish friends who do not believe in Jesus, but they all agree with on this Matthew issue.

With this Christian background, I invented the word Herod Complex. As you know, King Herod ordered all new-born babies be killed after hearing that a new king was born. Most of my physics colleagues were No. 1 boys or girls in their respective high schools. If you are the No. 1, you cannot afford another No. 1 in your class. I know I have this complex, and I am able to understand atrocious behavior of my colleagues in terms of the Herod complex. In this way, I live in this world more comfortably.

Koreans are also hard workers. The United States rewards those who work hard. This was particularly true at Carnegie Tech which is still known as a brute-force school.

Studying in my dormitory room at Carnegie Tech (1958).

While I was a student at Carnegie Tech, I had excellent professors. I can mention three names. They were Lincoln Wolfenstein, Michel Baranger, and Hugh Young. Wolfenstein's ideology was that you have to produce the numbers which can be compared with experiments. At that time, I had to use slide rules to calculate the numbers, and I am still good at it. I took the first-year quantum mechanics course from Michel Baranger. Because of his influence, I was able to pinpoint the mistake Dashen made in his calculation of the n-p mass difference. This aspect of my professional life is on my webpage. Hugh Young is widely known for his textbooks, and was a dedicated teacher when I was a student. I have images of Baranger and Wolfenstein on my website. I would like to visit Hugh Young at CMU to have a photo with him.

Indeed, Carnegie Tech gave me a very solid education during my first four years in the United States. Pittsburgh is a very interesting city, and I have my Pittsuburgh page.

When I was a student (1958-61) and a postdoc (1961-62), I enjoyed being associated with many physics and non-physics friends. There was a student from the Soviet Union (unthinkable at that time) named Alexander Merkolov. He brought with him a Soviet-made shaving machine which operated on winding spring. He complained Americans do not sing at parties. Indeed, Russians sing passionately at their parties.

Princeton commencement ceremony (1961). I was sitting on the third row, and was able to have a close-up view of Dean Rusk, Henry Cabot Lodge, and Mary Bunting who were among the recipients of the honorary degrees.

I frequently had arguments with those students who were planning to be in foreign service. One of them told me and others that the best way to handle China (hostile to the U.S. at that time) is to let Japanese invade China by giving weapons to them. I then asked him whether Japanese were going to fight for Americans. American foreign policies are not always consistent with true Americanism. You can see why.

In addition to those non-physics people, I think I was popular among my physics friends and professors. I was not John A. Wheeler's student, but he still remembers me and likes to talk with me. Indeed, Princeton was very nice to me. Recently, I started developing my own Princeton page. It is not secret, and you are welcome to visit my Princeton page still under construction.

I frequently asked stupid questions to Eugene Wigner, and I now contend that I understand his physics better than anyone else in this world. I also initiated and developed an international conference series known as the "Wigner Symposium."

As a consequence, there is a tendency in the physics community to associate my name with Eugene Wigner, and it is often said that Wigner was my advisor when I was a student at Princeton. This was not the case. In order to clarify this issue, I would like to attach the following paper of mine in which I clearly spell out my association with Wigner.

For details, you may read a review article, entitled

Wigner's Last Papers on Spacetime Symmetries

in the Proceedings of the 4th International Wigner Symposium (Guadalajara, Mexico, August 1995), edited by N. Atakishiyeb, T. H. Seligman, and K. B. Wolf (World Scientific, Singapore, 1996).

For your convenience, the first two sections of this article are included in this file.

My photo with Wigner. This photo was taken while Professor Wigner was visiting the Univ. of Maryland in 1986.

I. Introduction

I met Eugene Wigner while I was a graduate student at Princeton University from 1958 to 1961. I stayed there for one more year as a post-doctoral fellow before joining the faculty of the University of Maryland in 1962. My advisor was Sam Treiman, and I wrote my PhD thesis on dispersion relations. However, my extra-curricular activity was on Wigner's papers, particularly on his 1939 paper on representations of the Poincar\'e group [1]. It is not uncommon for one's extra-curricular activity to become his/her life-time job. Indeed, by 1985, I had completed the manuscript for the book entitled "Theory and Applications of the Poincar\'e Group" [2] with Marilyn Noz who has been my closest colleague since 1970.

After writing this book, I approached Wigner again and asked him whether I could start working on edited volumes of all the papers he had written, but he had a better idea. Wigner told me that he was interested in writing new papers and that he had been looking for a younger person who could collaborate with him. This was how I was able to publish seven papers with him. Today, I would like to talk about two of those papers. They constitute a re-interpretation of Wigner's original paper on the Poincar\'e group.

Why is this paper so important? Where does it stand in the history of physics? From the principles of special relativity, Einstein derived the relation E = mc2 in 1905. This formula unifies the momentum-energy relations for both massive and massless particles, which are E = p2/2m and E = cp respectively. In his 1939 paper [1], Wigner observed that relativistic particles have their internal space-time degrees of freedom. For example, the spin of a particle at rest is a manifestation of the three-dimensional rotational symmetry. Wigner in his paper formulated space-time symmetries of relativistic particles in terms of the little groups of the Poincar\'e group. In this review talk, I would like to emphasize that Wigner's little group is a Lorentz-covariant entity and unifies the internal space-time symmetries of both massive and massless particles, just as Einstein's E = mc2 does for the energy-momentum relation.

On the other hand, Wigner did not reach this conclusion in 1939, and the above statement is based on many subsequent papers published on this subject during the period 1939-1990. In fact, his 1939 paper has a stormy history. This paper had been rejected by one of the prestigious mathematics journals before John von Neumann, then the editor of the Annals of Mathematics, invited Wigner to submit it to his journal. It is not uncommon even these days to hear the comment that the paper does not have anything to do with physics. Today, I would like to clarify this issue.

In Sec. II, I give a brief review of the subject and explain why Wigner's paper is essential in understanding modern physics. In order to give a more transparent interpretation of his paper, I give a geometrical interpretation of his work based on the paper which Wigner published with me in 1987 and 1990 [3,4]. The purpose of these two papers was to translate all the earlier works on this subject into a geometrical language. The main conclusion of these papers is that the E(2)-like little group does not share the same geometry as the E(2) group whose geometry is quite transparent to us. The geometry of the little group is that is the cylindrical group dealing with the surface of a circular cylinder [5]. The cylindrical axis is parallel to the momentum.

Also shown in these two papers is that the O(3)-like little group, which can be described in terms of a sphere in the rest frame, becomes continuously deformed into the symmetry group describing a point moving on the cylindrical surface as the momentum/mass ratio becomes large. For the case of electromagnetic four-potential satisfying the Lorentz condition, the rotation around the axis corresponds to helicity, while the translation along the direction of the axis corresponds to a gauge transformation.

In Sec. III, we discuss the three-dimensional rotation group and its contractions to the cylindrical and the two-dimensional Euclidean groups. It is shown that both of these contractions can be combined into a single four-by-four representation. In Sec. IV, the generators of the little groups are discussed in the light-cone coordinate system. It is shown that these generators are identical with the combined geometry of the cylindrical group and the Euclidean group discussed in Sec. V. The geometry of Sec. VI therefore gives a comprehensive description of the little groups for massive and massless particles.

II. Historical Review of Wigner's Little Groups

In 1939, Wigner observed that internal space-time symmetries of relativistic particles are dictated by their respective little groups [1]. The little group is the maximal subgroup of the Lorentz group which leaves the four-momentum of the particle invariant. He showed that the little groups for massive and massless particles are isomorphic to the three-dimensional rotation group and the two-dimensional Euclidean group respectively. Wigner's 1939 paper indeed gives a covariant picture massive particles with spins, and connects the helicity of massless particle with the rotational degree of freedom in the group E(2). This paper also gives many homework problems for us to solve.

First, like the three-dimensional rotation group, E(2) is a three- parameter group. It contains two translational degrees of freedom in addition to the rotation. What physics is associated with the translational-like degrees of freedom for the case of the E(2)-like little group?

Second, as is shown by Inonu and Wigner [6], the rotation group O(3) can be contracted to E(2). Does this mean that the O(3)-like little group can become the E(2)-like little group in a certain limit?

Third, it is possible to interpret the Dirac equation in terms of Wigner's representation theory [7]. Then, why is it not possible to find a place for Maxwell's equations in the same theory?

Fourth, the proton was found to have a finite space-time extension in 1955 [8], and the quark model has been established in 1964 [9]. The concept of relativistic extended particles has now been firmly established. Is it then possible to construct a representation of the Poincar\'e group for particles with space-time extensions?

The list could be endless, but let us concentrate on the above four questions. As for the first question, it has been shown by various authors that the translation-like degrees of freedom in the E(2)-like little group is the gauge degree of freedom for massless particles [10]. As for the second question, it is not difficult to guess that the O(3)-like little group becomes the E(2)-like little group in the limit of large momentum/mass [11]. However, the non-trivial result is that the transverse rotational degrees of freedom become gauge degrees of freedom [12].

Then there comes the third question. Indeed, in 1964 [13], Weinberg found a place for the electromagnetic tensor in Wigner's representation theory. He accomplished this by constructing from the SL(2,c) spinors all the representations of massless fields which are invariant under the translation-like transformations of the E(2)-like little group. Since the translation-like transformations are gauge transformations, and since the electromagnetic tensor is gauge-invariant, Weinberg's construction should contain the electric and magnetic fields, and it indeed does.

Next question is whether it is possible to construct electromagnetic four-potentials. After identifying the translation-like degrees of freedom as gauge degrees of freedom, this becomes a tractable problem. It is indeed possible to construct gauge-dependent four-potentials from the SL(2,c) spinors [14]. Yes, both the field tensor and four- potential now have their proper places in Wigner's representation theory. The Maxwell theory and the Poincar\'e group are perfectly consistent with each other.

The fourth question is about whether Wigner's little groups are applicable to high-energy particle physics where accelerators produce Lorentz-boosted extended hadrons such as high-energy protons. The question is whether it is possible to construct a representation of the Poincar\'e group for hadrons which are believed to be bound states of quarks [2,15]. This representation should describe Lorentz-boosted hadrons. Next question then is whether those boosted hadrons give a description of Feynman's parton picture [16] in the limit of large momentum/mass. These issues have also been discussed in the literature [2,17].

The application of the Poincar\'e group is not limited to relativistic theories of particles. This group plays many important roles in classical mechanics, the theory of superconductivity, as well as in quantum optics. This new trend makes it more urgent to understand correctly Wigner's papers on the Lorentz group. The following sections are based on Wigner's last papers on this subject [3,4] where his 1939 paper was translated into a geometrical language.

References

  1. E. P. Wigner, Ann. Math. 40, 149 (1939).
  2. Poincar\'e Group (Reidel, Dordrecht, 1986).
  3. Y. S. Kim and E. P. Wigner, J. Math. Phys. 28, 1175 (1987).
  4. Y. S. Kim and E. P. Wigner, J. Math. Phys. 31, 55 (1990).
  5. For an earlier discussion of the cylindrical geometry in connection with Maxwell's equations, see L. J. Boya and J. A. de Azcarraga, An. R. Soc. Esp. Fis. y Quim. A 63, 143 (1967).
  6. E. Inonu and E. P. Wigner, Proc. Natl. Acad. Scie. (U.S.A.) 39, 510 (1953).
  7. V. Bargmann and E. P. Wigner, Proc. Natl. Acad. Scie (U.S.A.) 34, 211 (1948).
  8. R. Hofstadter and R. W. McAllister, Phys. Rev. 98, 217 (1955).
  9. M. Gell-Mann, Phys. Lett. 13, 598 (1964).
  10. A. Janner and T. Jenssen, Physica 53, 1 (1971); ibid. 60, 292 (1972); J. Kuperzstych, Nuovo Cimento 31B, 1 (1976); D. Han and Y. S. Kim, Am. J. Phys. 49, 348 (1981); J. J. van der Bij, H. van Dam, and Y. J. Ng, Physica 116A, 307 (1982); D. Han, Y. S. Kim, and D. Son, Phys. Rev. D 26, 3717 (1982).
  11. S. P. Misra and J. Maharana, Phys. Rev. D 14, 133 (1976); S. Ferrara and C. Savoy, in Supergravity 1981, S. Ferrara and J. G. Taylor, eds. (Cambridge Univ. Press, Cambridge, 1982), p.151; D. Han, Y. S. Kim, and D. Son, J. Math. Phys. 27, 2228 (1986); P. Kwon and M. Villasante, J. Math. Phys. 29 560 (1988); ibid. 30, 201 (1989).
  12. D. Han, Y. S. Kim, and D. Son, Phys. Lett. 131B, 327 (1983).
  13. S. Weinberg, Phys. Rev. 134, B882 (1964); ibid. 135, B1049 (1964).
  14. D. Han, Y. S. Kim, and D. Son, Am. J. Phys. 54, 818 (1986).
  15. R. P. Feynman, M. Kislinger, and F. Ravndal, Phys. Rev. D 3, 2706 (1971); Y. S. Kim, M. E. Noz, and S. H. Oh, J. Math. Phys. 20, 1341 (1979).
  16. R. P. Feynman, in High Energy Collisions, Proceedings of the Third International Conference, Stony Brook, New York, C. N. Yang et al., eds. (Gordon and Breach, New York, 1969); J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969).
  17. Y. S. Kim and M. E. Noz, Phys. Rev. D 15, 335 (1977); Y. S. Kim, Phys. Rev. Lett. 63, 348-351 (1989).