Further Contents of Wigner's 1939 Paper on His Little Groups
 Where does Eugene Wigner stand in the history of physics?
 Albert Einstein was an important historical figure. Eugene Paul Wigner
was also an important Princeton figure.
 What is the connection between Einstein and Wigner? Wigner published
his paper on the Lorentz group in 1939. What does this paper tell
about Einstein?
 During the latter half of the 20th Century, highenergy accelerators routinely produced protons moving as the speed very close to that of light. What does Wigner's 1939 paper about those ultrafast protons?
 Albert Einstein was an important historical figure. Eugene Paul Wigner
was also an important Princeton figure.
 In developing new physical theories, they started with point particles. Newton
formulated his gravitational law for the point particles. It took him
20 years to develop a new mathematical tool do deal with particles with
nonzero radii, such as the sun and earth.
During the early years of the 20th Century, Albert Einstein developed his special theory of relativity for point particles, while Niels Bohr was worrying about the electron orbit of the hydrogen atom. Then, the question is how would this atom appear to moving observers. The hydrogen atom is not a point particle. It has a welldefined internal spacetime structure. Click here for a story.

Dirac (left) and Wigner.
Dirac's wife was Winger's younger sister. Click here for a story.  Eugene Paul Wigner (19021995) developed the mathematical tool to deal with this
problem in his paper on the Lorentz group published in 1939. The purpose of
this page is to talk about what happened since then.
 I am not the first person to worry about moving hydrogen atoms.
Paul A. M Dirac published a number of key papers on this issue. It is fun
to integrate those papers. Click here
to see how we can integrate those papers. In order to carry out this
integration, it requires a thorough understanding of the physics contained
Wigner's 1939 paper.
 Let us first look at Wigner's 1939 paper on his little groups governing
internal spacetime symmetries of particles.
Click here.
This paper is one of the most difficult papers to read. Indeed, this paper was rejected by three different journals before the Annals of Mathematics accepted for publication. According to Wigner, this journal accepted this paper only because its editor at that time was John von Neumann, who was Wigner's hometown friend. Wigner and von Neumann came from the same high school in Budapest (Hungary).
Wigner (left) and von Neuman at the entrance to the main auditorium of
their high school in Budapest (Hungary).
Photo by Y.S.Kim (1997).
I started reading this paper in 1959 when I was a graduate student at Princeton. I asked Professor Wigner why his little groups have three degrees of freedom. He told me that the Lorentz group has six degrees of freedom. If the momentum with its three degrees of freedom is fixed, there are only three degrees of freedom left in the Lorentzian world.
 I became an assistant professor in 1962 at the University of Maryland in
the Greater Washington area. I could not devote my full time to Wigner's
paper because I had to follow the current trend (what others were doing)
to keep my job and to get promoted.
1965
 In 1965, I totally lost confidence in the current trend.
At that time, the physics world was dominated by a superstition
generated by Princeton and Berkeley that physics comes from
singularities in the complex plane. I had to write papers
on this subject. In 1965, I found out
how fallacious this superstition was.
 Eugene Wigner was totally isolated from Princeton's
superstition, even though he was respected as No. 2 man after
Einstein. The reason was that nobody in Princeton was able to decode
Wigner's 1939 paper on the Lorentz group.
 At Princeton, the person in charge of this paper was
Arthur Wightman.
In 1961, while I was still a graduate student, I was in his class where
he gave his own interpretation of Wigner's 1939 paper. His lecture was
published in French in
From Wightman, I learnd the twobytwo representation of the Lorentz group, and it was a beautiful mathematics. However, I was not satisfied with Wightman's interpretation of the E(2)like little group for massless particles. In later years, he used to tell me and tell others that my interpretation of this problem was wrong. It is now safe to say that Wigtman was not capable of handling Wigner's E(2) issue.

What is the E(2) issue? Let us go back to 1939.
1939
 Thus, I went back to 1939, and I started looking at Wigner's 1939 paper again.
In 1963, Wigner
received the Nobel prize for his contributions to the symmetry problems in
physics, but not for his 1939 paper which was dearest to Wigner's heart.
I studied many papers on this subject. but I was not completely satisfied. The best way to approach this problem was to do it by myself. As Wigner told me in 1959, the particle has six degrees of freedom in the Lorentzcovariant world. If its momentum is fixed, it has only three degrees of freedom. For instance, if the particle is at rest (with its fixed momentum), it has rotational three rotational degrees of freedom. This is what the spin (internal angular momentum) is all about. This aspect is well known.
 On the other hand, the massless particle (like photon) cannot be brought
to its rest frame. Wigner, in his 1939 paper, pointed out that its internal
symmetry also has three degrees of freedom, The symmetry group is like the
two dimensional Euclidean group with one rotational and two translational
degrees of freedom. It is easy to associate the rotational degree of
freedom with the helicity of this massless particle.
Next, what physics is associated with the two translational degrees of freedom? This question has a stormy history, and they were finally found to correspond to gauge transformations.
1987
 It is not difficult to associate the rotational degree of freedom with
the helicity of the massless particle. Then what physics is associated with
the translational degrees of freedom?
 This question was not completely addressed until I published a paper
with Wigner in 1987.
Click here to see the paper. The true symmetry of the massless particle in that of a cylinder, not of the twodimensional plane, as illustrated in this figure. The updown translation along the cylindrical surface corresponds to gauge transformation.
 Indeed, from 1939 to 1987 (almost 50 years), many authors attempted to
resolve this issue. It is thus appropriate to list some of those papers dealing
with this problem.
 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, page 1 (1971), and
Physica 60, page 292 (1972).
 J. Kupersztych,
Nuovo Cimento B 31, page 1 (1976).
 D. Han, Y .S. Kim, and D. Son,
Phys. Rev. D [25], 461  463 (1982).
 Steven Weinberg talking to Wigner (far left).
Photo by Dieter Brill (1957).
 S. Weinberg,
Phys. Rev. 133, B1318 (1964).
1990
 Einstein's E = mc^{2} means the energymomentum relation
for moving particles, as shown in the following table. The question is
whether the O(3)like little group for the massive particle becomes the
cylidrical group when the particle moves with the speed of light or
speed close to it. We can formulate this problem with the following
table.
Einstein and Wigner Massive/Slow between Massless/Fast Energy
MomentumE=p^{2}/2m Einstein's
E=(m^{2} + p^{2})^{1/2}E=p Spin
Helicity, GaugeS_{3}
S_{1} S_{2}Wigner's
Little GroupsHelicity
Gauge Trans.  This table tells that the O(3)like symmetry for the massive
particle should become the E(2)like symmetry when the particle
moves with that of light or very close to it. This
problem was completely solved in
this paper of 1990.
 This figure illustrates how the O(3) symmetry of massive
particles become the cylindrical symmetry of massless particles in
the ultraspeed limit.

In 1990, I was fortunate enough to publish Wigner saying that the
the O(3) symmetry of massive particle can become that of
the cylindrical symmetry of massless particle in the largespeed or
massless limit. This procedure is illustrated in the following figure.
 This problem also has a history. In 1953, Inonu and Wigner
considered a localized area on the spherical surface in the
largeradius limit. This is the contraction of O(3) to E(2).
Click here for a story.
For this problem, I published a number of papers with my younger colleagues, before seeing Wigner in 1986.
 D. Han, Y. S. Kim, and D. Son,
Gauge transformations as LorentzBoosted rotations,
Phys. Lett. B [131], 327  329 (1983). 
D. Han Y. S. Kim, and D. Son,
Eulerian Parametrization of Wigner's Little Groups and Gauge Transformations in term of Rotations of Twocomponent Spinors,
J. Math. Phys. [27], 2228  2236 (1986).
 D. Han, Y. S. Kim, and D. Son,
Let us go back to 1965.
GellMann's quark model in 1964, and Feynam's parton picture in 1969.
 Like the hydrogen atom, the proton
is a bound state of more fundamental particles called "quarks."
However, when this boundstate proton moves at a speed close to that of light,
it appears as a collection of an infinite number of partons whose propertie
are totally different from those of the quarks in the proton.
 This problem can be traced back to BohrEinstein issue. One hundred years ago,
Bohr was worrying about the electron orbit in the hydrogen atom, while Einstein
was interested in how things appear to moving observers. Then, the question is
how the hydrogen atom appears to moving observers. If Bohr and Einstein discussed
this problem, there are no written records. Thus, the resolution of the quarkparton
puzzle constitute that of the BohrEinstein issue of moving hydrogen atoms.

1970. Feynman's harmonic oscillators
 In 1970, in his invited talk given at the Washington meeting of the American
Physical Society, Feynman suggested using harmonic oscillator wave
functions for the proton and other hadrons.
Later, with his younger coauthors, Feynman published his 1970 talk in the Physical Review D [3] 27062732 in 1971. The authors of this paper used harmonicoscillator wave functions for the Lorentzian world. In this world, the timeseparation variable should be included along with the spaceseparation variable. Their Gaussian wave functions are Lorentzinvariant, but not normalizable in the timeseparation variable. It is unfortunate because, in 1970, there were normalizable wave functions in the literature, and Feynman et al. quoted one of them.

In 1977, using the normalizable Lorentzcovariant wave function,
I published a paper in
Phys. Rev. D telling that the proton in
the quark model and Feynman's parton picture for ultrafast proton
are two different ways of looking at one Lorentzcovariant entity.
Marilyn Noz was my coauthor for this paper.
The following figure illustrates the content of this paper.
 This wave function is not derivable from the present form of quantum
field theory. Then, where does this Lorentzcovariant wave function fit
into the genealogy of physics? The answer is in Wigner's 1939 paper on
the little groups. This wave oscillator wave function is a representation
of the O(3)like little group for massive particles. Together with my
younger colleagues, in 1979, I published a
paper on this aspect.
Einstein's Genealogy Massive/Slow between Massless/Fast Energy
MomentumE=p^{2}/2m Einstein's
E=(m^{2} + p^{2})^{1/2}E=p Spin
Helicity, GaugeS_{3}
S_{1} S_{2}Wigner's
Little GroupsHelicity
Gauge Trans.Proton
HadronGellMann's
Quark ModelCovariant
Harmonic Osc.Feynman's
Parton Picture  This table leads to
This table can be translated into
 Click here for a more detailed story about Feynman's parton picture.

 copyright@2021 by Y. S. Kim, unless otherwise specified.
 Click here for his home page.
 His photobiography.
 His Princeton page.
 His Einstein page.
I received my PhD degree from Princeton in 1961, seven years after high school graduation in 1954. This means that I did much of the ground work for the degree during my high school years.