Further Contents of Wigner's 1939 Paper on Little Groups.

• Where does Eugene Wigner stand in the history of physics?
  1. Albert Einstein was an important historical figure. Eugene Paul Wigner was also an important Princeton figure.

  2. 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?

  3. During the latter half of the 20th Century, high-energy accelerators routinely produced protons moving as the speed very close to that of light. What does Wigner's 1939 paper about those ultra-fast protons?

• 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 non-zero radii, such as the sun and earth.

  

  During the early years of the 20-th 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 well-defined internal space-time 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 (1902-1995) 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 space-time symmetries of particles. Click here.

  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).

  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).

  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 no-body 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
  A. S. Wightman, L'invariance dans la mecanique quantique relativiste , Relations de dispersion et particules elementaires, edited by C. DeWitt and R. Omnes, (John Wiley & Sons Inc New York 1962) pp 150 - 226.

  From Wightman, I learnd the two-by-two 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

It is easy to associte the O(3) symmetry of a massive particle with its spin. For massless a particle, it is also easy to associate the rotation in the xy plane with its helicity. However, what physics is associted the translations in the xy plane? This question has a stormy history until the complete solution is found in 1990. The two translational degrees of freedom on the xy plane collapse into one translational degree along the z axis, corresponding to the un-observable gauge transformation.

• 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 Lorentz-covariant 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 two-dimensional plane, as illustrated in this figure. The up-down 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.

  Steven Weinberg talking to Wigner (far left). Photo by Dieter Brill (1957).

  1. S. Weinberg, Phys. Rev. 133, B1318 (1964).
     
  2. S. Weinberg, Phys. Rev. 134, B882 (1964).
     
  3. S. Weinberg, Phys. Rev. 135, B1049 (1964).

  4. A. Janner and T. Jenssen, Physica 53, page 1 (1971), and Physica 60, page 292 (1972).

  5. J. Kupersztych, Nuovo Cimento B 31, page 1 (1976).

  6. D. Han, Y .S. Kim, and D. Son, Phys. Rev. D [25], 461 - 463 (1982).

1990

• Einstein's E = mc² means the energy-momentum 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 Momentum E=p2/2m Einstein's E=(m2 + p2)1/2 E=p

  Spin Helicity, Gauge S3 S1 S2 Wigner's Little Groups Helicity 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 ultra-speed 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 large-speed 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 large-radius 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.

  1. D. Han, Y. S. Kim, and D. Son,
     Gauge transformations as Lorentz-Boosted rotations,
     Phys. Lett. B [131], 327 - 329 (1983).

  2. D. Han Y. S. Kim, and D. Son,
     Eulerian Parametrization of Wigner's Little Groups and Gauge Transformations in term of Rotations of Two-component Spinors,
     J. Math. Phys. [27], 2228 - 2236 (1986).
     

Let us go back to 1965.

Gell-Mann's quark model in 1964, and Feynam's parton picture in 1969.

Are they talking about the same thing in different frames?

• Like the hydrogen atom, the proton is a bound state of more fundamental particles called "quarks." However, when this bound-state 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 Bohr-Einstein 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 quark-parton puzzle constitute that of the Bohr-Einstein issue of moving hydrogen atoms.

  

1970. Feynman's harmonic oscillators

In April of 1970, I took this photo in a dark lecture hall, while Feynman was giving his invited talk at the APS spring meeting in Washinton, DC, USA.

• 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 co-authors, Feynman published his 1970 talk in the Physical Review D [3] 2706-2732 in 1971. The authors of this paper used harmonic-oscillator wave functions for the Lorentzian world. In this world, the time-separation variable should be included along with the space-separation variable. Their Gaussian wave functions are Lorentz-invariant, but not normalizable in the time-separation 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 Lorentz-covariant wave function, I published a paper in Phys. Rev. D telling that the proton in the quark model and Feynman's parton picture for ultra-fast proton are two different ways of looking at one Lorentz-covariant entity. Marilyn Noz was my co-author 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 Lorentz-covariant 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 Momentum E=p2/2m Einstein's E=(m2 + p2)1/2 E=p

  Spin Helicity, Gauge S3 S1 S2 Wigner's Little Groups Helicity Gauge Trans.

  Proton Hadron Gell-Mann's Quark Model Covariant 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.

  1. Click here for his home page.
  2. His photo-biography.
  3. His Princeton page.
  4. 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.