Maxwell and Wigner

Let us go back to my webpage on Maxwell and Minkowski. There I said
  1. 1n 1908, Minkowski proved Maxwell's equations are covariant under Lorentz transformations, thus are consistent with Einstein's special relativity.

  2. Sam Treiman was my thesis advisor when I was a graduate student at Princeton (1958-61). Steven Weinberg was Treiman's first student and received his PhD degree in 1957.

  3. When I was a student, Weinberg was not famous, but I had to read his thesis before writing mine.

So what? What scientific significance does this carry?

During the years 1961-65, every physicist had to do bootsrap, N/D method, or S-matrix theory. I was also following the trend, and publsihed some nontrivial papers on this subject.

Weingberg was somewhat different. He was interested in possible applications of Wigner's little groups [1] to S-matrix theory. Some of you will recall Weinberg's papers entitled Feynman Rules Any Spin [2,3].

At that time, many physicists said Weinberg was wasting his time because Wigner's 1939 paper on the Lorentz group has nothing to do with physics. In particular, Wigner's representation theory cannot accommodate Lorentz-covariant Maxwell fields.

I knew from Wigner that his representation theory, as presented in his original paper [1], cannot fully explain the four-potential for photons. But I did not think Weinberg was wasting his time, thanks to my educational background which I bragged about before.

I am very happy to show you an old photo of Weinberg asking a question to Wigner. What do you think his question was? Most probably about Wigner's 1939 paper on his little groups dictating internal space-time symmetries of elementary particles. This photo was taken while Weinberg was a student at Princeton. He finished his degree in 1957 and went to Columbia. Daniel Sperber (also shown in the photo) was still working for his degree when I went to Princeton in 1958. He told me what I should do and what I should not do as a graduate student. I took Sam Treiman's "Advanced Quantum Mechanics" (field theory) during my first year.

I also became interested in Wigner's 1939 paper when I was a graduate student, but was not able to get anywhere at that time, and my classmates laughed at me. Indeed, Weinberg's papers, especially on massless particles, revitalized my interest in Wigner's little groups. During the years 1981-86, I published many papers on this subject with my younger colleagues, D. Han and D. Son. Some of them [5,6,7,8] are listed for your information.

Wigner's Little Group:
  • Lorentz-boosted rotation for massive particles.

  • Gauge transformations for massless particles.

This figure explains what we did. Let us start with a particle moving along a given direction. It can be Lorentz-boosted (red line). Then its momentum will have a different magnitude in a different direction. How about its helicity (angular momentum along the direction of momentum)? It does not remain invariant under this boost.

As far as the helicity is concerned, it remains invariant when boosted along the direction of momentum, as specified by a green line. It remains invariant also under the rotation, as shown in the figure. Indeed, to each Lorentz boost, there is a unique helicity-conserving transformation consisting of a boost followed by a rotation.

The difference between these two transformations becomes the loop transformation shown in this figure, and it leaves the momentum invariant. This is precisely Wigner's little group. As defined by Wigner, the little group is a Lorentz-boosted rotation for massive particles. A massive particle can first be Lorentz-boosted to the rest frame. In this frame, it can be rotated without changing its momentum. Then the system can be boosted back to the original momentum.

Yes, it is possible to construct four-by-four transformation matrix corresponding to the loop, and it is also possible to write it as a Lorentz-boosted rotation. What happens if the particle mass becomes zero? Then the loop becomes a gauge transformation.

In quantum electrodynamics, we start with the four-potential without longitudinal and time-like components. The problem is whether this vector field can be unitarily transformed into a new Lorentz frame. The answer is yes if it is accompanied by a gauge transformation.


Massive/Slow between Massless/Fast
Energy
Momentum 		E=p2/2m Einstein's
E=(m2 + p2)1/2 		E=p
Spin, Gauge,
Helicity 		S3
S1 S2 		Wigner's
Little Group 		S3
Gauge Trans.

We constructed this table in 1985, and publshed in one of our papers [8]. In the meantime, I went to Princeton to show this table to Wigner. He became very happy and asked me whether he could write new papers.

Wigner said the symmetry group is isomorphic to the two-dimensional Euclidean group with two translation-like degrees of freedom. His question was how these two degrees of freedom could explain one gauge variable.

I thought I could write papers with Wigner on this subject. Wigner's 1939 paper [1] was very important to him. Equally dear to his heart was the Inonu-Wigner paper of 1953 on group contractions [9]. These authors started with a sphere. They then constructed the two-dimensional Euclidean group from a plane tangent to a sphere. We observed that a tangential cylinder can also be considered, and that this is also a content of the Lorentz group. Then, there is only one gauge dimension (up and down along the cylinder), as illustrated in this figure. This paper was published in 1990 [11], but a pdf version of its preprint is available.

This paper is not easy to follow. I hope I could someday translate its content into a show-and-tell webpage for your entertainment. The point is that this paper completes Wigner's interpretation of Maxwell's equations. Wigner's little group dictactes the internal space-time symmetries of elementary particles (like photons) and composite particles (like hadrons).

In addition to Weinberg's papers, I and my colleagues benefited greatly from the papers of Janner and Jenssen [12] who say explicitly Wigner's E(2)-like little group generates gauge transformations. Kuperzstych [13] in 1976 used the concept of loop transformations before we did in 1981.

It is very cumbersome to carry out those Lorentz-group calculations with four-by-four matrices applicable to the Minkowskian space-time. In 2005, S. Baskal and I used two-by-two matrices of the SL(2,C) group to calculate angles associated with Lorentz boosts, including those coming from the kinematical configuration given in the present webpage [14]. Furthermore, those two-by-two matrices are directly applicable to many optical processes [15].

If you think I should mention your papers on this subject, please let me know. It is very easy to make corrections and additions to webpages.

  1. E. Wigner, Ann. Math. 40 149 (1939).

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

  5. D. Han and Y. S. Kim, Am. J. Phys. 49, 348 (1981).
  6. D. Han, Y. S. Kim, and D. Son, Phys. Rev. D 26, 3717 (1982).
  7. D. Han, Y. S. Kim, and D. Son, Phys. Rev. D 31, 328 (1985).
  8. D. Han, Y. S. Kim, and D. Son, J. Math. Phys. 27, 2228 (1986).

  9. E. Inonu and E. P. Wigner, Proc. Natl. Acad. Sci. (U.S.) 39, 510 (1953).
    Click here for further information.

  10. Y. S. Kim and E. P. Wigner, J. Math. Phys. 28, 1175 (1987).
  11. Y. S. Kim and E. P. Wigner, J. Math. Phys. 31, 55 (1990).
    Click here for a pdf file.

  12. A. Janner and T. Jenssen, Physica 53, 1 (1971); ibid. 60, 292 (1972).
  13. K. Kuperzstych, Nuovo Cimento B 31, 1 (1976).

  14. S. Baskal and Y. S. Kim, J. Phys. A 38 6545(2005), and http://www.arxiv.org/abs/math-ph/0401032.
  15. Click here for applications of SL(2,c) to optical sciences.

Y. S. Kim (2008.11.15)