Eugene Paul Wigner: Poincaré Group after Einstein

In 1905, Einstein formulated his special relativity for point particles. What happens if the particle has intrinsic space-time structure. For instance, Isaac Newton formulated his universal gravity law for two point particles. It took him twenty years to invent a new mathematics to take care of extended particles like the sun and earth.

Of course, there were physicists who worried about relativity applicable to rigid bodies, like rigid spheres or rigid potatoes, and there still may be those worrying about this problem. However, quantum mechanics changed the world. Rigid bodies are now wave packets and standing waves. OK! Do we then understand those wave objects in the Lorentz-covariant world? We can talk about this problem later.

Before tackling this difficult problem, we must study the basic internal space-time symmetries of relativistic particles. In 1939, Eugene Wigner published an article in Annals of Mathematics while he was at the University of Wisconsin (Madison). This paper had been rejected by three different journals until John von Neumann (then the editor of the Annals of Mathematics) decided to publish it in his journal.

In this paper, Wigner considers a free particle whose four-momentum is known. Then, are there other degrees of freedom for this particle? The Lorentz group has six degrees of freedom (three boosts and three rotations). If we fix the four-momentum, we are freezing three degrees of freedom (three, not four because the energy is determined by Einstein's energy-momentum relation). What happens to the remaining three degrees of freedom.

If a particle has its nonzero mass, like the proton or the earth, there is a Lorentz frame in which this particle is at rest. In this case, this particle has three rotational degrees of freedom. These degrees of freedom are associated with the dynamical quantity called "spin."

In group theory terms, Wigner was constructing the maximal subgroup of the Lorentz group whose transformations leave the four-momentum of the given particle invariant. This subgroup is called Wigner's little group. This little group dictates the internal space-time symmetry of relativistic particles.

If the particle in question is massless, there are no Lorentz frames in which the particle is at rest. The best we can do is to specify the direction and magnitude of the momentum. It also takes up three degrees of freedom. Thus, the massless particle also has three internal degrees of freedom. In his 1939 paper, Wigner also worked out the little group for massive particle. He showed that rotations around the momentum leaves the momentum invariant. This symmetry is associated with the helicity. Wigner found two additional symmetry operations which leave the four-momentum invariant. From the group theoretical point of view, these symmetry operations are like translations in two-dimensional Euclidean space while the helicity corresponds to the rotation around the origin in this two-dimensional space.

In his 1939 paper, Wigner was interested in unitary representations of the little groups, and noted that these translation-like operations cannot be associated with measurable quantities. Thus, he suggested restricting representations which freeze these translational degrees of freedom, as we can see in this page of Wigner's paper.

I started reading Wigner's 1939 paper while I was a student at Princeton (1958-61), but did not completely understand its content until I became much older. Of course, the key issue was what physics can be associated with those translation-like degrees of freedom for massless particles.

In 1980, D. Han started working with me. We performed a combination of rotations and boosts which would preserve the four-momentum of a massless particle and applied this transformation to the electromagnetic four-vector.

While this was going on, there were some people very unhappy with what we were doing. Their contention was that Weinberg never said the little group transformations are gauge transformations in his 1964 papers. They said further that the translation-like transformation cannot exist in Weinberg's construction, because he starts with the representation with zero eigenvalue of those translation-like operators.

This opposition was strong enough to affect my professional advancement. This was not the first experience of this kind in my research life. I went through the storm known today as the Dashen-Frautschi fiasco, and I even know how to enjoy them. As is widely known these days, in his 1995 book on on quantum field theory, Weinberg explicitly says translation-like transformations correspond to the gauge degree of freedom.

The story is not over yet. When we talk about gauge transformations on photons we talk about one degree of freedom. There are two translation-like degrees of freedom in Wigner's little group for photons. How can we collapse those two into one. During the period 1985-91, I used to go to Princeton regularly to tell Professor Wigner the stories he wanted to hear. In 1985, he was 83 years old, but he was eager to write new papers. He was very happy to hear that those two translation-like degrees correspond to gauge transformation. He then raised the question of how two translations become one gauge transformation. We worked hard, and published a paper in 1987 providing a solution to this problem. The point is that the little-group is only iomorphic to the two-dimensional Euclidean group. The little group takes the form of transformations on a cylindrical surface consisting of rotations (helicity) and up-down translations (gauge transformations).

Many people say Wigner was my thesis advisor when I was a student at Princeton. This was not the case. My advisor was Sam Treiman, who is also known as Weinberg's advisor. Weinberg was four years ahead of me, and I read his thesis before writing mine. I know how to read Weingberg's papers.

Y. S. Kim (2005.5.12)