Einstein and Group Contractions

When Einstein introduced Lorentz covariance for Newtonian particle dynamics, the question was how he could explain the fact that Newton's law is perfectly consistent with the Galilei system. The explanation was and still is that Lorentz transformations become Galilei transformations in the limit of small velocity or large velocity of light.

Erdal Inonu giving the talk at the workshop (1997).

This is of course very easy to say and write down the formulas, but the precise group theoretical treatment was not achieved until 1953 when Erdal Inonu and Eugene Wigner published a paper on group contractions [Proc. Nat. Aca. Sci. 39, 510 (1953)]. They explained precisely how the six generators of the Lorentz group become the six generators of the Galilei group. You may be interested in Inonu's review paper given at the Workshop on Quantum Groups, Deformations and Contractions (Istanbul, Turkey, 1997).
In the same paper, Inonu and Wigner give a detailed exposition of the contraction of the O(3) rotation group into the E(2) group, namely the Euclidean group in two-dimensional space consisting of rotations around the origin and translations in two orthogonal directions. We shall see what effect this has in studying internal space-time symmetries of relativistic particles.
In his 1939 paper on the inhomogeneous Lorentz group [Ann. Math. 40, 149], Wigner showed that the internal space-time symmetries of massive and massless particles are isomorphic to O(3) (three-dimensional rotation group) and E(2) (two-dimensional Euclidean group) respectively. They are known as Wigner's little groups.

The O(3)-like symmetry for a massive particle corresponds to the spin of the particle. As for the E(2)-like symmetry for a massless particle, it is not difficult to associate the rotational degree of freedom to the helicity. After some stormy history, it has been now established that the two translational degrees of freedom of E(2) correspond to the gauge degree of freedom.

Then the following question arises. Can the E(2)-like little group for massless particles be obtained from the O(3)-like little group by a group contraction procedure. The answer to this question is YES. You may be interested in the following papers.

Y. S. Kim and E. P. Wigner, "Cylindrical group and massless particles," J. Math. Phys. Vol. 28, pages 1175-1179 (1987).

Y. S. Kim and E. P. Wigner, "Space-time geometry of relativistic particles," J. Math. Phys. Vol. 31, pages 55-60 (1990).

The following figure summarizes how the E(2)-like little group can be obtained from the O(3)-like little group by group contraction or the high-speed limit.

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.

You may visit a separate webpage for detailed explanation of this table.

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).

Finally it is my pleasure to show my photos with the main characters of group contractions. One is with Eugene Wigner and the other is with Erdal Inonu.

with Wigner (1986).
with Inonu (1997). The other gentleman in the photo is Nikolaj Gromov from Russia. He now holds the world championship on group contractions.

Y.S.Kim (2005.10.21)

Why is he with Einstein?

copyright@2005 by Y. S. Kim, unless otherwise specified. Wigner portrait by Bulent Atalay (1978).
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