Poincaré and Einstein

Henri Poincaré was born in 1854, 25 years before Einstein. He had a cousin named Raymond who was the president of France (1913-1920). Bertrand Russell of England once said Henri, not Raymond, was the greatest man France had ever produced.

Russell was not the only one who had a great admiration for Henri Poincaré. You must also have an admiration for him in your own way. I also respect him, and I express my admiration in the following way. Since 1973, all of my published and unpublished papers have something to do the Poincaré group. In collaboration with Marilyn Noz, I published a book entitled "Theory and Applications of the Poincaré" group. Here is a review of this book written by Marino del Olmo.

In spite of my life-long dedication to one of his mathematical instruments, I am not the best person to tell you about Henri Poincaré and his role in Einstein's formulation of special relativity.

Raymond Streater is not only an outstanding physicist but also a very critical history writer. I would like to invite you to visit his web page on Henri Poincaré. According to Streater, Poincaré formulated the Lorentz-covariant space-time symmetry before Einstein completed special relativity as a new physical theory.

The Poincaré group was later called "inhomogeneous Lorentz group" by Eugene Wigner in his fundamental paper of 1939. The word "inhomogeneous" was added because the group takes into account space-time translations in addition to Lorentz transformations. In the mathematical language, this inhomogeneous group is a "semi-direct product" of the space-time translation group and the four-dimensional Lorentz group.

Wigner's approach to the Poincaré group is often called "induced representation" by mathematicians, because he starts with subgroups which are called Wigner little groups. The little group is the maximal subgroup of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. In this way, Wigner formulated the concept of internal space-time symmetries of relativistic particles. These days, there are many young people interested in inventing a new physics applicable to internal space-time structure of the particle. Their job could become easier if they first study the internal space-time symmetry dictated by Wigner's little groups.

This web page would be meaningless if I tell you I have nothing to say about Poincaré while I spent more than 30 years on the Poincaré group. I can say many things, including applications of the Lorentz group to optical sciences. However, if I am forced to say the single most important work I did on this subject, I would tell you the following story.

After Einstein's 1905, the most revolutionary development in physics was the formulation of quantum mechanics where particles can also be regarded as waves. Since then many laudable efforts have been made to make quantum mechanics consistent with special relativity. The formulation of quantum field theory was of course a giant step, but the present form of field theory based on the S-matrix and Feynman diagrams cannot deal with localized wave functions for bound states.

One way to tackle the standing-wave problem in the Lorentz-covariant regime is to observe that

Quantum mechanics is largely a physics of harmonic oscillators.
Special relativity is a physics of the Poincaré group.

Therefore, the first step toward solving the standing-wave problem is to construct a representation of the Poincaré using harmonic oscillators. I can say more, but let me stop here.

I would like to thank Marcel Froissart for bringing Raymond Streater's article to my attention and urging me to add a Poincaré page to my Einstein page which is still under construction.

I met both Froissart and Streater while I was at Princeton as a graduate student (1958-61) and a post-doc (1961-62). I am very happy to mention their names on this webpage. Here is a photo of Froissart which I took in 1962 using my Canon camera (made by an obscure company at that time). Ray Streater maintains his own website. He has a very comprehensive list of physicists of our time. Please visit his site to check whether he mentions your name. You might be interested in visiting my own Princeton page, Many people say it is a fun page.

Y. S. Kim (2005.4.12)