Henri Poincaré

This photo of Henri Poincaré (1854-1912) is from
www-groups.dcs.st-and.ac.uk/
~history/Mathematicians/Poincare.html.

You might also visit the Wikipedia page http://en.wikipedia.org/wiki/Henri_Poincar%C3%A9 about Henri Poincaré.

Poincaré made fundamental contributions to many different subjects in physics and mathematics. I am not able to discuss all of his contributions, but I can talk about his pioneering work on the Lorentz group and about his Poincaré sphere for polarization optics. I would like to point out that the Poincaré sphere is a representation of the Lorentz group. The Lorentz group is not restricted polarization optics. We shall see how Poincaré's Lorentz group is applicable to many other areas of optics.

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 Since 1973, all of my published and unpublished papers have something to do the Poincaré group.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 addition, I have shown that the Lorentz group, pioneered by Poincaré is the basic mathematical language for optical sciences.

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.

Louis Michel and Eugene Wigner (1984). The difference between the Poincaré and Lorentz groups.

While Poincaré and Lorentz were interested in velocity additions or Lorentz boosts of velocities, Einstein was working on the Lorentz boost of energy and momentum variables. In this way Einstein ended up with his energy-momentum relation.

Then what are the issues after Poincaré and Einstein? The Lorentz group is generated by six generators, namely for three rotations and three boosts. There are two Casimir operators commuting with all of these generators, but there do not seem to correspond to any dynamical quantities.

In the Lorentz-covariant world, we also have to consider space and time translations. There are thus four additional generators, and the total number of generators becomes ten. Poincaré also considered this group and this ten-generator group is called the Poincaré group. Also for this group, there are two Casimir operators which commute with all those ten generators. These Casimir operators correspond to the mass and the spin of the particle.

As Louis Michel pointed out in 1962 in one of his lecture notes, there are confusions in the literature about the difference between the Lorentz group and the Poincaré group. Even these days, the level of confusion remains the same if not worse. In my 1979 paper with Noz and Oh, we were able to amplify Michel's point using harmonic oscillators. I became very happy after learning this difference from Michel.

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's 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. I should stop here because I already talked too much about this subject. Click here if you are interested in what I talked about in the past.

Poincaré and Optics

This picture is associated with squeezed states of light, but you will see from the ellipse above that it is also a representation of the Poincaré group.

If you studied the polarization of light, you must have seen this sphere, called the Poincaré sphere. Poincaré was quite fond of expressing his ideas using curved surfaces, particularly spherical surfaces.

Poincaré originally constructed this sphere to understand polarization of light waves. These days, it serves as the basic instrument for understanding the density matrix and decoherence mechanism mentioned very frequently in the physics literature.

Did you know that this sphere is a representation of the Poincaré's Lorentz group? You may be interested in my paper with Sibel Baskal on this subject in Journal of Physics A (2006), or ArXiv.


Since I spent some years on this subject, and I was invited to write a chapter on this subject for a book on mathematical methods in optics. I am now making preparations for writing this Chapter with Sibel Baskal who published many papers with me on this subject. If you think your papers should be mentioned in this Chapter, do not hesitate to contact me at yskim@umd.edu.

The Lorentz group in optics is not restricted to polarizations. The squeezed state in quantum optics is a representation of the Lorentz group. For wave optics, there is the two-by-two matrix called the ABCD matrix. This matrix plays the pivotal role in lens optics, laser cavities, an multilayer optics. Did you know that the ABCD matrix is a representation the Lorentz group? Click here for some of the papers on this subject.

Since both the internal space-time symmetry and optical sciences share the same mathematics, it is possible to study internal space-time symmetries using opticcal instruments. You may be interested in my latest conference talk on this subject.

It is fun to combine optics and particle symmetries to one subject.

Click here for Photos from Paris.