Poincaré's Ding-an-Sich

Princeton's Graduate College

When I was a graduate student (1958-61), one of the philosophy majors came to me and told me what quantum mechanics is. I was not happy with his attitude and asked him whether he knows how to add waves. I asked specifically what happens when we add two sound waves with different frequencies. He then told me that this is a mathematical question, and that it does not carry its own substance. I thought he was hopeless and told him to go away.

If Bertrand Russell (1872-1970) of England had said the same thing to me about mathematics, I could not have told him to go away. Russel was one of the great thinkers of the past century and I like his history books. Russell told Herni Poincaré (1854-1912) that mathematics is only a tool of logic and nothing else, but Poincaré repeatedly disagreed.

Did Poincaré clearly define what mathematics is to Russell? We do not know, but we can try to figure out what Poincaré had in mind. We can do this not necessarily because we are smarter than he was, but because we are equipped with tons of progresses in physics made since Poincaré's death in 1912.

Like Einstein, Poincaré was heavily influenced by Immanuel Kant. One object or event could appear differently to different observers, but there is one and only one absolute entity. Kant used the word "Ding-an-Sich" (thing in itself). From Poincaré's point of view, mathematics could have been his "Ding-an-Sich."
If Russel and Poincaré are both right, nothing is everything. Even without those two great names, we usually complain that nothing is everything according to philosophers.
In preparation for this webpage, I made a trip to Paris 2012 in order to have a photo of myself at Poincaré's grave. He was buried at the Montparnasse Cemetery 1912.
1. This is his grave, one hundred years old in 2012. Many people came here and left their used Metro tickets on the stone.
2. Henri Poincaré was buried in his family grave.
3. I was there in 2012.
4. This young Australian came to this cemetery to visit the graves of the people dear to her heart. Her list should have included
5. Guy de Maupassant, who wrote a number of important stories about women.
6. Jean-Paul Sartre was also buried there, together with Simone de Beauvoir. I like Sartre and I have a webpage dedicated to him.
Where is the Montparnasse Cemetery? It is one block east of Montparnasse Tower.
1. The Cemetery seen from the top of the Montparnasse Tower.
2. The Montparnasse Tower is one of the tallest structures in Paris. This is a non-traditional buildings in Paris.
3. From the top of the Arch of Triumph, we can see how tall this structure is compared with other buildings in Paris. Underneath of the golden dome seen in this photo is Napoleon's gasket. The building is called "Hotel des Invalides" meaning hospital for wounded soldiers.
4. The Hotel des Invalides seen from the top of the Montparnasse Tower.
5. The Eiffel Tower seen from the top of the Montparnasse Tower. Modern buildings in the district of Defense are also seen. These buildings are for banks and other money-managing institutions.
6. The Basilica of the Sacré Coeur can also be seen from the top of this Tower. This photo was taken on a cloudy day, and became an impressionist photo.
7. From the Sartre Square, the Montparnasse Tower looks like this. The Sartre square is the area with the Cafe Flores and the Cafe Deux Magots where Sartre used to preach his philosophy. Click here for the Sartre Square in Paris.

Let us go back to physics. Speaking of the role of mathematics in physics, Poincaré started with the three-by-three rotation matrix applicable to the three-dimensional space. He expanded it to four-by-four by augmenting the time variable. He then came up with five-by-five matrices in order to take into account the space-time translational variables. This is the way in which he completed the formulation of the Poincaré group.

Here is what Raymond Streater says about Poincaré. Go to his web page on Henri Poincaré. According to him, Poincaré formulated the Lorentz-covariant space-time symmetry before Einstein completed special relativity as a new physical theory. Streater is not only an outstanding physicist but also a very critical history writer. I had two overlapping years with him in Princeton (1960-62).

Poincaré and Einstein

Einstein's Relativity. While many people were worrying about transformation properties of velocities, Einstein was interested in transformation properties of momentum and energy, and came-up with the concept of four-momentum. In so doing he obtained one mathematical formula the energy-momentum relation, leading to his celebrated E = mc².
By 1904, Lorentz and Poincaré proved the equations of electromagnetism, with the velocity of light as an invariant constant, are invariant under Lorentz transformations. Then, the question is why Newton's equation is not. Here, Einstein had to face a Hegelian problem: how to make Newton's equation and Maxwell's equations obey the same transformation law. He solved this problem and came up with a relativistic form Newton's equation. During this process, Einstein found out the momentum and energy can become a four-vector transforming like the space-time four-vector.
Hermann Minkowski was born in Lithuania and studied in Koenigsberg. He was a devoted Kantianist, and a also a creative mathematician. He was one of Einstein's teachers. In 1907, he published a paper where he gave a geometrical interpretation Lorentz transformations. He was particularly interested in expressing everything in terms of quadratic forms.
Lorentz transformations in the four-dimensional Minkowskian space include both rotations and squeeze transformations. Rotations are quite familiar to physicists. The squeeze is one of the standard deformations in engineering, but this transformation is still strange to physicists.
Let us go back to Kant. It is easy to say that Einstein's relativity is inconsistent with Kant's Ding-an-Sich which is the absolute coordinate frame. Let us see what Einstein did more closely. He divided many different observations into two groups (as in the case of Taoism). He then combined these two into one. Yes, Einstein was a Kantianist with the Lorentz-covariant energy-momentum relation as his Ding-an-Sich. Einstein's Ding-an-Sich is different from Kant's Ding-an-Sich. It appears as a mathematical formula.
Click here for an illustration of how Einstein reached his Ding-an-Sich. He had to go through Taoism and Hegelianism.
1. This image will illustrate how Kantianism and Taoism were developed from the same geographical environment. Kantianism is a product of the geographical condition of Koenigsberg. Like Venice, Koenigsberg was the traffic center for the ships navigating in the Baltic Sea. Many people came there with different ideas. The city had to find common ground to entertain those people. This is what Kant's Ding-an-Sich is all about.
  China was created as a collection of many isolated population pockets with different backgrounds. They came to one place with different ideas. They divided those ideas into two opposing groups. This is what Taoism is all about.
2. Two-party system of American government was developed in the same way as Taoism was developed in ancient China.
From these illustrations, we now have a clearer picture of what Kantianism is all about. From Poincaré's point of view, Einstein's Ding-an-Sich is mathematics. The mathematical expression of Einstein's energy-momentum relation is much more than an instrument of logic.

Poincaré Group after Einstein

When Einstein formulated his special relativity in 1905 and derived his energy-momentum relation, he did not take into consideration of the fact that particles have internal space-time symmetries. Then how can we define these symmetries?

How about momentum? Indeed, Einstein's four-momentum is associated with space-time translations in the Minkowskian space. If the momentum variable is to play a role, we have to resort to the inhomogeneous Lorentz group where the translational degrees of freedom are augmented to the Lorentz group with the rotation and boost generators. This inhomogeneous Lorentz group is commonly called the Poincaré group.

If we want to study the internal space-time symmetry of a particle, we should consider the symmetry for a fixed value of its momentum. This is necessarily a subgroup of the Lorentz group with three degrees of freedom. It was Eugene Wigner who addressed this issue in 1939.

In order to discuss this problem, we need many webpages. You may go to this page for a brief survey of this field. You will note there that this subject has been my main business since 1973.

Poincaré's Geometry and Topology

Poincaré was quite fond of drawing pictures when he was thinking. He used the surface of a sphere to describe the polarization of light. This sphere is known as the Poincaré sphere, and discussed in every optics textbooks. Like the Eulier angles, this sphere carries three variables.
The sphere has its radius. If it is allowed to vary, the sphere has four variables. In this case, did you know that the sphere can be used for representations of the Lorentz group which Poincaré formulated? I talked about this aspect in one of my earlir.
1. Click here for one of the webpages.
2. Click here for a paper I presented at the Fedorov Memorial Symposium: International Conference "Spins and Photonic Beams at Interface," dedicated to the 100th anniversary of F. I. Fedorov (Minsk, Belarus, 2011). Academian Fedorov was one on the pioneers in applications of the Lorentz group in optical sciences.
3. One mathematics for two different branches of physics. The damped harmomic oscillator and the LCR circuit are two different manifestations of the second-degree differential equation. Likewise, special relativity and polarization phyiscs are two different manifestations of the Lorentz group.
  The Lorentz group is applicable to many other branches of optical sciences. For a list my paper on this subject, go to this page.
4. Here again, Poincaré had enough reason to think mathematics is his Ding-an-Sich.
Poincaré was also interested in circles. He was particularly interested in under what circumstances we can shrink a circle continuously to a point.
1. Is there a physical application of this topology? Click here.
2. We would not be working against his wishes if we worry about how a circle can be deformed while preserving its area. It can become an ellipse and eventually become a straight line. Click here for the ellipse, and here for a detailed story. Here again, we are talking about the Lorentz group formulated first by Poincaré.
3. Here again, in the spirit of Poincaré, we can worry about to what extent a hyperbola is the same as an ellipse. I studied this aspect in this webpage. We know one can be continuously transformed into the other, but art they the same?
  
  Let us start with a hyperbola which can be written as
  We can write this as
  Here, if (x + y) becomes larger, (x - y) has to becomes smaller. What does this mean?
  Let us start with a circle
  which can also be written as
  If (x + y) becomes larger, (x - y) has to becomes smaller, it is an elliptic deformation of the above equation. Thus, the circle is the same as the hyperbola to the extent that the first hypergolic equation leads to an elliptic deformation of the cricle. Click here for a story. What physics can you do with this? Click here.
  I am very happy to say that I was able to reach this conclusion based on analytic geometry I learned during my high-school senior year (1953-54) in Korea.

Two and Three

webpage

another webpage

However, I am not the first one to see this point. Henri Poincaré was the first one to recognize the significance of this number. I am encouraged, and I have to do some more work to understand my own papers before making a meaningful statement about Poincaré's work on this subject. Please come again.

Conclusion

Since I have been writing papers for more than fifty years, I am entitled to organize my publications within some framework. It appears that they could fit into Poincaré's world. Thus, Henri Poincaré is my Ding-an-Sich.

It is very easy for young physicists to get lost these days. If you think you are lost in physics, find out your own Ding-an-Sich. If you thinks you are an established physicist, but you are not sure about whether your work will remain in history, find out what your Ding-an-Sich is.

Other Poincaré Papers

Wigner's Little Groups dictate internal space-time symmetries of elementary particles. Wigner's work is based on the Poincaré group governing both Lorentz transformations and translations. This group is called the inhomogenous Lorentz group.
Raymond Poincaré was Henri's cousin. He was the president of France from 1913 to 1920, covering the period of World War I. I am not able to write a story about him. Perhaps we can rely on his Wikipage.

Why is he with Einstein?

The photo of the Poincaré sphere on this webpage came from Christian Brosseau's book entitled "Fundamentals of Polarized Light, A Statistical Optics Approach" (Wiley, New York, 1998). I am grateful to Professor Brosseau for sending me a copy of this book.