# Hermann Minkowski (1864-1909)

Hermann Minknowski was born in Kaunas (Lithuania), and studied in Königsberg, Germany. After World War II, this city became the Russian city of Kaliningrad. This city was the geographical environment for Immanuel Kant to produce his philosophy, which served as the philosophical base for Einstein. You may click on the following links for While Königsberg was a port city like Venice, Kaunas was in its hinterland. Here are two maps telling where they are.

The University of Königsberg (Albertina University) had many mathematicians interested in physics. They were particularly interested in Maxwell's equations. We are now using the form of Maxwell's equations developed there. Minkowski continued his research in those equations after he left Königsberg.

He was a professor of mathematics and physics at the University of Zurich, while Einstein was a student there. There he completed the proof that Maxwell's equations are covariant under Lorentz transformations. He published his result in 1908.

Is this the end of the story of the Lorentz covariance of Maxwell's equations? No. This problem was not completely settled until 1990. We shall talk about this next time.

Another important contribution Minkowski made is the Minkowski space where

x2 + y2 + z2 - t2

is a Lorentz-invariant quantity. If we do not change the x and y variables,

z2 - t2 = constant.

We are quite familiar with the two-dimensional geometry associated with hyperbolic condition. But this geometry is still is strange to us. I am not the first one to feel in this way. It was Paul A. M. Dirac who invented the light-cone coordinate system to deal with this problem. However, the problem with Dirac was that he never draws pictures in his papers, unlike John A. Wheeler.

Let us write the above formula as

(t + z) (t - z) =constant.

If we introduce the "light-cone" variables

u =(t + z) and v = (t - z),

u v = constant,

the Minkowskian geometry becomes the geometry of rectangles, with u and v for two perpendicular sides. If the area of the rectangle remains constant, one side becomes contracted when the other side expands. This is the geometry of squeeze.

I learned this geometry during my high-school years. Since 1973, most of my papers are based on this squeeze geometry. This geometry is not only useful for Lorentz-boosted particles, but also provides the basic mathematical language for optical sciences.

• Sam Treiman was born in Chicago, but his parents came from Lithuania. He did his undergraduate work at Northwestern University, and the graduate study at the University of Chicago. Enrico Fermi was his advisor.

Sam Treiman was a very important person to me. He was my thesis advisor at Princeton University. I got my degree in 1961. Four years before, in 1957, a brilliant man named Steven Weinberg got his PhD degree under his supervision. This is a photo of Weinberg and Treiman taken in 1985.

When my son went to Princeton as a freshman in 1983, Treiman was the chairman of the department. He helped my son in many ways. Here is a photo of my family with the Treimans taken in 1987 at the graduation reception.

As you see, Treiman was a very sharp-looking person. When I went to Princeton in 1958, many people told me to work for John A. Wheeler, but I decided to work for Sam Treiman because he was so handsome and precise.

Among the many advantages of having Sam Treiman as the advisor, I was forced to read Steven Weinberg's thesis before writing mine. Weinberg was not famous at that time. I used this training to read the papers Weinberg wrote on the Lorentz group in the early 1960s, and to continue my own program on the same subject. I did enough work to tell the stories Wigner liked to hear in 1985.

The issue was whether Maxwell's equations are consistent with Wigner's little groups which dictate internal space-time symmetries of elementary particles. Hermann Minkowski was interested in whether the Maxwell equations are consistent with Lorentz, not Galilei, transformations.

## Recent Photos from Lithuania

I went in September (2008) to Vilnius to attend the 12th International Conferece on Quantum Optics and Quantum Information. As usual, I brought my camera with me and took a number of photos. I would like share some of them with you.

• Vilnius, City of Harmony consisting of communist statues, capitalist skyscrapers, and catholic churches.
1. This Stalin-style Tower tells Lithuania had a Soviet-dominated communist past.
2. Socialist Housing Complex. These buildings were presumably built after 1990, but it is very difficult to build individual houses.
3. Capitalistic Hotels and Office Buildings. Lithuanians are very busy in constructing their capitalism. Where do they get all the money?
4. Shopping Mall in the new capitalistic region of Vilnius.
5. Gift Shop in this shopping mall.

• City of Churches. A church every block.
1. Pope John-Paul II visited this city in 1995. I like him and I respect him.
2. Church Service. Lithuanians believe in Jesus rigorously.
3. Russian Orthodox Church. There are still many Russians in Lithuania. They also believe in Jesus.
4. Greek Orthodox Church. If there enough Catholic churches, there are some orthodox churches.
5. Jewish Synagogue and the plate explaining its history. Since I have many Jewish friends and colleagues, I visit Jewish communities wherever I go. I was particularly interested in the Lithuanian Jewish community because many talented Jewish scholars and artists came from this area. Indeed, there were in the past more than 100 synagogues in Vilnius, perhaps as many as Catholic churches. However, there is only one remaining. I dropped in one of the Jewish community centers and started talking to the attendant. He was old and could not speak English. When I asked him to have a photo with me, he declined. Instead, he offered to take this photo for me.

• Mothers with their babies. Vilnius is a city with life.
1. Children's Playground near a residential apartment complex.
2. Children's Bicycle Race on the City Hall square.
3. Rock Concert for Children in front of the City Hall.
4. High School Students going home after school.
5. Art Exhibition at the city-center square of Vilnius. This event is for all ages.

• Lithuanian Student going home after working during the summer in South Carolina (U.S.A.). This photo was taken at Warsaw's Frederic Chopin Airport (September 2008).
1. Students at the University of Vilnius.
2. Music Students at Vilnius Conservatory. They believe in music.
3. Students taking a rest at a fountain. They all speak fluent English.
4. Lithuanian Student taking opinion polls from visitors from different countries. She was very happy with my answers to her questions, and leaned toward me when this photo was taken.

• Gintaras is the Lithuanian word for amber. This is one of the amber shops in the Gintaras district of Vilnius. The Baltic Coast is the amber capital of the world, and Lithuanians makes money by exporting their amber products and by selling directly to foreign visitors.
1. In this Gintaras store, these ladies are very happy with me because I spent \$160 at their store.
2. Lithuanian Restaurant in the Gintaras district, with traditional menu.
3. Potato Dish. Potato is the staple for the Baltic area, as rice is for Asian countries. Thus, Lithuanians developed the art of making potato dishes beautiful and tasteful.
4. Lithuanian Pears are stretched.
5. Samsung. I was very happy to see this sign even in Vilinius. When I left Korea in 1954, Samsung was an obscure candy company. As far as electronics is concerned, Korea barely started making soldering irons. I was very happy to use a Korean-made soldering iron then.

Y. S. Kim (2008.10.10)

copyright@2008 by Y. S. Kim, unless otherwise specified. The portrait of Hermann Minkowski is from http://library.thinkquest.org/05aug/01273/whoswho.html.