by Eberhard Zeidler
There are problems mathematicians have been seeking the solution to for thousands of years, including Faust's question ,,whatever holds the world together in its inmost folds". One of the most profound interrelations known today can be outlined with the words
force = curvature
Since ancient times, physicists have been trying to understand the forces (or interactions) in the world while mathematicians have attempted to describe the curvature of curves and surfaces. Today, we know that physicists and mathematicians have been pursuing the same objective for a long time without being aware of it. For instance, Gauss surveyed land in the Kingdom of Hanover from 1821 to 1825 with great physical effort. He then asked himself whether it is possible to determine the curvature of a surface such as the surface of the earth just by making measurements on it, but without using the space surrounding it. Gauss only found an affirmative answer after making long and futile attempts in the form of a complicated formula that he termed the theorema egregium, i.e. the wonderful theorem. When Bernhard Riemann (1826-1866) was reading his post-doctoral paper in 1854 in the presence of his venerable teacher Gauss, he demonstrated how the Gaussian theory of surfaces can be generalised to higher dimensions. After his speech, Gauss said he was extremely impressed by what Riemann had said. More than 60 years later, Einstein used Riemann's deliberations to formulate his general theory of relativity replacing Newton's gravitational force with the curvature of the 4-dimensional space-time continuum . All of the gravitational processes in the universe can be described with one single magic formula:
This is the Einstein-Hilbert principle of the least action (or critical action, to put it more generally) for the universe. Here, R is the curvature of the universe, and encodes the distribution of mass and energy of the stars and galaxies. Coding the diversity of the processes taking place in the universe in one single formula is a remarkable achievement of the human spirit. This also includes the existence and properties of black holes and the expansion of the universe after the Big Bang. The mathematician can demonstrate that the magic formula (1) represents the simplest conceivable formula on a sufficiently high level of abstraction. Throughout his entire life, Einstein searched for a unified theory of matter, unfortunately without success. Today we know the standard model of particle physics where the principle of "force = curvature" is realised on the scale of the elementary particles. In the context of the standard model, we can use the magic curvature formula
on a sufficiently high level of abstraction to describe the electromagnetic interaction, the weak interaction responsible for radioactive decay and the strong interaction that keeps the atomic nuclei. Long before the standard model was postulated, the magic formula was known to the mathematicians as Cartan's structural formula that generalises the Gaussian theorema egregium. Two physicists, Wu from Harvard University and Yang from State University New York, Stony Broke, only discovered the connection between the different paths that mathematician and physicists take, in 1975. Both of them published a short dictionary in the Physical Reviews that made it possible to translate the different terms developed by mathematicians and physicists in the framework of the gauge field theory. One of the greatest unsolved problems of modern physics is finding a comprehensive theory encompassing Einstein's general theory of relativity and the standard model of elementary particles as approximations with the appropriate scale of energy. Many scientists all over the world are working on this problem intensively and there is a fruitful flow of ideas between physics and very abstract parts of modern mathematics. A case in point is the string theory, where elementary particles are replaced with oscillating strings as on a violin. The physicist Edward Witten (born in 1951) who works at the Institute for Advanced Study in Princeton has used his physical intuition and abundance of ideas to bring completely new deliberations into mathematics. In 1990, he received the highest mathematical award, the Fields Medal, that was conferred upon Gerd Faltings (born in 1954) from the Max Planck Institute for Mathematics in Bonn four years before.
One of mathematics' strengths is the high degree of abstraction of modern mathematics that enables us to write complicated things in a simple and clearly expressed form. Unfortunately, this is also something that estranges mathematicians from many people who are not capable of decoding these formulas. This is the reason why we cannot use formulas to communicate on mathematics with the public at large. There is a world congress of mathematicians every four years and there was an easily understandably written essay on mathematicians and mathematics by the poet Hans Magnus Enzensberger at the world congress in Berlin in 1998. It bore the title "Drawbridge up: Mathematics - a cultural anathema". In it, Enzensberger points out that people rarely brag that they don't understand anything about painting, poetry or music. However, one thing most people agree about is the fact that they don't like mathematics. And the mathematician probably has a fair share of the blame for this. The German Association of Mathematicians undertakes great efforts to improve the image of mathematics in the public eye. There has to be mathematical education in schools that schoolchildren enjoy and that is focused on interesting and everyday questions in spite of the difficulty of the material. Einstein once said that we should make things as simple as possible, but not simpler. If we take a look at a score of Johann Sebastian Bach, the first thing you see is the formal language of notes that many people simply don't understand, a cornucopia of formal structures as in a fugue. But, Bach's music is a great deal more. It makes your soul vibrate, and that's how the mathematician feels about his or her science. Mathematics is a valuable part of human culture just like painting, poetry and music.
It is not only the fundamental forces that play a key role in nature, but also passing on information in such things as hereditary information. In 1948, the American electrical engineer and mathematician Claude Shannon (1916-2001) created a new branch of mathematics, namely information theory. Shannon dealt with the question of efficiently transferring information in communication channels. The magic formula
is at the top of the standard model of information theory, that strangely enough is based upon the notion of probability. The numberk is a normalizing constant. An experiment in probability is carried out in this model that has exactly n possible outcomes , realised with the corresponding probabilities . For n = 2 we could imagine tossing a coin with the outcomes of "heads" or "tails" and they have the probability . Shannon maintains that you have gained the informationS when you know the outcome of the experiment. However, that doesn't seem to have a great deal to do with our intuitive idea of what information is. But, it can be shown that this concept of information is a measure of the average number of questions with yes-no answers necessary for finding out what the outcome of the experiment is. For instance, it is sufficient in our coin experiment to pose one single question. Physicists have been familiar with the concept of information for more than one hundred years, but in a completely different framework. There they called it entropy. In 1824, the French engineer Sadi Carnot (1796-1832) calculated the optimal efficiency of steam engines when transforming heat energy into mechanical energy. In this process, he came upon a concept that Rudolf Clausius (1822-1888) called entropy in 1865. The second main theorem of thermodynamics formulated by Clausius states that the entropy of the universe strives towards a maximum, meaning that all structures disintegrate. An example was given of this recently when astrophysicists simulated the behaviour of a white dwarf star with a supernova explosion on computers using methods of turbulence theory so that supernovae explosions of white dwarves can be used to determine the distance of galaxies in the depths of the universe. This also allows us to measure the red shift in the spectra of galaxies. The observations ascertained by the Hubble Telescope in this framework show that our universe will accelerate its expansion and never contract again. There have been about 15 billion years since the Big Bang. All stars will have been extinguished in 100,000 billion years, which satisfies the requirements of Clausius' thermal death. Of course, there is the theoretical possibility that a new Big Bang might be ignited in a far-off future if there are sufficiently great quantum fluctuations of the ground state of our universe.
About the Autor
Prof. Dr. Eberhard Zeidler wurde 1940 in Leipzig geboren. Dort studierte er Mathematik und Physik. 1974 wurde er zum ordentlichen Professor für Analysis an die Universität Leipzig berufen. Zusammen mit Prof. Dr. Jürgen Jost und Prof. Dr. Stefan Müller gründete er 1996 das Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig und war von 1996 bis 2003 dessen geschäftsführender Direktor. Er ist Mitglied der Deutschen Akademie der Naturforscher Leopoldina. Für sein Lebenswerk erhielt er den Alfried Krupp Wissenschaftspreis 2006 der Alfried Krupp von Bohlen und Halbach-Stiftung.
Prof. Zeidler starb im November 2016.
Figure 1: Einstein's Cross