At the Second World Congress of Mathematicians in Paris in 1900, in a seminal lecture, Hilbert formulated his famous 23 open problems.50 The hundredth anniversary of Hilbert’s lecture was celebrated in Paris, in the
49These sums correspond to Feynman functional integrals.
50See F. E. Browder (Ed.) Mathematical Developments Arising from Hilbert’s Problems, Amer. Math. Soc., New York, 1976, and B. Yandell, The Honors Class:
Hilbert’s Problems and Their Solvers, Peters, Natick, Massachusetts, 2001.
“Amphith´eatre” of the Coll`ege de France, on May 24, 2000. The Scientific Advisory Board of the newly founded Clay Mathematics Institute (CMI) in Cambridge, Massachusetts, U.S.A., selected seven Millennium prize prob- lems. The Scientific Advisory Board consists of Arthur Jaffe (director of the CMI, Harvard University, U.S.A), Alain Connes (Institut des Hautes ´Etudes Scientifiques (IH´ES) and Coll`ege de France), Andrew Wiles (Princeton Uni- versity, U.S.A.), and Edward Witten (Institute for Advanced Study, Prince- ton, U.S.A.). The CMI explains its intention as follows:
Mathematics occupies a privileged place among the sciences. It embodies the quintessence of human knowledge, reaching into every field of human endeavor. The frontiers of mathematical understanding evolve today in deep and unfathomable ways. Fundamental advances go hand in hand with discoveries in all fields of science. Technological applications of mathemat- ics underpin our daily life, including our ability to communicate thanks to cryptology and coding theory, our ability to navigate and to travel, our health and well-being, our security, and they also play a central role in our economy. The evolution of mathematics will remain a central tool to shap- ing civilization. To appreciate the scope of mathematical truth challenges the capabilities of the human mind.
In order to celebrate mathematics in the new millennium, the CMI has named seven “Millennium prize problems”. The Scientific Advisory Board of the CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a $ 7 million prize fund to these problems, with $ 1 million allocated to each.
The seven Millennium prize problems read as follows:
(i) The Riemann conjecture in number theory on the zeros of the Riemann zeta function and the asymptotics of prime numbers.
(ii) The Birch and Swinnerton–Dyer conjecture in number theory on the relation between the size of the solution set of a Diophantine equation and the behavior of an associated zeta function near the critical points= 1.
(iii) The Poincar´e conjecture in topology on the exceptional topological structure of the 3-dimensional sphere.
(iv) The Hodge conjecture in algebraic geometry on the nice structure of projective algebraic varieties.
(v) The Cook problem in computer sciences of deciding whether an answer that can be quickly checked with inside knowledge, may without such help require much longer to solve, no matter how clever a program we write.
(vi) The solution of the turbulence problem for viscous fluids modelled by the Navier–Stokes partial differential equations.
(vii) The rigorous mathematical foundation of a unified quantum field theory for elementary particles.
A detailed description of the problems can be found on the following Internet address:
http://www.claymath.org/prize−problems/
For a detailed discussion of the seven prize problems, we refer to K. De- vlin, The Millennium Problems: the Seven Greatest Unsolved Mathematical Puzzles of Our Time, Basic Books, New York, 2002.
Elementary Particles
First Law of Progress in Theoretical Physics: You will get nowhere by crunching equations.
Second Law:Do not trust arguments based on the lowest order of pertur- bation theory.
Third Law: You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you’ll be sorry.1
Steven Weinberg, 1983 For the motivation and convenience of the reader, let us sketch important basic ideas of elementary particle physics in this chapter. It is our philosophy that before studying a complex subject, one should know the main goals in nontechnical terms. As an introduction to particle physics, we recommend the textbooks by Nachtmann (1990), Coughlan and Dood (1991), Sibold (2001), and Seiden (2005). For the history of elementary particle physics, see the beautiful books by the two Nobel laureates Steven Weinberg (1983), (1995) and Martinus Veltman (2003).
In the 1960s and early 1970s, Gell-Mann and Fritzsch, Glashow, Salam, and Weinberg founded the so-called Standard Model of particle physics which is of fundamental importance for modern physics.2 The Standard Model of particle physics is based on
• the principle of critical action, and
• the principle of local symmetry (gauge theory).
The Lagrangian density of the Standard Model will be thoroughly studied in Volume III. It turns out that the Standard Model of particle physics is governed by the same mathematical approach as Einstein’s theory of general relativity on gravitation from 1915. The common mathematical tool is the theory of curvature in modern differential geometry (called gauge theory in physics). A survey on the Standard Model in particle physics and its possible generalizations along with a summary of the most important literature can be found in
1 S. Weinberg, Why the renormalization group is a good thing. In: A. Guth, K.
Huang, and R. Jaffe (Eds.), Asymptotic Realms of Physics: Essays in Honor of Francis Low, MIT Press, Cambridge, Massachusetts, 1983, pp. 1–19 (reprinted with permission).
2 The Nobel prize in physics was awarded to Gell-Mann in 1969 and to Glashow, Salam, and Weinberg in 1979.
J.Rosner, Resource letter SM-1: The standard model and beyond, 2003.
Internet:http://arXiv:hep-ph/0206176
This serves as a survey on modern physics including the following topics:
quarks and leptons, the Higgs mass, CP violation, strong CP problem and massless axions, dynamics of heavy quarks, precision electroweak measure- ments, neutrino oscillations and neutrino masses, grand unification of inter- actions and extended gauge groups, proton decay, baryon asymmetry of the universe, supersymmetry, the riddle of dark matter and dark energy in the present universe, and string theory. Up-dated particle data are summarized by
Particle Data Group. Internet:http://pdg.lbl.gov
For the cosmic microwave background radiation and its information on the early universe see
NASA home page, WMAP, Internet:http://www.nasa.gov/home/
The WMAP (Wilkinson Microwave Anisotropy Probe) satellite experiment of NASA allows us to see the state of the universe at the age of 400 000 years after the Big Bang. In particular, the WMAP experiment shows that our universe is 13.7ã109 years old. The five ages of our expanding universe starting from the Big Bang are studied in Adams and Laughlin (1997), (1999) (inside the physics of eternity). The five ages of our universe read as follows:
• the primordial era (from the Big Bang until the age of 105 years),
• the stelliferous era of the present universe (106–1014 years),
• the degenerate era (1015–1039years) (brown and white dwarfs, neutron stars and black holes dominate the universe),
• the black hole era (1040–10100 years) (black holes dominate the universe), and
• the dark era (>10101 years).
It turns out that the expansion of our universe is accelerated. In the final dark era, the energy density of the universe goes to zero after the vaporization of the last black holes by Hawking radiation. However, it is possible that a new Big Bang is generated by large quantum fluctuations of the ground state (vacuum) of the universe. For modern astrophysics and cosmology, we refer to Schutz (2003) (phenomenology), Shore (2003) (tapestry of astrophysics), B¨orner (2003) (early universe), and Straumann (2004) (general relativity and astrophysics).