Equations for universal truth

July 28, 2000

Arthur Jaffe takes an energetic look at what makes mathematical physics the queen bee of science

Isaac Newton put mathematical physics on the map. Ever since, British scientists have played a pivotal role in its development, which has recently seen enormous change. From the tradition of Faraday, Maxwell and Dirac, to the modern interplay between physics and mathematics, science has witnessed one of the deepest intellectual syntheses imaginable.

Now the International Association of Mathematical Physics has chosen Imperial College, London, as the site of its year 2000 Congress, where about 900 specialists are speaking to each other and to the public.

The explosion of activity in this subject touches every nook and cranny of thinking on mathematics and theoretical physics. One hears energetic discussions of familiar and not so familiar themes of the quantum Hall effect, the foundations of quantum theory, quantum computing, quantum fields, renormalisation, membranes and strings, quantum geometry, gravity, invariants of manifolds, biophysics and even the theory of arithmetic.

Mathematical physicists seek to unravel the laws at the foundation of the universe, especially how one can unify quantum theory with the forces of nature including electromagnetism, elementary particles forces and gravity. Similarly, in statistical and condensed matter physics, scientists attempt to distil macroscopic physics from the microscopic. Ultimate mathematical truth emerges from these pictures and shapes the future of the "Queen of the Sciences".

While the individual lectures at this meeting tend to be specific, the sum total experiences cover the waterfront. The lecturers range from Nobel laureate Gerard 't Hooft, speaking on the nature of the quantum, to the Fields medallist Simon Donaldson, newly appointed as professor at Imperial, speaking on how ideas from physics lead mathematicians to new geometric concepts.

The varied lectures touch on new areas (often through cross-fertilisation of fields), they sometimes bridge the gap between the esoteric and the practical (as exemplified by talks on quantum computing) or they look to formulate new fundamental laws of nature (as in the theory of strings). Advances lead to new mathematics, to new theories and, in the best cases, both.

"Nature demonstrates that nothing is too beautiful to be true," said Fields medallist Alain Connes in his lecture for the general public on Tuesday evening. Connes sketched how early quantum theory inspired a recent revolution in geometrical thinking, showing that our classical notions of space and time could be complemented by a more subtle interpretation called "non-commutative geometry", in which one builds quantum theory into the fundamental notion of space.

Amazingly, this picture leads to the old and the new falling together into place. Connes described how the theory of renormalisation (owing much to Imperial physicist Abdus Salam in the 1950s and developed in the 1970s for modern gauge theories by 't Hooft and M. Veltman) turns out to be unified with traditional mathematical themes from Hopf, Riemann and Hilbert, ultimately connecting with the frontiers of the modern theory of numbers.

The striking point is that a few years ago, experts regarded these mathematical subjects as separate ones, and they were individually viewed to be as distant from the influence of physics as one could possibly imagine. While the unification of ideas has dominated the landscapes of major progress in both physics and mathematics for the past 40 years, it has progressed further than any early visionary of such progress imagined.

Given the complexity and the far-reaching nature of the subject - ranging from the mathematical embodiment of all physical laws to the discovery of new and abstract mathematical structures - the organisers of this congress have made a special effort to communicate the excitement.

First, they planned three special lectures designed to explain fundamental ideas to the general public, complementing the lecture of Connes with talks by David Ruelle and Sir Michael Atiyah, other leaders in the field.

Second, they have put on a series of symposia especially for young researchers, to present the personal experiences of senior scientists to serve as inspiration for younger workers.

It is heartening that there are so many vibrant researchers probing the unknown mysteries in mathematical physics. And it is certain that the spin-offs from this process will be a central theme in future science and will play a role in shaping the 21st century.

Arthur Jaffe, professor of mathematics and physics at Harvard University, is president of the Clay Mathematics Institute and past president of both the American Mathematical Society and the International Association of Mathematical Physics.

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