The interaction between geometry and physics. (English) Zbl 1118.58002

Etingof, Pavel (ed.) et al., The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31–September 4, 2003. Boston, MA: Birkhäuser (ISBN 0-8176-4076-2/hbk). Progress in Mathematics 244, 1-15 (2006).
This is a slightly extended version of the lecture delivered at Harvard. The lecture represents the unity of mathematics and physics – “the unreasonable effectiveness of mathematics in physics” and “the unexpected effectiveness of physics in mathematics”.
The author starts by reviewing rapidly the parts of geometry and of physics which have been involved in the interaction. He discusses the dimensional hierarchy in the Kaluza-Klein theory; rigidity theorems, moduli spaces of bundles, moduli spaces of curves, intersection theory and mirror symmetry in quantum field theory. The most striking application of quantum field theory in three dimensions was undoubtedly Witten’s interpretation of the polynomial knot invariants discovered by Vaughan Jones. The author analyzes Donaldson invariants. As part of some very general ideas of duality in quantum field theories, Seiberg and Witten produced a quite different theory which was expected to be equivalent to Donaldson theory. There are a large number of important areas where quantum theories yields topological results. The author discusses some aspects of topological theories.
The connection between geometry and physics extends to string theories. It should be clear that quantum theory, in its modern form, has had profound consequences for mathematics and in particular for geometry. The author tries to peer into future to see what kind of mathematics might emerge from the current geometry-physics interface. Trying to forecast the physics is even harder and the author offers a different perspective. As we know, the holy grail in current fundamental physics is how to combine Einstein’s theory of general relativity with quantum theory. These two theories operate very effectively but at quite different scales, GR at cosmic distances and QM at subatomic scales. The difficulty in combining the two theories is both conceptual and technical.
As is well known, Einstein dreamt of a unified geometric theory, extending GR, and he never accepted the philosophical foundations of QM, with its uncertainty principle. In the long debate on this controversy between Einstein and Bohr the general verdict of the physics community was that Einstein lost and that his idea of a unified field theory was a hopeless pipe dream. With the remarkable success of the standard model of elementary particles, incorporating geometrically the electromagnetic, the weak force and the strong force, Einstein’s ideas were given new life. Now, with string theory offering the hope of the ultimate unification it might appear that the old controversy between Einstein and Bohr has been resolved, with the honours more equally split. Unification is perhaps being achieved, but QM has persisted. But it is at least worth exploring alternative scenarios.
There are in particular two attractive ideas that have their devotees. The first (in historical precedence) is Roger Penrose’s twistor theory. On the one hand this has, as a technical mathematical tool, proved its worth in a number of problems. It is also related to supersymmetry and duality. Links with string theory are being explored. But beyond these mathematical technicalities there lies a deeper philosophical idea. Penrose is an Einsteinian who believes that in the hoped-for marriage between GR and QM it is the latter that must give the most, adapting itself to the beauty of GR. Twistors are thought of as a first step to achieving this goal. Moreover, Penrose speculates that the mysterious role of complex numbers in QM should ultimately have a geometric origin in the natural complex structure of the base of the light-cone in Minkowski space. So far, it has to be conceded that the weight of evidence is not in Penrose’s favor, but that does not mean that he may not ultimately be vindicated.
A completely different scenario is offered by Alain Connes’ noncommutative geometry, a theory with a rich mathematical background and a promising future. Links with physics exist and new ones are being discovered. In a sense Connes takes off from the Heisenberg commutation relations, in a definitely non-Einsteinian direction. However, he tries to keep the geometric spirit by using the same concepts and terminology. It is certainly possible that the final version of M-theory may use Connes’ framework for its formulation.
For the entire collection see [Zbl 1083.00015].


58-02 Research exposition (monographs, survey articles) pertaining to global analysis
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory