Intellectica 2009/1, n° 51

The Mathematical Continuum. New Conceptions, New Challenges

Marc Lachièze-Rey

The Continuity of Physics

 Abstract: The Newtonian physics introduces continuous space and time, but considers matter, under the form of corpuscles, as discontinuous. The theories of relativity also represent the space-time as a continuum. The nature of light was discussed since before the XVIIth century : is it made of particles, as suggested by Newton, or of waves ? The latter conception, however, involved an electromagnetic ether whose properties remained elusive. This was the electromagnetic version of the continuous / discontinuous opposition. At the beginning of the XIXth century, interferometry suggested to see the light as a wave, more precisely as an electromagnetic wave. This appeared as a victory of the continuous vision. But in the beginning of the XXth century, the quantum revolution introduced a completely new vision which discarded both [classical] waves and corpuscles for the light : it is now described by new entities, wave functions or quantum fields. Subsequently, it appeared that the same applied to matter. In some sense, this resolved the debate  by showing that continuity and discreteness (under the form of quantization) were both present in matter and radiation. From the mathematical point of view, this situation could be described either geometrically, through non commutative geometry or algebraically with the help of operators. This was not the end of the story. Various arguments suggest that our present description of the nature is not satisfactory, and that we must search for a more complete and unified physical theory. Most physicists estimate that it should involve, in one form or an other, a quantization of gravitation, that may be provided by a quantization of geometry itself. This motivates an important part of present research, which considers various approaches like quantum geometry, loop quantum gravity, spin networks and causal networks, dynamical triangulations. Each of them introduces new conceptions of space and time, were these entities also appear as quantified : like matter in the original quantum theory, they incorporate continuous and discontinuous characteristics. We do not know by now what will be the best theory. Work is in progress.

 Key words: Light, matter, radiation, space, time, space-time, quantum physics, relativity theory, quantum gravity, non commutative geometry.