Intellectica 2009/1, n° 51

The Mathematical Continuum. New Conceptions, New Challenges

Alain Comtet

Particles and Fields, Two Figures of the Discrete and the Continuum in Physical Theories

 Abstract: The theoretical framework within which modern physics has been built relies on the concept of field. We show how this concept which, in the first instance serves as a substitute for that of action at a distance, has gradually come to play a central role in the construction of relativistic quantum theories. The development of this framework has brought about a change in the meaning of the concepts of locality, causality and symmetry. Particles and fields appear as two aspects of the discrete and the continuum in terms of which the discrete versus continuum question can be formulated. We put forward the view that there is not just one continuum but rather a multiplicity of continua associated with the different layers of physical reality. How can this view be reconciled with that of elementarity ? We recall how this question arises in quantum field theory and how it is resolved in terms of notions developed in the study of critical phenomena. We discuss the conflict between two descriptions of the physical universe : one based on a reductionnist approach with the aim of uncovering the laws of elementarity, the other emphasising the notion of collective behaviour and that of emerging properties.

 Key words: Continuum-discrete antinomy, elementarity, emergence, fields and particles.