Bernhard Riemann, a(rche)typical mathematical-physicist?

Abstract

The work of Bernhard Riemann is discussed under the perspective of present day mathematics and physics, and with a prospective view toward the future, too. Against the (unfortunately rather widespread) trend—which predominantly dominated national scientific societies in Europe during the last century—of strictly classifying the work of scientists with the aim to constrain them to separated domains of knowledge, without any possible interaction among those and often even fighting against each other (and which, no doubt, was in part responsible for the decline of European in favor of American science), it will be here argued, using Riemann as a model, archetypical example, that good research transcends any classification. Its uses and applications arguably permeate all domains, subjects and disciplines one can possibly define, to the point that it can be considered to be universally useful. After providing a very concise review of the main publications of Bernhard Riemann on physical problems, some connections between Riemann's papers and contemporary physics will be considered: (1) the uses of Riemann's work on the zeta function for devising applications to the regularization of quantum field theories in curved space-time, in particular, of quantum vacuum fluctuations; (2) the uses of the Riemann tensor in general relativity and in recent generalizations of this theory, which aim at understanding the presently observed acceleration of the universe expansion (the dark energy issue). Finally, it will be argued that mathematical physics, which was yet not long ago a model paradigm for interdisciplinary activity—and had a very important pioneering role in this sense—is now quickly being surpassed by the extraordinarily fruitful interconnections which seem to pop up from nothing every day and simultaneously involve several disciplines, in the classical sense, including genetics, combinatorics, nanoelectronics, biochemistry, medicine, and even psychology, with such intriguing issues that include, e.g., artificial life and the modelization of the process of consciousness.