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Abstract
The work of Bernhard Riemann is discussed under the perspective of present day mathematics and physics, and with a prospective view towards 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: (i) 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; (ii) 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 ps
Highlights
A proper discussion of Riemann’s work would no doubt need much more space and time than the few pages at disposal here
FINAL CONCLUSIONS Let us finish this short overview of Riemann’s work and its important uses in modern Physics—a clear example of the extraordinarily fruitful interrelation between the worlds of Physics and Mathematics—with a touching sentence that appears in a letter written by Albert Einstein and addressed to Arnold Sommerfeld, of the year 1912—this means, some 60 years after the celebrated Habilitationschrifft of Bernhard Riemann—where Einstein comments on the efforts he is doing in trying to understand Riemannian Geometry: “Aber eines ist sicher, dass ich mich im Leben noch nicht annähend so geplagt habe und dass ich große Hochachtung vor der Mathematik eingeflößt bekommen habe, die ich bis jetzt in ihren subtileren Teilen in meiner Einfalt für puren Luxus gehalten habe!”
I could have chosen other icons, that are by general agreement much more archetypical examples for a mathematical physicist
Summary
A proper discussion of Riemann’s work would no doubt need much more space and time than the few pages at disposal here. Ueber die Fläche vom kleinsten Inhalt bei gegebener Begrenzung, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13 (1868) Six among these fifteen papers (namely, those with numbers 2, 3, 8, 9, 10, 14) are the ones that I have selected because they directly address issues of theoretical and experimental physics. This paper is generally considered to incorporate the main results of Riemann’s physical (and philosophical) ideas on the “unification” of gravity, electricity, magnetism, and heat It contains his observation on how a theory of electricity and magnetism is closely related with those for the propagation of light and heat radiation. This is, in my view, to push to an extreme the opinion that I defended in the introduction, which was a lot more moderate and faceted
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