Mathematical Physics

Although Riemann is mainly remembered for his groundbreaking work in creating geometric function theory, his larger interests suggest that he was first and foremost a mathematical physicist. This is not only apparent in the many connections with his predecessors -- Gauss, Dirichlet, and Wilhelm Weber -- but also in Riemann's teaching activity, as well as several papers he published on physical topics. Rather than attempting to describe these works, the present chapter charts a course through several of the most important fields of physics that informed Riemann's overall picture of the physical world. It begins with the central role of potential theory, which Gauss first cast into a general mathematical theory. Although a quintessentially physical theory that applied to physical forces in space, Riemann quickly grasped its relevance for the 2-dimensional case in the context of his geometric theory of complex functions. He only learned somewhat later about Dirichlet's contributions to this theory, which motivated him to dub his key tool for establishing the existence of uniquely defined analytic functions the Dirichlet principle. Potential theory was also directly applicable to the force laws for magnetism as well as for electrically charged bodies. Terrestrial magnetism had long been at the center of the collaborative work undertaken by Gauss and Weber, a tradition Riemann quickly absorbed studying under Benjamin Goldschmidt. When Riemann returned to Göttingen in 1849, he continued to pursue this interest alongside ongoing experiments connected with Weber's theory of electrodynamics. Thus, throughout the early 1850s, geomagnetism and electrical phenomena dominated his research interests, complemented by speculative ideas relating to microphysics that he hoped to develop into a mathematical theory complementing Newtonian mechanics. These latter interests remained largely invisible during his lifetime, however, since he never published any of his more speculative writings. On the other hand, Riemann's physical speculations went hand-in-hand with his reflections on geometrical space, as laid out in his Habilitation lecture. William Kingdon Clifford immediately seized on the significance of this text, which he translated for Nature. He later expounded on this new dynamical conception of space as a framework for a new physics, but unfortunately Clifford, too, died at a young age.