Mathematical Physics

Abstract

The interplay between mathematics and physics needs no emphasis. What may need to be emphasized is that mathematics is not merely a tool with which the presentation of physics is facilitated, but the only medium in which physics can survive. Just as language is the means by which hu- mans can express their thoughts and without which they lose their unique identity, mathematics is the only language through which physics can ex- press itself and without which it loses its identity. And just as language is perfected due to its constant usage, mathematics develops in the most dra- matic way because of its usage in physics. The quotation by Weyl above, an approximation to whose translation is “In this drama of mathematics and physics—which fertilize each other in the dark, but which prefer to deny and misconstrue each other face to face—I cannot, however, resist playing the role of a messenger, albeit, as I have abundantly learned, often an unwel- come one,” is a perfect description of the natural intimacy between what mathematicians and physicists do, and the unnatural estrangement between the two camps. Some of the most beautiful mathematics has been motivated by physics (differential equations by Newtonian mechanics, differential ge- ometry by general relativity, and operator theory by quantum mechanics), and some of the most fundamental physics has been expressed in the most beautiful poetry of mathematics (mechanics in symplectic geometry, and fundamental forces in Lie group theory).