Geometrical Methods of Mathematical Physics

The recent most impressive developments in theoretical physics, i.e. gauge and string theories with their far- reaching relations to basic concepts of present day mathematics brought a renewed and intense exchange between mathematicians and physicists. In a way the past few years might be compared to the great period of the mid-twenties with its really decisive mathematical contributions to the foundations of both quantum theory as well as general relativity .. Among others, this far-reaching exchange led H.Weyl in 1929 to the formulation of his (second) gauge principle (concerning the phase of the quantum mechanical wave function) which led, in turn (through the most natural generalization by C.N.Yang and R.Mills in 1954) to present day gauge theory which brought, first of all, a real breakthrough with respect to a theoretical inter- pretation of an enormous realm of data in modern particle physics. It led, in fact, to the well-known unification of electromagnetism with weak interactions (Weinberg-Salam theory) and, at the same time, to a most natural unifica- tion of particle and nuclear physics through QCD which deals with the basic quark-gluon structure of heavy matter, e.g. hadrons.

The recent most impressive developments in theoretical physics, i.e. gauge and string theories with their far- reaching relations to basic concepts of present day mathematics brought a renewed and intense exchange between mathematicians and physicists. In a way the past few years might be compared to the great period of the mid-twenties with its really decisive mathematical contributions to the foundations of both quantum theory as well as general relativity .. Among others, this far-reaching exchange led H.Weyl in 1929 to the formulation of his (second) gauge principle (concerning the phase of the quantum mechanical wave function) which led, in turn (through the most natural generalization by C.N.Yang and R.Mills in 1954) to present day gauge theory which brought, first of all, a real breakthrough with respect to a theoretical inter- pretation of an enormous realm of data in modern particle physics. It led, in fact, to the well-known unification of electromagnetism with weak interactions (Weinberg-Salam theory) and, at the same time, to a most natural unifica- tion of particle and nuclear physics through QCD which deals with the basic quark-gluon structure of heavy matter, e.g. hadrons.

Have you ever wondered why the language of modern physics centres on geometry? Or how quantum operators and Dirac brackets work? What a convolution really is? What tensors are all about? Or what field theory and lagrangians are, and why gravity is described as curvature?