Supersymmetric Quantum Theory and ( Non - Commutative ) Differential Geometry

In this paper we describe an approach to differential topology and geometry rooted in supersymmetric quantum theory. We show how the basic concepts and notions of differential geometry emerge from concepts and notions of the quantum theory of non-relativistic particles with spin, and how the classification of different types of differential geometry follows the classification of supersymmetries. Historically, the development of geometry has been closely linked to that of classical physical theory. Geometry was born, in antique Greece, out of concrete problems in geodesy. Much later, the invention and development of differential geometry took place in parallel to the development of classical mechanics and electromagnetism. A basic notion in differential geometry is that of parameterized curves on a manifold. This notion is intimately related to the one of trajectories of point particles central in classical mechanics: Tangent vectors correspond to velocities, vector fields to force laws. There are many further examples of such interrelations. To mention one, the work of de Rham and Chern was inspired and influenced by Maxwell's theory of electromagnetism, a classical field theory. In this century, the foundations of physics have changed radically with the discovery of quantum mechanics. Quantum theory is more fundamental than classical physics. It is therefore natural to ask how differential geometry can be rediscovered – starting from the basic notions of quantum theory. In this work, we outline a plausible answer to this question. As a payoff, we find natural generalizations of differential geometry to non-commutative differential geometry, in the sense of Connes. Connes' theory naturally leads to a far-reaching generalization of geometry which makes it possible to study highly singular and genuinely non-commutative spaces as geometric spaces. Such generalizations are called for by deep problems in quantum mechanics, quantum field theory and quantum gravity. Quantum mechanics entails that phase space must be deformed to a non-commutative space; moreover, attempts to reconcile quantum theory with general relativity naturally lead to the idea that space-time is non-commutative. The idea to reconsider and generalize algebraic topology and differential geometry from the point of view of operator theory and quantum theory is a guiding principle in Connes' work on non-commutative geometry, the development and results of which are summarized in detail in [Co1 −4 ] and the references therein. Connes' work is basic for the approach presented in this paper. That supersymmetry and supersymmetric quantum field theory provide natural formulations of problems in differential …