The Interface of Noncommutative Geometry and Physics

As a mathematical theory, noncommutative geometry (NCG) is by now well established. From the beginning, its progress has been crucially influenced by quantum physics: we briefly review this development in recent years. The standard model of fundamental interactions, with its central role for the Dirac operator, led to several formulations culminating in the concept of a real spectral triple. String theory then came into contact with NCG, leading to an emphasis on Moyal-like algebras and formulations of quantum field theory on noncommutative spaces. Hopf algebras have yielded an unexpected link between the noncommutative geometry of foliations and perturbative quantum field theory. The quest for a suitable foundation of quantum gravity continues to promote fruitful ideas, among them the spectral action principle and the search for a better understanding of “noncommutative spaces”.