Geometry at the Quantum Scale

This book presents a geometric perspective on the ever-expanding landscape of (often unfalsifiable) theories of fundamental physics that have emerged from the beginning of the twentieth century until the present day. The authors present a summary of the landmark results, and the assumptions needed to obtain them, viewed with the common philosophy that many difficulties in fundamental physics could be elucidated by an understanding of the geometrical language of quantum physics, motivated initially by the geometric implications of the 't Hooft-Veltman renormalization of gauge fields. The geometrical language underpinning this worldview is affine geometry, a common feature of the four known fundamental interactions, where affinity between observables is mediated by a fundamental interaction, as in the example of local gauge fields in quantum field theory. The same can be said of Einstein's gravitational field based on the (pseudo-)Riemannian geometry where the affinity comes from the metric. Such notions allow one to interpret affinity at the quantum scale of observations, and consider questions such as: What is the meaning of the Planck regime? What is the meaning of the cosmological principle? What is quantum gravity? Is the Newtonian gravitational constant really universal? Is information lost in quantum gravity? The book reviews fundamental background material, presents manifold insights into the geometrical nature of quantum physics and cosmology, suggests promising avenues for future research, and in places even proposes experimental tests. It will be of interest to high energy physicists, cosmologists, mathematicians, and philosophers of science.