Geometry at the Quantum Scale: Understanding the Geometric Language of Relativistic Quantum Mechanics

This book presents a single geometric language for the fundamental physics that has been discovered from the beginning of the twentieth century to the present day. The first two chapters give a brief summary of the four known fundamental forces, namely the three known gauge forces (the electromagnetic and the two nuclear forces) and the gravitational force. These chapters also summarize the main experimental results from high energy physics, including the implications of the existence of the Higgs particle discovered at the LHC, and the implications of recent astronomical observations. The subsequent chapters include quantum gravity based on the consequences of the't Hooft-Veltman renormalization theorem for gauge fields and the non-renormalization of Einstein's gravitational field of General Relativity.

The main focus of the book is to show that the three gauge forces are defined in the quantum domain, while Einstein's gravitation remains an essentially classical interaction. However, the main implication of Einstein's reasoning is built into the Einstein-Hilbert action principle, which is independent of the source of gravitation. As such, the Einstein-Hilbert principle by itself does not depend on any specific source, leading to a gravitational field that is far more general than that defined by General Relativity, while retaining all its geometrical characteristics. This makes a key difference in the sense that all four fundamental interactions may now be defined in the same quantum domain. Einstein's classical theory of General Relativity becomes a particular case of the quantum gravitational field defined by the Einstein-Hilbert action.

At the quantum scale, the direct sum of the resulting four geometries produces an 11-dimensional  Riemannian geometry whose curvature defines a new quantum cosmology without hierarchies, so that all fundamental interactions contribute effectively to the evolution of the universe, a result that is of interest to high energy physicists, cosmologists, mathematicians, philosophers and all those who seek a plausible explanation for the physical world.