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

by Marcos D. Maia, Edmundo M. Monte

Overview

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.

Product Details

ISBN-13:	9783031611964
Publisher:	Springer Nature Switzerland
Publication date:	10/04/2025
Edition description:	2024
Pages:	130
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Marcos Duarte Maia was born in October 1940 in the Brazilian state of North Rio Grande. He studied Engineering at the State University of Goia's (Brazil) from 1962 to 1964 and obtained a BSc degree in Physics in 1966 at the University of Brasilia under the supervision of Jayme Tiomno.
He moved to London in 1967 to work with Hermann Bondi on gravitational waves as a research student at the Kings's College of the University of London. Soon after, Bondi moved to the ESRO program in Vienna, and his supervision was transferred to Professor Clive Kilmister. In 1971, he defended a thesis on the subject of Eddington's Fundamental Theory.
After his return to Brazil, he was hired by the Department of Mathematics of the University of Brasilia in 1971, where he stayed for the next 20 years, before moving to the Physics Institute of the same university.

In 1985 he was a visiting scientist at the University of Washington, Seattle; a visiting Researcher at the Institute of Theoretical physics at Santa Barbara, California, and a Visiting Researcher at Stanford Linear Accelerator (SLAC). Finally in 1993 he was a visiting Scientist at the NASA-Fermilab Astrophysics Center near Chicago. His present position is that of Senior Research Associate at the Institute of Physics of the University of Brasilia.

Edmundo M. Monte has a Bachelor's degree in Mathematics from the Federal University of Paraíba (UFPB), Brazil, a Master's degree in Atmospheric Sciences from UFPB, Brazil, and a PhD in Mathematics from the University of Brasília (UnB), Brazil, with a thesis in General Relativity. He held a post-doctorate position in Relativity at the Department of Geometry and Topology, Granada University (UGR), Spain, and a Visiting Professorship with a Mathematics Research Fellowship at the International Center for Theoretical Physics (ICTP), Italy; He was also a Visiting Professor at UGR, Spain. He entered the UFPB through competition as a Professor in the Department of Mathematics at UFPB. He is currently a Full Professor at the Department of Physics at UFPB, João Pessoa, Brazil. His research experience is in the area of Physics and Mathematics, with an emphasis on Fundamental Geometric Problems in Spacetime Immersions, working mainly on the following topics: immersions of pseudo-Riemannian manifolds, space-time immersions, geometry and topology of space-times, special and general relativity, cosmology and gravitation.

Table of Contents

Chapter 1 A Brief Review on Gauge Forces.- Chapter 2 Gravitation and Cosmology.- Chapter 3 Quantum Gravity.- Chapter 4 Renormalizable Einstein’s Gravitation.- Chapter 5 The Renormalizable Einstein’s Gravitation.- Chapter 6 Symmetry Mixing.- Chapter 7 The Equations of the Standard Model.- Chapter 8 The Relativistic Poincaré Conjecture.- Chapter 9 The Higgs Roulette.- Chapter 10 Quantum Black Holes at the LHC.