Unifying Classical and Quantum Physics

From the triviality problem in scalar field theories to the role of quantization in addressing singularities, this book connects fundamental mathematical concepts with pressing questions in modern physics. This book presents a comprehensive exploration of affine quantization and its implications for quantum mechanics, quantum field theory, and even quantum gravity. Canonical quantization has been a cornerstone of quantum mechanics, but it struggles with certain fundamental problems, such as the quantization of a particle in a box. A new approach, affine quantization, offers a more robust framework, addressing challenges that canonical methods cannot resolve. Drawing on the results of our collaboration, we develop a self-contained and rigorous exposition, complete with detailed proofs. Our goal is to provide a valuable reference for graduate students, mathematicians, and physicists seeking deeper insights into unresolved issues of the Standard Model—particularly the Higgs sector—and quantum gravity. From the triviality problem in scalar field theories to the role of quantization in addressing singularities, this book connects fundamental mathematical concepts with pressing questions in modern physics. By presenting both theoretical developments and numerical results, we offer readers a structured approach to exploring novel quantization methods and their impact on contemporary physics.