Modern Mathematical Methods of Quantum Theory: An Introduction with Functional Analysis, Banach Algebras, and Functional Calculus (Synthesis Lectures on Engineering, Science, and Technology) - Hardcover

Beyer, Horst Reinhard

Hardcover

ISBN 10: 3032401534 ISBN 13: 9783032401533

Publisher: Springer, 2026

Synopsis

This book focuses on providing the functional analytic basis for quantum theory, including the functional calculi for densely-defined, linear, and self-adjoint operators in complex Hilbert spaces. The field of functional analysis focuses on the study of infinite dimensional vector spaces. Since its creation in the beginning of the 20th century, it has developed into a vast field, providing the mathematical foundation for quantum theory, including the theory of relativistic quantum fields, the basis for operator theory, the theory of partial differential equations, and the theory of distributions. The author uses an approach that employs Banach algebra methods, which provides a foundation for the theory of operator algebras as well as applications in the theory of relativistic quantum fields. The book develops the basic theory of Banach algebras up to the proof of the Gelfand-Naimark theorem and uses the latter theorem for the definition of the continuous functional calculus for bounded normal operators. Full solutions to the presented problems are included, and the majority of the corresponding results are used throughout the book.

About this title

About the Author

Horst R. Beyer, Ph.D., is currently affiliated with the University of Tuebingen Institute for Astronomy and Astrophysics, Theoretical Astrophysics Division. Dr. Beyer has written nine books and 39 published articles. His research interests include mathematical physics, in particular the applications of operator theory in quantum field theory, general relativity, astrophysics, and the engineering sciences.

From the Back Cover

This book focuses on providing the functional analytic basis for quantum theory, including the functional calculi for densely-defined, linear, and self-adjoint operators in complex Hilbert spaces. The field of functional analysis focuses on the study of infinite dimensional vector spaces. Since its creation in the beginning of the 20th century, it has developed into a vast field, providing the mathematical foundation for quantum theory, including the theory of relativistic quantum fields, the basis for operator theory, the theory of partial differential equations, and the theory of distributions. The author uses an approach that employs Banach algebra methods, which provides a foundation for the theory of operator algebras as well as applications in the theory of relativistic quantum fields. The book develops the basic theory of Banach algebras up to the proof of the Gelfand-Naimark theorem and uses the latter theorem for the definition of the continuous functional calculus for bounded normal operators. Full solutions to the presented problems are included, and the majority of the corresponding results are used throughout the book.

In addition, this book:

• Discusses the process of quantization of a classical physical system and the measuring process in quantum theories
• Presents Banach algebra methods and provides a basis for subsequent study of operator algebras
• Includes a functional analytic foundation for learning quantum theories, including relativistic quantum field theories

Edition details

• Publisher: Springer
• Publication date: 2026
• Language: English
• ISBN 10: 3032401534
• ISBN 13: 9783032401533
• Binding: Hardcover
• Number of pages: 339