Modern Mathematical Methods of Quantum Theory

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.