Synthesis Lectures on Engineering, Science, and Technology Quantum Spin Representations of the Poincaré Group, Part I

This book discusses how relativistic quantum field theories must transform under strongly continuous unitary representations of the Poincaré group. This book discusses how relativistic quantum field theories must transform under strongly continuous unitary representations of the Poincaré group. The focus is on the construction of the representations that provide the basis for the formulation of current relativistic quantum field theories of scalar fields, the Dirac field, and the electromagnetic field. Such construction is tied to the use of the methods of operator theory that also provide the basis for the formulation of quantum mechanics, up to the interpretation of the measurement process. In addition, since representation spaces of primary interest in quantum theory are infinite dimensional, the use of these methods is essential. Consequently, the book also calculates the generators of relevant strongly continuous one-parameter groups that are associated with the representations and, where appropriate, the corresponding spectrum. Part I of Quantum Spin and Representations of the Poincaré Group specifically addresses: conventions; basic properties of SO(2) and SO(3); construction of a double cover of SO(3); SU(2) spinors; continuous unitary representation of SU(2); basic properties of the Lorentz Group; unitary representation of the restricted Lorentz Group; an extension to a strongly continuous representation of the restricted Poincaré Group; and an extension to a unitary/anti-unitary representation of the Poincaré Group. In addition, this book: Connects mathematical results with their applications in physics, particularly in quantum field theory Provides mathematical rigor, introduces physical constants, and presents the dimensions of physical quantities Discusses how the use of methods from operator theory have become an indispensable tool for quantum field theory 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 8 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. This book discusses how relativistic quantum field theories must transform under strongly continuous unitary representations of the Poincaré group. The focus is on the construction of the representations that provide the basis for the formulation of current relativistic quantum field theories of scalar fields, the Dirac field, and the electromagnetic field. Such construction is tied to the use of the methods of operator theory that also provide the basis for the formulation of quantum mechanics, up to the interpretation of the measurement process. In addition, since representation spaces of primary interest in quantum theory are infinite dimensional, the use of these methods is essential. Consequently, the book also calculates the generators of relevant strongly continuous one-parameter groups that are associated with the representations and, where appropriate, the corresponding spectrum. Part I of Quantum Spin and Representations of the Poincaré Group specifically addresses: conventions; basic properties of SO(2) and SO(3); construction of a double cover of SO(3); SU(2) spinors; continuous unitary representation of SU(2); basic properties of the Lorentz Group; unitary representation of the restricted Lorentz Group; an extension to a strongly continuous representation of the restricted Poincaré Group; and an extension to a unitary/anti-unitary representation of the Poincaré Group.