This book focuses on unstable systems both from the classical and the quantum mechanical points of view and studies the relations between them. The first part deals with quantum systems. Here the main generally used methods today, such as the Gamow approach, and the Wigner-Weisskopf method, are critically discussed. The quantum mechanical Lax-Phillips theory developed by the authors, based on the dilation theory of Nagy and Foias and its more general extension to approximate semigroup evolution is explained. The second part provides a description of approaches to classical stability analysis and introduces geometrical methods recently developed by the authors, which are shown to be highly effective in diagnosing instability and, in many cases, chaotic behavior. It is then shown that, in the framework of the theory of symplectic manifolds, there is a systematic algorithm for the construction of a canonical transformation of any standard potential model Hamiltonian to geometric form, making accessible powerful geometric methods for stability analysis in a wide range of applications.

This book deals with underlying basic concepts in relativity. The fundamental work of Stueckelberg, who formulated a consistent relativistic classical and quantum dynamics, generalized for application to many-body systems by Horwitz and Piron (SHP theory), is explained with emphasis on its conceptual content. The two-body bound state and scattering theory are also discussed. The ideas are involved in the Lindner experiment showing interference in time and the proposed experiment of Palacios et al. searching for the persistence of entanglement at unequal times is discussed. The meaning of the Newton-Wigner position operator and the Landau-Peierls construction in terms of relativistic dynamics is given. Finally, the embedding of the SHP theory into the framework of general relativity, providing a canonical structure with particle coordinates and momenta, is studied, carrying with it new concepts in relativistic dynamics.

This book focuses on unstable systems both from the classical and the quantum mechanical points of view and studies the relations between them. The first part deals with quantum systems. Here the main generally used methods today, such as the Gamow approach, and the Wigner-Weisskopf method, are critically discussed. The quantum mechanical Lax-Phillips theory developed by the authors, based on the dilation theory of Nagy and Foias and its more general extension to approximate semigroup evolution is explained. The second part provides a description of approaches to classical stability analysis and introduces geometrical methods recently developed by the authors, which are shown to be highly effective in diagnosing instability and, in many cases, chaotic behavior. It is then shown that, in the framework of the theory of symplectic manifolds, there is a systematic algorithm for the construction of a canonical transformation of any standard potential model Hamiltonian to geometric form, making accessible powerful geometric methods for stability analysis in a wide range of applications.