An Algebraic Characterization of Vacuum States in Minkowski Space. III. Reflection Maps

Employing the algebraic framework of local quantum physics, vacuum states in Minkowski space are distinguished by a property of geometric modular action. This property allows one to construct from any locally generated net of observables and corresponding state a continuous unitary representation of the proper Poincare group which acts covariantly on the net and leaves the state invariant. The present results and methods substantially improve upon previous work. In particular, the continuity properties of the representation are shown to be a consequence of the net structure, and surmised cohomological problems in the construction of the representation are resolved by demonstrating that, for the Poincare group, continuous reflection maps are restrictions of continuous homomorphisms.Comment: 20 pages; change of title, reference added; version as to appear in Commun. Math. Phy

Geometric modular action and spontaneous symmetry breaking

We study spontaneous symmetry breaking for field algebras on Minkowski space in the presence of a condition of geometric modular action (CGMA) proposed earlier as a selection criterion for vacuum states on general space-times. We show that any internal symmetry group must commute with the representation of the Poincare group (whose existence is assured by the CGMA) and each translation-invariant vector is also Poincare invariant. The subspace of these vectors can be centrally decomposed into pure invariant states and the CGMA holds in the resulting sectors. As positivity of the energy is not assumed, similar results may be expected to hold for other space--times.Comment: Dedicated to the memory of Siegfried Schlieder. 17 pages, no figures. Revised version: simplified arguments and improved results; as to appear in Annales H. Poincar

Further Representations of the Canonical Commutation Relations

We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (i.e. the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize which such perturbations lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, resp. quasi-equivalent, with Fock, coherent or quasifree representations

Local Primitive Causality and the Common Cause Principle in Quantum Field Theory

If \{A(V)\} is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V_1 and V_2 are spacelike separated spacetime regions, then the system (A(V_1),A(V_2),\phi) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A \in A(V_1), B \in A(V_2) correlated in the normal state \phi there exists a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V_1 and V_2 and disjoint from both V_1 and V_2, a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system (A(V_1),A(V_2),\phi) with a locally normal and locally faithful state \phi and open bounded V_1 and V_2 satisfies the Weak Reichenbach's Common Cause Principle.Comment: 14 pages, Late

Quantum Probability Theory

The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.Comment: 28 pages, LaTeX, typos removed and some minor modifications for clarity and accuracy made. This is the version to appear in Studies in the History and Philosophy of Modern Physic

Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools

We provide a brief overview of the basic tools and concepts of quantum field theoretical scattering theory. This article is commissioned by the Encyclopedia of Mathematical Physics, edited by J.-P. Francoise, G. Naber and T.S. Tsun, to be published by the Elsevier publishing house.Comment: 14 pages, no figure

Robust decision making for agile systems development Part 2: a decomposition and analysis

The need for agility in operational systems within the defence enterprise and procurement domains has been identified by many authors, and over time, there have been a number of initiatives and programmes that have sought to identify the nature of agility, and the means by which it can be defined and employed within individual cases and scenarios. These have identified impediments to the successful realization of agile practices and methods, particularly the resilience of agile decision making throughout the conceptual understanding, design and implementation of the operational system. To further investigate the extent to which this process can be implemented in a robust and reliable manner, Cranfield University created the ‘Robust Enterprise-based Approach for Agility in Capability Through-life (REA2CT)’ framework, which provides a number of functional steps to institute a systems development lifecycle approach to producing agile solutions for use in networked systems and systems-of-systems. This paper builds upon the description of the framework[1] by applying the Axiomatic Design (AD) theory to identify where complexity exists within the requirements and design activities that underpin the framework. Using this analysis, this paper identifies ‘pain points’ within the REA2CT framework, and suggests necessary improvements to facilitate the implementation of agility throughout the systems development lifecycle

Robust decision making for agile systems development Part 1: exploring the paradigm

The need for agility in operational systems within the defence enterprise and procurement domains has been identified by many authors, and over time, there have been a number of initiatives and programmes that have sought to identify the nature of agility, and the means by which it can be defined and employed within individual cases and scenarios. These have identified impediments to the successful realization of agile practices and methods, particularly the resilience of agile decision making throughout the conceptual understanding, design and implementation of the operational system. To further investigate the extent to which this process can be implemented in a robust and reliable manner, Cranfield University created the ‘Robust Enterprise-based Approach for Agility in Capability Through-life (REA2CT)’ framework, which provides a number of functional steps to institute a systems development lifecycle approach to producing agile solutions for use in networked systems and systems-of-systems. This paper briefly examines the Customer Need (CN) for the enterprise-based delivery of system (of systems) agility into the operational domain. Axiomatic Design (AD) theory is used to describe the REA2CT framework, identifying Functional Requirements (FRs) which might satisfy the CN for agility. Initial Design Parameters (DPs) are proposed to satisfy the FRs