Abstract: The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of massive free field theory in s = 1,2 and 3 spatial dimensions. Not quite unexpectedly, one obtains for s = 2,3 in the scaling (short distance) limit the algebra of local observables in massless free field theory. The case s = 1 offers, however, some surprises. There the algebra of observables acquires in the scaling limit a non-trivial center and describes charged physical states satisfying Gauss ’ law. The latter result is of relevance for the interpretation of the Schwinger model at short distances and illustrates the conceptual and computational virtues of the method. 1

. We re-examine various notions of statistical independence presently in use in algebraic quantum theory, establishing alternative characterizations for such independence, some of which are also valid without assuming that the observable algebras mutually commute. In addition, in the context which holds in concrete applications to quantum theory, the equivalence of three major notions of statistical independence is proven. Introduction The notions of "independence of two systems" are legion in quantum theory. This is quite understandable, since the physical concept is central in many aspects of quantum theory, and the various formalizations of independence capture different qualitative and quantitative aspects for possibly different ends. 1 Those notions which have appeared in the literature and have formulations in algebraic quantum theory have been extensively reviewed in [24], where their logical interrelationships have been discussed. However, some issues of logical relation wer...

Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the formalism of algebraic quantum "eld theory (AQFT) provides a rigorous framework within which to analyse entanglement in the context of a fully relativistic formulation of quantum theory. What emerges from the analysis are new practical and theoretical limitations on an experimenter's ability to perform operations on a "eld in one spacetime region that can disentangle its state from the state of the "eld in other spacelike-separated regions. These limitations show just how deeply entrenched entanglement is in relativistic quantum "eld theory, and yield a fresh perspective on the ways in which the theory di!ers conceptually from both standard non-relativistic quantum theory and classical relativistic "eld theory. � 2001 Elsevier

Reichenbach's principle of a probabilistic common cause of probabilistic correlations is formulated in terms of relativistic quantum field theory and the problem is raised whether correlations in relativistic quantum field theory between events represented by projections in local observable algebras A(V1) and A(V2) pertaining to spacelike separated spacetime regions V1 and V2 can be explained by finding a probabilistic common cause of the correlation in Reichenbach's sense. While this problem remains open, it is shown that if all superluminal correlations predicted by the vacuum state between events in A(V1) and A(V2) have a genuinely probabilistic common cause, then the local algebras A(V1) and A(V2) must be statistically independent in the sense of C*-independence.

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), f) is said to satisfy the Weak Reichenbach’s Common Cause Principle iff for every pair of projections A ¥ A(V 1), B ¥ A(V 2) correlated in the normal state f 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), f) with a locally normal and locally faithful state f and suitable bounded V 1 and V 2 satisfies the

The Hughston-Jozsa-Wootters (HJW) theorem entails that any finite ensemble compatible with a given density operator can be prepared from a fixed initial state by operations on a spacelike separated system. In this paper, we generalize the HJW theorem to the case of arbitrary measures on the state space of a von Neumann algebra with hyperfinite commutant. In doing so, we also show that every POV measure with range in a hyperfinite von Neumann algebra induces a local, completely positive instrument. I.

The analyzability of the universe into subsystems requires a concept of the "independence" of the subsystems, of which the relativistic quantum world supports many distinct notions which either coincide or are trivial in the classical setting. The multitude of such notions and the complex relations between them will only be adumbrated here. The emphasis of the discussion is placed upon the warrant for and the consequences of a particular notion of subsystem independence, which, it is proposed, should be viewed as primary and, it is argued, provides a reasonable framework within which to sensibly speak of relativistic quantum subsystems.

An overview is given of what mathematical physics can currently say about the vacuum state for relativistic quantum field theories on Minkowski space. Along with a review of classical results such as the Reeh–Schlieder Theorem and its immediate and controversial consequences, more recent results are discussed. These include the nature of vacuum correlations and the degree of entanglement of the vacuum, as well as the striking fact that the modular objects determined by the vacuum state and algebras of observables localized in certain regions of Minkowski space encode a remarkable range of physical information, from the dynamics and scattering behavior of the theory to the external symmetries and even the space–time itself. In addition, an intrinsic characterization of the vacuum state provided by modular objects is discussed. 1

. We provide an overview of the connections between Bell's inequalities and algebraic structure. 1. Introduction Motivated by the desire to bring into the realm of testable hypotheses at least some of the important matters concerning the interpretation of quantum mechanics evoked in the controversy surrounding the EinsteinPodolsky -Rosen paradox [18][5], Bell discovered the first example [3][4] of a family of inequalities which are now generally called Bell's inequalities. These inequalities provide an upper bound on the strength of correlations between systems which are no longer interacting but have interacted in the past. Stated briefly, Bell showed that if the correlation experiments can be modelled by a single classical probability measure, then the strength of the correlations must satisfy a bound which is violated by certain quantum mechanical predictions (and, as has been verified experimentally, by nature - for a review of this original application of Bell's inequalities and ...

We propose some formulations of the notion of "operational independence" of two subsystems S1; S2 of a larger quantum system S and clarify their relation to other independence concepts in the literature. In addition, we indicate why the operational independence of quantum subsystems holds quite generally, both in nonrelativistic and relativistic quantum theory.