A condition of geometric modular action is proposed as a selection principle for physically interesting states on general space-times. This condition is naturally associated with transformation groups of partially ordered sets and provides these groups with projective representations. Under suitable additional conditions, these groups induce groups of point transformations on these space-times, which may be interpreted as symmetry groups. The consequences of this condition are studied in detail in application to two concrete spacetimes -four-dimensional Minkowski and three-dimensional de Sitter spaces -for which it is shown how this condition characterizes the states invariant under the respective isometry group. An intriguing new algebraic characterization of vacuum states is given. In addition, the logical relations between the condition proposed in this paper and the condition of modular covariance, widely used in the literature, are completely illuminated.

We reexamine 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. © 1997 American Institute of Physics. ͓S0022-2488͑97͒00703-2͔ I. INTRODUCTIONThe 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. ͑See Refs. 1-3 for recent applications of these notions.͒ Those notions which have appeared in the literature and have formulations in algebraic quantum theory have been extensively reviewed in Ref. 4, where their logical interrelationships have been discussed. However, some issues of logical relation were left open in that review. One of the goals of this paper is to settle the few remaining conjectures in Ref. 4.Representing the algebras of observables associated to the two subsystems by A and B, respectively, one of the most commonly used expressions of independence is the requirement that the algebras mutually commute elementwise. Another is expressed heuristically in the condition that each system can be prepared independently of the other. It is known 5-9 that this latter notion of statistical independence is logically independent of the requirement that the algebras A and B commute. In Ref. 4 the various versions of statistical independence utilized in the literature were discussed almost exclusively in the context of commuting pairs of algebras. Another purpose of this paper is to provide further information about statistical independence in the more general circumstance that the observable algebras do not necessarily commute.We shall provide alternative characterizations of C*-and W*-independence ͑see below for definitions͒, which are also valid if the algebras do not necessarily commute. In addition, we shall prove that in the category of commuting pairs of von Neumann algebras on separable Hilbert spaces, where all three notions are applicable, C*-independence, strict locality, and W*-independence are equivalent. This is precisely the setting most often met in applications to quantum theory. Moreover, we shall furnish some new results about the notion of C*-independence in the product sense, show that W*-independence is strictly weaker than W*-independence in the product sense, and close with some comments concerning the question of additional conditions sufficient to conclude the mutual commutativity of a pair of C*-independent algebras.Throughout this paper a few assumptions will be made tacitly, since they obtain in the applications to theoretical physics known to us. All algebras are assumed to have id...

If the vacuum is passive for uniformly accelerated observers in anti-de Sitter space-time (i.e. cannot be used by them to operate a perpetuum mobile), they will (a) register a universal value of the Hawking-Unruh temperature, (b) discover a TCP symmetry, and (c) find that observables in complementary wedge-shaped regions are commensurable (local) in the vacuum state. These results are model independent and hold in any theory which is compatible with some weak notion of space-time localization.PACS numbers: 04.62.+v, 11.10.Cd, 11.30.Er Quantum field theory in anti-de Sitter space-time (AdS) has recently received considerable attention [1]. It seems therefore worthwhile to clarify in a model independent setting the universal properties of such theories which are implied by generally accepted and physically meaningful constraints.We report here on the results of an investigation of this question [2] which brought to light the fact that stability properties of the vacuum (the impossibility of operating a perpetuum mobile) are at the root of (a) the Hawking-Unruh effect in AdS, (b) a TCP-like symmetry in AdS, and (c) local commutativity properties of the observables. As will be explained, these results hold in any quantum theory in AdS which allows one in principle to specify the spacetime localization properties of observables.We consider here AdS of any dimension n ≥ 3. It can conveniently be described in terms of Cartesian coordinates in the ambient space R n+1 as the quadric AdS n = {x ∈ R n+1

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