Essential self-adjointness of Wick squares in quasi-free Hadamard representations on curved spacetimes J. Math. Phys. 53, 042502 (2012) A simplicial gauge theory J. Math. Phys. 53, 033501 (2012) Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation J. Math. Phys. 52, 103703 (2011) A symmetric approach to the massive nonlinear sigma model Bell's inequalities are briefly presented in the context of order-unit spaces and then studied in some detail in the framework of C *-algebras. The discussion is then specialized to quantum field theory. Maximal Bell correlationsp(tP,d(&I), d(&2») for two subsystems localized in regions &1 and &2 and constituting a system in the state tP are defined, along with the concept of maximal Bell violations. After a study of these ideas in general, properties of these correlations in vacuum states of arbitrary quantum field models are studied. For example, it is shown that in the vacuum state the maximal Bell correlations decay exponentially with the product ofthe lowest mass and the spacelike separation of &1 and &2' This paper is also preparation for the proof in Paper II [So J. Summers and R. Werner, J. Math. Phys. 28, 2448 (1987)] that Bell's inequalities are maximally violated in the vacuum state.

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Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel's strict deformations of C * -dynamical systems with automorphic actions of R n , whenever the latter are presented in a covariant representation. Moreover, the device can be used for the deformation of relativistic quantum field theories by adjusting the convolutions to the geometry of Minkowski space. The resulting deformed theories still comply with pertinent physical principles and their Tomita-Takesaki modular data coincide with those of the undeformed theory; but they are in general inequivalent to the undeformed theory and exhibit different physical interpretations.

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.

Under weak technical assumptions on a net of local von Neumann algebras {stf(&)} in a Hubert space 3tf, which are fulfilled by any net associated to a quantum field satisfying the standard axioms, it is shown that for every vector state φ in 2f? there exist observables localized in complementary wedgeshaped regions in Minkowski space-time that maximally violate Bell's inequalities in the state φ. If, in addition, the algebras corresponding to wedgeshaped regions are injective (which is known to be true in many examples), then the maximal violation occurs in any state φ on &(3tif) given by a density matrix.