The formalism of covariant conditional expectations is described as leading to an operational definition of generalized observables in quantum mechanics, wide enough to account for the fuzziness inherent in actual measurement processes, relative to a multidimensional physical continuum. As an application, a position operator for the photon is defined and its intrinsic fuzziness is discussed.

This paper is intended to provide the axiomatic study of nonequilibrium quantum statistical mechanics with some simple and rigorously solvable models. The models considered here are obtained as generalizations of the Ising model. They illustrate and allow a rational discussion of the following concepts relevant to the theory of irreversible phenomena: coarse-graining and time-smoothing, ergodicity, recurrences, impossibility of a Markovian description of the approach to equilibrium for some physical systems, justification of the various random phase assumptions, properties of the interaction responsible for the approach to equilibrium, master equations, etc.

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A strictly ordered hierarchy of eight causal properties encountered in General Relativity is reviewed for the explicit case of the gravitational plane waves. Illustrative proofs are given to the effect that the place of these space-times is precisely known in the hierarchy: they are causally continuous, but not causally simple. The other conditions of the hierarchy are also discussed separately, as are some causality conditions that belong outside the hierarchy. The investigation relies on the following tools: (1) the symplectic structure appearing in certain matrix differential equations of the Riccati type; and (2) the global properties that can be derived from the isometry group generated by the Killing vector fields of the metric. Connections to some of the techniques of singularity theory are also pointed out.

An axiomatic framework for a quantum mechanical extension to the theory of Anosov systems is constructed, and it is shown that this retains some of the characteristic features of its classical counterpart, e.g., nonvanishing Lyapunov exponents, a vectorial K-property, and exponential clustering. The effects of quantization are investigated on two prototype examples of Anosov systems, namely, the iterations of an automorphism of the torus (the ‘‘Arnold Cat’’ model) and the free dynamics of a particle on a surface of negative curvature. It emerges that the Anosov property survives quantization in the case of the former model, but not of the latter one. Finally, we show that the modular dynamics of a relativistic quantum field on the Rindler wedge of Minkowski space is that of an Anosov system.