Interaction, Locality and Measurement

Given two systems with configuration spaces X and Y, we consider their joint description on the configuration space given by the set product \(X \times Y\). In the formalism of ensembles on configuration space, this description requires a probability distribution P(x, y) defined over the joint configuration space, the corresponding conjugate quantity S(x, y), and an ensemble Hamiltonian \(\mathcal{H}_{XY}[P,S]\). Once a composite system is defined, it becomes necessary to introduce a number of new concepts which must be defined carefully. For example, such systems may consist of subsystems which are independent or entangled, non-interacting or interacting, and one must give a precise mathematical formulation of these properties. Issues of locality must be taken into consideration. Observables which are ascribed to one of the subsystems (and are therefore initially defined on only one of the initial configuration spaces, X or Y) must be extended to the joint ensemble, but this can not be done in an arbitrary way. These concepts play an important role in the description of composite systems, and we address them in the first sections of this chapter. The remaining sections are devoted to a description of interactions between subsystems that model measurements, starting with basic measurement procedures followed by more elaborate procedures that describe weak measurements and measurement-induced collapse.