We develop various Ehrenfeucht–Fraïssé games for distances between metric structures. We study two forms of distances: pseudometrics stemming from mapping spaces onto each other with some form of approximate isomorphism, and metrics stemming from measuring the distances between two spaces isometrically embedded into a third space. Using an infinitary version of Henson's positive bounded logic with approximations, we form Scott sentences capturing fixed distances to a given space. The Scott sentences of separable spaces are in Lω1ω for 0-distances and in Lω2ω for positive distances.

Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product 〈x0|Kt|x1〉 of the eigenvector |x0〉 of the position operator with eigenvalue x0 and Kt(|x1〉), where Kt is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and straightforward calculations give the wrong value. We look at this phenomenon, and then complement this by showing how to instead correctly calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new.

Abstract We define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is dp-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we give a new example of such a class based on B. Zilber’s approximations of Weyl algebras. We introduce a way to measure the dependence of a tuple a from a set B over another set A. We prove basic properties of the notion, e.g., that a is independent of B over A in the usual sense of homogeneous model theory if and only if the measure of dependence is < ε for all ε > 0. In well behaved situations, the measure corresponds to the distance to a free extension. As an example of our measure of dependence we show a connection between the measure and entropy in models from quantum mechanics in which the spectrum of the observable is discrete. As an application, we show that weak simplicity implies a very strong form of simplicity and study the question of when the dependence inside a set of all realisations of some type can be seen to arise from a pregeometry in cases when the type is not regular. In the end of the paper, we demonstrate our notions and results in one more example: a class built from the p-adic integers.