Geometry of Schroedinger Space-Times II: Particle and Field Probes of the Causal Structure

TL;DR

This paper investigates the global causal structure of z=2 Schrödinger space-times, revealing how scalar fields and particles probe their properties, and demonstrating the possibility of defining a well-behaved quantum Hilbert space despite their unusual causal features.

Contribution

It provides a new isometric embedding for Schrödinger space-times and analyzes their causal structure through particle and scalar field probes, highlighting the existence of a well-defined quantum evolution.

Findings

Global coordinates are naturally derived from the embedding.

Despite non-distinguishing causality, a Hilbert space of scalar modes is well-defined.

Scalar Wightman functions encode the Galilean causal structure.

Abstract

We continue our study of the global properties of the z=2 Schroedinger space-time. In particular, we provide a codimension 2 isometric embedding which naturally gives rise to the previously introduced global coordinates. Furthermore, we study the causal structure by probing the space-time with point particles as well as with scalar fields. We show that, even though there is no global time function in the technical sense (Schroedinger space-time being non-distinguishing), the time coordinate of the global Schroedinger coordinate system is, in a precise way, the closest one can get to having such a time function. In spite of this and the corresponding strongly Galilean and almost pathological causal structure of this space-time, it is nevertheless possible to define a Hilbert space of normalisable scalar modes with a well-defined time-evolution. We also discuss how the Galilean causal structure is reflected and encoded in the scalar Wightman functions and the bulk-to-bulk propagator.