Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Matthias Blau, Jelle Hartong, Blaise Rollier

TL;DR

This study analyzes the global geometry of the $z=2$ Schrödinger space-time, introducing a codimension-2 embedding to realize global coordinates and then examining causality through both point-particle and scalar-field probes. While there is no globally defined time function and the spacetime is not stably causal, a globally defined time coordinate $T$ provides the best possible time labeling, and scalar fields with fixed lightcone momentum $m$ exhibit a well-posed evolution with well-defined Wightman functions and propagators. The work reveals a distinctly Galilean causal structure encoded in field-theoretic quantities and establishes a groundwork for holographic considerations via bulk-to-boundary propagators, highlighting both the similarities and key differences with AdS. It also clarifies geometric aspects through the embedding and points toward future exploration of other dynamical exponents and perturbations in Schrödinger spacetimes.

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.

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

TL;DR

This study analyzes the global geometry of the Schrödinger space-time, introducing a codimension-2 embedding to realize global coordinates and then examining causality through both point-particle and scalar-field probes. While there is no globally defined time function and the spacetime is not stably causal, a globally defined time coordinate provides the best possible time labeling, and scalar fields with fixed lightcone momentum exhibit a well-posed evolution with well-defined Wightman functions and propagators. The work reveals a distinctly Galilean causal structure encoded in field-theoretic quantities and establishes a groundwork for holographic considerations via bulk-to-boundary propagators, highlighting both the similarities and key differences with AdS. It also clarifies geometric aspects through the embedding and points toward future exploration of other dynamical exponents and perturbations in Schrödinger spacetimes.

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.

Paper Structure

This paper contains 18 sections, 99 equations.

Table of Contents

  1. Introduction
  2. Global Schrödinger geometry
  3. Isometries, timelike Killing vectors and global coordinates
  4. Isometric embeddings and global coordinates
  5. Point particle probes of the causal structure
  6. Time functions and time coordinates
  7. Galilean-like causal structure
  8. Scalar field probes of the causal structure
  9. Canonical analysis
  10. Time evolution
  11. Wightman functions and Green's functions
  12. Bulk-to-boundary propagator
  13. Discussion
  14. Schrödinger algebra and isometric embeddings
  15. Schrödinger invariants
  16. ...and 3 more sections