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The Schroedinger operator in Newtonian space-time

Katarzyna Grabowska, Janusz Grabowski, Paweł Urbański

Abstract

The Schroedinger operator on the Newtonian space-time is defined in a way which is independent on the class of inertial observers. In this picture the Schroedinger operator acts not on functions on the space-time but on sections of certain one-dimensional complex vector bundle over space-time. This bundle, constructed from the data provided by all possible inertial observers, has no canonical trivialization, so these sections cannot be viewed as functions on the space-time. The presented framework is conceptually four-dimensional and does not involve any ad hoc or axiomatically introduced geometrical structures. It is based only on the traditional understanding of the Schroedinger operator in a given reference frame and it turns out to be strictly related to the frame-independent formulation of analytical Newtonian mechanics that makes a bridge between the classical and quantum theory.

The Schroedinger operator in Newtonian space-time

Abstract

The Schroedinger operator on the Newtonian space-time is defined in a way which is independent on the class of inertial observers. In this picture the Schroedinger operator acts not on functions on the space-time but on sections of certain one-dimensional complex vector bundle over space-time. This bundle, constructed from the data provided by all possible inertial observers, has no canonical trivialization, so these sections cannot be viewed as functions on the space-time. The presented framework is conceptually four-dimensional and does not involve any ad hoc or axiomatically introduced geometrical structures. It is based only on the traditional understanding of the Schroedinger operator in a given reference frame and it turns out to be strictly related to the frame-independent formulation of analytical Newtonian mechanics that makes a bridge between the classical and quantum theory.

Paper Structure

This paper contains 7 sections, 2 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Newtonian space-time
  3. The Schrödinger operator
  4. The explicit transformations
  5. The Schrödinger bundle
  6. Relation to Newtonian mechanics
  7. Concluding remarks

Key Result

Theorem 4.1

The map ${\mathbb R}^3\ni v\mapsto T_v$, where $T_v$ is a linear operator in $C^\infty_{\mathbb C}({\mathbb R}^3\times{\mathbb R})$ defined by is a representation of ${\mathbb R}^3$ in $C^\infty_{\mathbb C}({\mathbb R}^3\times{\mathbb R})$ leaving invariant the Schrödinger operator (S1).

Theorems & Definitions (2)

  • Theorem 4.1
  • Theorem 5.1