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The Schroedinger operator in Newtonian space-time

Katarzyna Grabowska·Janusz Grabowski·Paweł Urbański
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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.

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Katarzyna Grabowska, Janusz Grabowski, Paweł Urbański. 2006-11-18. The Schroedinger operator in Newtonian space-time. https://arxiv.org/abs/math-ph/0611044

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