The Schroedinger operator as a generalized Laplacian

TL;DR

This paper presents a geometric framework for Schroedinger operators as generalized Laplacians on a principal bundle, unifying classical and quantum mechanics in a frame-independent manner.

Contribution

It introduces a novel geometric formulation of Schroedinger operators as Laplace-Beltrami operators on a principal bundle, independent of inertial frames.

Findings

Schroedinger operators act on sections of a line bundle rather than functions.

A natural differential calculus for wave forms is developed.

The framework links classical Newtonian mechanics with quantum theory.

Abstract

The Schroedinger operators on the Newtonian space-time are defined in a way which make them independent on the class of inertial observers. In this picture the Schroedinger operators act not on functions on the space-time but on sections of certain one-dimensional complex vector bundle -- the Schroedinger line bundle. This line bundle has trivializations indexed by inertial observers and is associated with an U(1)-principal bundle with an analogous list of trivializations -- the Schroedinger principal bundle. For the Schroedinger principal bundle a natural differential calculus for `wave forms' is developed that leads to a natural generalization of the concept of Laplace-Beltrami operator associated with a pseudo-Riemannian metric. The free Schroedinger operator turns out to be the Laplace-Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schroedinger principal bundle. The presented framework is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton-Jacobi equations, that makes a bridge between the classical and quantum theory.