Conformal symmetry of an extended Schrodinger equation and its relativistic origin

TL;DR

This paper demonstrates that any power of the Schrödinger Lagrangian exhibits non-relativistic conformal symmetry, which originates from the conformal isometry of a higher-dimensional Klein-Gordon Lagrangian, linking non-relativistic and relativistic symmetries.

Contribution

It establishes a general connection between non-relativistic conformal symmetry of Schrödinger equations and relativistic conformal symmetry of Klein-Gordon equations in higher dimensions.

Findings

Any power p of the Schrödinger Lagrangian has non-relativistic conformal symmetry.

The symmetry involves a phase and a conformal factor depending on dimension and exponent.

The non-relativistic symmetry originates from the conformal isometry of a higher-dimensional Klein-Gordon Lagrangian.

Abstract

In this paper two things are done. We first prove that an arbitrary power $p$ of the Schrodinger Lagrangian in arbitrary dimension always enjoys the non-relativistic conformal symmetry. The implementation of this symmetry on the dynamical field involves a phase term as well as a conformal factor that depends on the dimension and on the exponent. This non-relativistic conformal symmetry is shown to have its origin on the conformal isometry of the power $p$ of the Klein-Gordon Lagrangian in one higher dimension which is related to the phase of the complex scalar field.