Dual Quantum Mechanics

TL;DR

This paper proposes a duality-based extension of quantum mechanics incorporating gravity, linking position and momentum representations through transformations derived from conformal gravity in the world-line formalism.

Contribution

It introduces a novel duality framework that relates position-dependent and momentum-dependent fields in quantum mechanics with gravity, supported by conformal gravity formalism.

Findings

Duality transformations connect position and momentum field dynamics.

Supports a possible extension of quantum mechanics with gravity.

Provides a formal basis for a dual quantum gravity framework.

Abstract

We point out a possible complementation of the basic equations of quantum mechanics in the presence of gravity. This complementation is suggested by the well-known fact that quantum mechanics can be equivalently formulated in the position or in the momentum representation. As a way to support this complementation, starting from the action that describes conformal gravity in the world-line formalism, we show that there are duality transformations that relate the dynamics in the presence of position dependent vector and tensor fields to the dynamics in the presence of momentum dependent vector and tensor fields.