# Underlying Geometric Flow in Hamiltonian Evolution

## TL;DR

This paper introduces a geometric approach to quantum mechanics using a perturbed Ricci flow and analyzes system stability.

## Contribution

A novel geometric formulation of quantum dynamics using perturbed Ricci flow and a new notion of quantum stability.

## Key findings

- A perturbed Ricci flow is constructed from Heisenberg equations for quantum dynamics.
- A theorem characterizes stability in quantum systems using Ho¨lder space topology.
- Quantum stability is defined based on tensor metric operator evolution.

## Abstract

In this paper, an underlying perturbed Ricci flow construction is made within the metric operator space, originating from the Heisenberg dynamical equations, to formulate a method which appears to provide a new geometric approach for the geometric formulation of the quantum mechanical dynamics. A quantum mechanical notion of stability and local instability is introduced within the quantum mechanical theory, based on the quantum mechanical dynamical equations governing the evolution of the tensor metric operator. The stability analysis is conducted in the topology of little Ho¨lder spaces of metrics which the tensor metric operator acts on. Finally, a theorem is introduced in an attempt to characterize the stability properties of the quantum mechanical system such that it brings the quantum mechanical dynamics into the analysis.