Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics

TL;DR

This paper introduces a geometric framework for quantum mechanics where evolution dimensions emerge naturally, linking Hamiltonians to Christoffel symbols and relating curvature, fidelity susceptibilities, and Berry curvatures to parallel transport.

Contribution

It presents a novel geometric formalism that interprets Hamiltonians as Christoffel-like operators and derives emergent evolution dimensions in quantum systems.

Findings

Hilbert space curvature is locally flat for closed systems

Fidelity susceptibilities relate to emergent parallel transports

Berry curvatures are connected to the geometric formalism

Abstract

Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.