On the Geodesic Nature of Wegner's Flow

TL;DR

This paper generalizes Wegner's flow equations, establishing conditions under which the flow follows a geodesic path in the quantum state space, highlighting its geometric optimality for generating stationary states.

Contribution

The paper derives a condition for geodesic flows in quantum state space, extending Wegner's method and demonstrating its geometric optimality.

Findings

Flow can be geodesic in the projective Hilbert space under certain conditions.

The method is shown to be geometrically optimal for generating stationary states.

Illustrated with physical examples demonstrating the theoretical results.

Abstract

Wegner's method of flow equations offers a useful tool for diagonalizing a given Hamiltonian and is widely used in various branches of quantum physics. Here, generalizing this method, a condition is derived, under which the corresponding flow of a quantum state becomes geodesic in a submanifold of the projective Hilbert space, independently of specific initial conditions. This implies the geometric optimality of the present method as an algorithm of generating stationary states. The result is illustrated by analyzing some physical examples.