# Geometry of quantum state manifolds generated by the Lie algebra operators

## TL;DR

This paper computes the Fubini-Study metrics of quantum state manifolds generated by various Lie algebra operators, providing a geometric understanding of quantum states in different algebraic contexts.

## Contribution

It derives explicit Fubini-Study metrics for manifolds generated by Heisenberg, so(3), and other Lie algebra operators, extending to arbitrary Lie algebras.

## Key findings

- Metrics for Heisenberg algebra-generated manifolds
- Metrics for so(3) algebra-generated manifolds
- Generalization to arbitrary Lie algebra operators

## Abstract

The Fubini-Study metric of quantum state manifold generated by the operators which satisfy the Heisenberg Lie algebra is calculated. The similar problem is studied for the manifold generated by the so(3) Lie algebra operators. Using these results we calculate the Fubini-Study metrics of state manifolds generated by the position and momentum operators. Also the metrics of quantum state manifolds generated by some spin systems are obtained. Finally, we generalize this problem for operators of an arbitrary Lie algebra.