Towards the Quantum Geometry of Saturated Quantum Uncertainty Relations: The Case of the (Q,P) Heisenberg Observables

TL;DR

This paper explores the geometric structures of quantum states that saturate the Heisenberg uncertainty relation, focusing on the case of position and momentum observables, and reviews properties of squeezed states as a foundation for future geometric analysis.

Contribution

It proposes a framework for identifying geometric structures associated with saturated quantum states, specifically analyzing the case of (Q, P) observables and squeezed states.

Findings

Review of properties of squeezed states

Identification of geometric structures related to saturated states

Preparation for quantum geometric and path integral analysis

Abstract

This contribution to the present Workshop Proceedings outlines a general programme for identifying geometric structures--out of which to possibly recover quantum dynamics as well--associated to the manifold in Hilbert space of the quantum states that saturate the Schr\"odinger-Robertson uncertainty relation associated to a specific set of quantum observables which characterise a given quantum system and its dynamics. The first step in such an exploration is addressed herein in the case of the observables Q and P of the Heisenberg algebra for a single degree of freedom system. The corresponding saturating states are the well known general squeezed states, whose properties are reviewed and discussed in detail together with some original results, in preparation of a study deferred to a separated analysis of their quantum geometry and of the corresponding path integral representation over such states.