Geometry of the State Space of Quantum Mechanics

Roger Boudet

Aalok Pandya

The geometry of the symplectic structures and Fubini-Study metric is discussed. Discussion in the paper addresses geometry of Quantum Mechanics in the classical phase space. Also, geometry of Quantum Mechanics in the projective Hilbert space has been discussed for the chosen Quantum states. Since the theory of classical gravity is basically geometric in nature and Quantum Mechanics is in no way devoid of geometry, the explorations pertaining to more and more geometry in Quantum Mechanics could prove to be valuable for larger objectives such as understanding of gravity.

Ingemar Bengtsson, Carlos Herdeiro, Roger Picken

Jincheng Zhang

In this paper, we introduce the Quantum Geometry Conjecture Generator (QGCG), a theoretical framework designed to bridge the gap between abstract quantum mechanics and differential geometry. Traditional geometry investigates spatial manifolds endowed with metrics and curvature, whereas quantum mechanics operates within complex Hilbert spaces governed by state vectors and operators. We propose a systematic methodology to construct a new mathematical discipline: Quantum Space Geometry. By mapping quantum state spaces, entanglement structures, and operator algebras onto geometric manifolds, QGCG automatically generates novel mathematical conjectures regarding the intrinsic curvature, topological invariants, and dualities of quantum systems. We formalize the foundational definitions, explore the interplay between quantum fidelity and metric tensors, and establish a series of unproven conjectures that outline a comprehensive roadmap for future research in quantum mathematics.

Prasenjit Deb

Frederik G. Scholtz, Biswajit Chakraborty

We show that when non-commutative quantum mechanics is formulated on the Hilbert space of Hilbert-Schmidt operators (referred to as quantum Hilbert space) acting on a classical configuration space, spectral triplets as introduced by Connes in the context of non-commutative geometry arise naturally. A distance function as defined by Connes can therefore also be introduced. We proceed to give a simple and general algorithm to compute this function. Using this we compute the distance between pure and mixed states on quantum Hilbert space and demonstrate a tantalizing link between statistics and geometry.