Spectral triplets, statistical mechanics and emergent geometry in non-commutative quantum mechanics

Abstract

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.

12 pages, no figures

References in corpus (4)

• Formulation, Interpretation and Application of non-Commutative Quantum Mechanics
• Spectrum of the non-commutative spherical well
• Particle Physics from Almost Commutative Spacetimes
• A unifying perspective on the Moyal and Voros products and their physical meanings

Cited by in corpus (10)

• On the role of Schwinger's SU(2) generators for Simple Harmonic Oscillator in 2D Moyal plane
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• Revisiting Connes' Finite Spectral Distance on Non-commutative Spaces : Moyal Plane and Fuzzy Sphere
• Pythagoras Theorem in Noncommutative Geometry
• Connes spectral distance and nonlocality of generalized noncommutative phase spaces
• Thermal effective potential in two- and three- dimensional non-commutative spaces
• Energy Spectrum and Phase Transition of Superfluid Fermi Gas of Atoms on Noncommutative Space
• Connes distance of $2D$ harmonic oscillators in quantum phase space
• Connes spectral distances, quantum discord and coherence of qubits