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

We show that when non-commutative quantum mechanics is formulated on the Hilbert space of Hilbert-Schmidt operators 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 Conne...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2013-03, Vol.46 (8), p.85204-16
Main Authors: Scholtz, F G, Chakraborty, B
Format: Article
Language:English
Subjects:
Algorithms
emergent geometry
Hilbert space
Mathematical analysis
Mathematical models
non-commutative geometry
Operators
Quantum mechanics
Spectra
Statistical mechanics
Statistics
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Summary:We show that when non-commutative quantum mechanics is formulated on the Hilbert space of Hilbert-Schmidt operators 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 algorithm to compute this function in generic situations. Using this we compute the distance between pure and mixed states on the quantum Hilbert space and demonstrate a tantalizing link between statistics and geometry.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8113/46/8/085204