scholarly journals Spectral distance on Lorentzian Moyal plane

2020 ◽  
Vol 17 (06) ◽  
pp. 2050089
Author(s):  
Anwesha Chakraborty ◽  
Biswajit Chakraborty
Keyword(s):  
Noncommutative Geometry ◽  
Coherent States ◽  
Euclidean Case ◽  
Spectral Triples ◽  
Spectral Distance ◽  
Moyal Plane ◽  
Pure States ◽  
Math Phys ◽  
Distance Formula

We present here a completely operatorial approach, using Hilbert–Schmidt operators, to compute spectral distances between time-like separated “events”, associated with the pure states of the algebra describing the Lorentzian Moyal plane, using the axiomatic framework given by [N. Franco, The Lorentzian distance formula in noncommutative geometry, J. Phys. Conf. Ser. 968(1) (2018) 012005; N. Franco, Temporal Lorentzian spectral triples, Rev. Math. Phys. 26(8) (2014) 1430007]. The result shows no deformations of non-commutative origin, as in the Euclidean case, if the pure states are constructed out of Glauber–Sudarshan coherent states.

scholarly journals Temporal Lorentzian spectral triples

2014 ◽  
Vol 26 (08) ◽  
pp. 1430007 ◽  
Author(s):  
Nicolas Franco
Keyword(s):  
Noncommutative Geometry ◽  
Minkowski Spacetime ◽  
Spectral Triple ◽  
Spectral Triples ◽  
Lorentzian Space ◽  
Pure States ◽  
Global Time ◽  
Distance Formula

We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3 + 1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a temporal Lorentzian spectral triple over a Moyal–Minkowski spacetime. We show that, when time is commutative, the algebra can be extended to unbounded elements. Using such an extension, it is possible to define a Lorentzian distance formula between pure states with a well-defined noncommutative formulation.

scholarly journals CONNES DISTANCE BY EXAMPLES: HOMOTHETIC SPECTRAL METRIC SPACES

2012 ◽  
Vol 24 (09) ◽  
pp. 1250027 ◽  
Author(s):  
JEAN-CHRISTOPHE WALLET
Keyword(s):  
Dirac Operator ◽  
Harmonic Oscillator ◽  
Infinite Number ◽  
Metric Spaces ◽  
Connected Components ◽  
Spectral Triples ◽  
Metric Properties ◽  
Spectral Distance ◽  
Physics Literature ◽  
Moyal Plane

We study metric properties stemming from the Connes spectral distance on three types of non-compact non-commutative spaces which have received attention recently from various viewpoints in the physics literature. These are the non-commutative Moyal plane, a family of harmonic Moyal spectral triples for which the Dirac operator squares to the harmonic oscillator Hamiltonian and a family of spectral triples with the Dirac operator related to the Landau operator. We show that these triples are homothetic spectral metric spaces, having an infinite number of distinct pathwise connected components. The homothetic factors linking the distances are related to determinants of effective Clifford metrics. We obtain, as a by-product, new examples of explicit spectral distance formulas. The results are discussed in detail.

scholarly journals MINIMAL LENGTH IN QUANTUM SPACE AND INTEGRATIONS OF THE LINE ELEMENT IN NONCOMMUTATIVE GEOMETRY

2012 ◽  
Vol 24 (05) ◽  
pp. 1250010 ◽  
Author(s):  
PIERRE MARTINETTI ◽  
FLAVIO MERCATI ◽  
LUCA TOMASSINI
Keyword(s):  
Noncommutative Geometry ◽  
Line Element ◽  
Geodesic Distance ◽  
High Energy ◽  
Minimal Length ◽  
Quantum Space ◽  
Spectral Distance ◽  
Moyal Plane ◽  
The One ◽  
Quantum Spacetime

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime (θ-Minkowski); on the other side, Connes' spectral distance in noncommutative geometry. Although in the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular, in the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d′L, which coincides exactly with the spectral distance dD on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator — together with their translations — d′L and dD coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d′L and dD as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.

Classical pure states are coherent states

1985 ◽  
Vol 111 (8-9) ◽  
pp. 409-411 ◽  
Author(s):  
Mark Hillery
Keyword(s):  
Coherent States ◽  
Pure States

scholarly journals Entanglement in massive coupled oscillators

2011 ◽  
Vol 11 (3&4) ◽  
pp. 278-299
Author(s):  
Nathan L. Harshman ◽  
William F. Flynn
Keyword(s):  
Coupled Oscillators ◽  
Coherent States ◽  
Degrees Of Freedom ◽  
Classical Model ◽  
Reduced Density Matrix ◽  
One Dimension ◽  
Atomic Position ◽  
Pure States ◽  
Non Gaussian ◽  
Reduced Density

This article investigates entanglement of the motional states of massive coupled oscillators. The specific realization of an idealized diatomic molecule in one-dimension is considered, but the techniques developed apply to any massive particles with two degrees of freedom and a quadratic Hamiltonian. We present two methods, one analytic and one approximate, to calculate the interatomic entanglement for Gaussian and non-Gaussian pure states as measured by the purity of the reduced density matrix. The cases of free and trapped molecules and hetero- and homonuclear molecules are treated. In general, when the trap frequency and the molecular frequency are very different, and when the atomic masses are equal, the atoms are highly-entangled for molecular coherent states and number states. Surprisingly, while the interatomic entanglement can be quite large even for molecular coherent states, the covariance of atomic position and momentum observables can be entirely explained by a classical model with appropriately chosen statistical uncertainty.

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