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

Scholtz, F. G.; Chakraborty, Biswajit

doi:10.1088/1751-8113/46/8/085204

Cited by 14 publications

(58 citation statements)

References 25 publications

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“…Since statistical mechanics plays an important role in understanding a system consisting of large number of particles, it is reasonable to assume that a study of statistical mechanics of these compact systems in the background of a non-commutative spacetime will be a viable option to understand the Planck scale physics. The effect of Planck scale physics, especially due to the presence of minimal length, in statistical mechanics has been reported by many in the literature [2][3][4][5][6][7][8][9]. Compact stars appears to be a potential source to study the effects of non-commutativity on statistical mechanics, due to its accessibility for observation.…”

Section: Introductionmentioning

confidence: 95%

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Compact stars in quantum spacetime

Harikumar

Zuhair

2018

J. Phys. Commun.

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We derive bounds on the deformation parameter of the κ-spacetime by analyzing the effect of noncommutativity on astrophysical model. We study compact stars, taken to be degenerate Fermi gas, in non-commutative spacetime. Using tools of statistical mechanics, we derive the degeneracy pressure of the compact star in κ-spacetime and from the hydrostatic equilibrium conditions we obtain a bound on the deformation parameter. We independently derive this bound using generalized uncertainty principle, which is a characteristic feature of quantum gravity approaches, strengthening the bound obtained.

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Section: Introductionmentioning

confidence: 95%

“…, which has the correct dimension and reduces to ÿ in the commutative limit a 0  . The possibility of such a modification was studied in [7].…”

Section: Degeneracy Pressure and Hydrostatic Equilibrium In κ-Spacetimentioning

confidence: 99%

Compact stars in quantum spacetime

Harikumar

Zuhair

2018

J. Phys. Commun.

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“…This non-Voros state will be useful if we are interested in computing Connes' spectral distance between a pair of such "points " i.e. pure states (see for example [11] and references there-in). However, since our interest is to compute the thermal correlation function between a pair of identical particles in Voros states, we shall not be concerned with such states in the rest of the paper.…”

Section: Computation Of Variance Matrix In the Non-commutative Casmentioning

confidence: 99%

“…In particular the Voros basis turns out to be a coherent state |z) representing a maximally localized state in the non-commutative plane. Indeed, it has been shown recently [11] that one can compute the spectral distance, a la Connes [12] between a pair of neighboring states |z) V and |z + dz) V to get a Euclidean geometry: d 2 (|z), |z + dz)) = 2θ 3 dzdz. But such a distance function cannot be assigned between the pair of neighboring states | x) M and | x + d x) M corresponding to the Moyal basis.…”

Section: Introductionmentioning

confidence: 99%

Thermal effective potential in two- and three-dimensional non-commutative spaces

Devi

Ghosh

Chakraborty

et al. 2013

J. Phys. A: Math. Theor.

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The issue of thermal correlation functions and the associated effective statistical potential in twodimensional Moyal space, arising in the twisted approach to implement rotational symmetry, has been revisited in an operatorial formulation where no explicit star product is used initially. The corresponding results using Moyal and Voros star products are then easily obtained by taking the corresponding overlap with Moyal and Voros bases. in contrast to the Moyal case where the concept of distance and, in particular, the relative separation between a pair of particles remain ambiguous when the Moyal star product is used, the Voros basis is more physical and the inter-particle distance can be introduced unambiguously. The forms of the correlation function and the effective potential are found to be same as the Moyal case except that the thermal wavelength undergoes a non-commutative deformation, ensuring that it has a lower bound of the order of √ θ. It is shown that in a suitable basis (called here quasi-commutative basis) in the multiparticle sector the thermal correlation function coincides with the commutative result both in the Moyal and Voros cases along with the restoration of the Pauli principle, except that in the Voros case the thermal wavelength, again, gets a non-commutative correction. Finally, we extend our result to three-dimensional non-commutative space and compute the correlation function and effective potential using both twisted and quasi-commutative bases in the Moyal and Voros cases. We find that there is SO(3) → SO(2) symmetry breaking in the effective potential, which also violates the Pauli principle, even for a pair of free particles, despite the fact that a deformed co-product is used to construct twisted symmetric/anti-symmetric basis. However, this SO(3) symmetry, along with Pauli principle, is restored once we use the quasi-commutative bases.

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“…Here we have introduced the density matrix ρ z ∈ H q , as viewed from H c and can be associated with the pure state ω ρz corresponding to the * -algebra H q = A M and defined as a linear functional on A M : ω ρz (a) ∈ C of norm one. As explained in detail in [13], here too we shall be working with normal states, so that the states can be represented by density matrices : ω ρz (a) = Tr Hc (ρ z a). For brevity, therefore, the states will be denoted just by density matrices themselves as ρ(a) = Tr (ρa).…”

Section: Introductionmentioning

confidence: 99%

Spectral distances on the doubled Moyal plane using Dirac eigenspinors

Kumar

Chakraborty²

2018

Phys. Rev. D

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We present here a novel method of computing spectral distances in doubled Moyal plane in a noncommutative geometrical framework using Dirac eigen-spinors, while solving the Lipschitz ball condition explicitly through matrices. The standard results of longitudinal, transverse and hypotenuse distances between different pairs of pure states have been computed and Pythagorean equality between them have been re-produced. The issue of non-unital nature of Moyal plane algebra is taken care of through a sequence of projection operators constructed from Dirac eigen-spinors, which plays a crucial role throughout this paper. At the end, a toy model of "Higgs field" has been constructed by fluctuating the Dirac operator and the variation on the transverse distance has been demonstrated, through an explicit computation.

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

Cited by 14 publications

References 25 publications

Compact stars in quantum spacetime

Compact stars in quantum spacetime

Thermal effective potential in two- and three-dimensional non-commutative spaces

Spectral distances on the doubled Moyal plane using Dirac eigenspinors

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