Spacetime singularity resolution in Snyder noncommutative space

We study the representations of the three-dimensional Euclidean Snyder-de Sitter algebra. This algebra generates the symmetries of a model admitting two fundamental scales (Planck mass and cosmological constant) and is invariant under the Born reciprocity for exchange of positions and momenta. Its representations can be obtained starting from those of the Snyder algebra and exploiting the geometrical properties of the phase space that can be identified with a Grassmannian manifold. Both the position and momentum operators turn out to have a discrete spectrum.

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“…( +1/2,1/2) (cos 2 ) ( , ) , (25) with = arcsin . As in the Snyder case, taking = ( + 1)/2, one recovers the eigenvalues (22 (14).…”

Section: The Momentum Representation IImentioning

confidence: 99%

Quantum Mechanics on a Curved Snyder Space

Mignemi

Štrajn

2016

Advances in High Energy Physics

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“…This relation can be easily deduced from (10) in which we have also noticed that dω ¼ 0 due to the Cartan formula as L 19,20]. So, the Liouville theorem is always satisfied on a symplectic manifold independent of a chart in which the physical system is considered.…”

Section: A Darboux Chartmentioning

confidence: 81%

“…Although Hamiltonian (21) is independent of the parameter λ, the new momentum p 0 is bounded as p 0 ∈ ½− 2 λ ; þ 2 λ Þ due to the transformation (20). Therefore, the Hamiltonian (21) should be counted distinct from the standard nondeformed Hamiltonian (7).…”

Section: B Noncanonical Chartmentioning

confidence: 96%

“…The two sets of modified Hamiltonian equations of motion (12) and (24) are in agreement with each other through the transformation (20). The Poisson bracket between two real valued functions F and G in this chart can be obtained by substituting the noncanonical structure (22) and the associated vector field (23) into the definition (13) with the result…”

Section: B Noncanonical Chartmentioning

confidence: 96%

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Thermostatistics of the polymeric ideal gas

Gorji¹,

Nozari²,

Vakili³

2014

Phys. Rev. D

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In this paper, we formulate statistical mechanics of the polymerized systems in the semiclassical regime. On the corresponding polymeric symplectic manifold, we set up a noncanonical coordinate system in which all of the polymeric effects are summarized in the density of states. Since we show that the polymeric effects only change the number of microstates of a statistical system, working in this coordinate is quite reasonable from the statistical point of view. The results show that the number of microstates decreases due to existence of an upper bound for the momentum of the test particles in the polymer framework. We obtain a corresponding canonical partition function by means of the deformed density of states. By using the partition function, we study thermodynamics of the ideal gas in the polymer framework and show that our results are in good agreement with those that arise from the full quantum consideration at high temperature, and they coincide with their usual counterpart in the limit of low temperature.

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“…This feature of LQC can also be realized from our setup through the well-known adiabatic condition for the universe (see also Ref. [39]),…”

Section: Cosmology: Dsr Versus Lqcmentioning

confidence: 99%

Early universe thermostatistics in curved momentum spaces

Gorji¹,

Hosseinzadeh²,

Nozari³

et al. 2016

Phys. Rev. D

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The theories known as doubly special relativity are introduced in order to take into account an observer-independent length scale and the speed of light in the framework of special relativity. These theories can be generally formulated on the de Sitter and also recently proposed anti-de Sitter momentum spaces. In the context of these theories, we study the statistical mechanics and to do this, we consider the natural measure on the corresponding extended phase space. The invariant measure on the space of distinct microstates is obtained by restriction of the natural measure of the extended phase space to the physical phase space through the disintegration theorem. Having the invariant measure, one can study the statistical mechanics in an arbitrary ensemble for any doubly special relativity theory. We use the constructed setup to study the statistical properties of four doubly special relativity models. Applying the results to the case of early universe thermodynamics, we show that one of these models that is defined by the cosmological coordinatization of anti-de Sitter momentum space, implies a finite total number of microstates. Therefore, without attribution to any ensemble density and quite generally, we obtain entropy and internal energy bounds for the early radiation dominated universe. We find that while these results cannot be supported by the standard Friedmann equations, they indeed are in complete agreement with the nonsingular effective Friedmann equations that arise in the context of loop quantum cosmology.

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Spacetime singularity resolution in Snyder noncommutative space

Cited by 26 publications

References 63 publications

Quantum Mechanics on a Curved Snyder Space

Quantum Mechanics on a Curved Snyder Space

Thermostatistics of the polymeric ideal gas

Early universe thermostatistics in curved momentum spaces

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