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The SoccerBall Problem?
"... Abstract. The idea that Lorentzsymmetry in momentum space could be modified but still remain observerindependent has received quite some attention in the recent years. This modified Lorentzsymmetry, which has been argued to arise in Loop Quantum Gravity, is being used as a phenomenological model ..."
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Abstract. The idea that Lorentzsymmetry in momentum space could be modified but still remain observerindependent has received quite some attention in the recent years. This modified Lorentzsymmetry, which has been argued to arise in Loop Quantum Gravity, is being used as a phenomenological model to test possibly observable effects of quantum gravity. The most pressing problem in these models is the treatment of multiparticle states, known as the ‘soccerball problem’. This article briefly reviews the problem and the status of existing solution attempts. Key words: Lorentzinvariance; quantum gravity; quantum gravity phenomenology; deformed special relativity 2010 Mathematics Subject Classification: 83A05; 83C45 1
Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2012
"... Research during the last decade demonstrates that effects originating on the Planck scale are currently being tested in multiple observational contexts. In this review we discuss quantum gravity phenomenological models and their possible links to loop quantum gravity. Particle frameworks, including ..."
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Research during the last decade demonstrates that effects originating on the Planck scale are currently being tested in multiple observational contexts. In this review we discuss quantum gravity phenomenological models and their possible links to loop quantum gravity. Particle frameworks, including kinematic models, broken and deformed Poincaré symmetry, noncommutative geometry, relative locality and generalized uncertainty principle, and field theory frameworks, including Lorentz violating operators in effective field theory and noncommutative field theory, are discussed. The arguments relating loop quantum gravity to models with modified dispersion relations are reviewed, as well as, arguments supporting the preservation of local Lorentz invariance. The phenomenology related to loop quantum cosmology is briefly reviewed, with a focus on possible effects that might be tested in the near future. As the discussion makes clear, there remains much interesting work to do in establishing the connection between the fundamental theory of loop quantum gravity and these specific phenomenological models, in determining observational consequences of the characteristic aspects of loop quantum gravity, and in further refining current observations. Open problems related to these developments are highlighted.
Some Implications of Two Forms of the Generalized Uncertainty Principle
"... Various theories of quantum gravity predict the existence of a minimum length scale, which leads to the modification of the standard uncertainty principle to the Generalized Uncertainty Principle (GUP). In this paper, we study two forms of the GUP and calculate their implications on the energy of t ..."
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Various theories of quantum gravity predict the existence of a minimum length scale, which leads to the modification of the standard uncertainty principle to the Generalized Uncertainty Principle (GUP). In this paper, we study two forms of the GUP and calculate their implications on the energy of the harmonic oscillator and the hydrogen atom more accurately than previous studies. In addition, we show how the GUP modifies the Lorentz force law and the timeenergy uncertainty principle.
Homogeneous Field and WKB Approximation in Deformed Quantum Mechanics with Minimal Length
"... In the framework of the deformed quantum mechanics with a minimal length, we consider the motion of a nonrelativistic particle in a homogeneous external field. We find the integral representation for the physically acceptable wave function in the position representation. Using the method of steepes ..."
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In the framework of the deformed quantum mechanics with a minimal length, we consider the motion of a nonrelativistic particle in a homogeneous external field. We find the integral representation for the physically acceptable wave function in the position representation. Using the method of steepest descent, we obtain the asymptotic expansions of the wave function at large positive and negative arguments. We then employ the leading asymptotic expressions to derive the WKB connection formula, which proceeds from classically forbidden region to classically allowed one through a turning point. By the WKB connection formula, we prove the BohrSommerfeld quantization rule up to O( 2 ). We also show that if the slope of the potential at a turning point is too steep, the WKB connection formula is no longer valid around the turning point. The effects of the minimal length on the classical motions are investigated using the HamiltonJacobi method. We also use the BohrSommerfeld quantization to study statistical physics in deformed spaces with the minimal length.
Minimal Length and the Existence of Some Infinitesimal Quantities in Quantum Theory and Gravity
"... It is demonstrated that provided a theory involves a minimal length, this theory must be free from such infinitesimal quantities as infinitely small variations in surface of the holographic screen, its volume, and entropy. The corresponding infinitesimal quantities in this case must be replaced by ..."
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It is demonstrated that provided a theory involves a minimal length, this theory must be free from such infinitesimal quantities as infinitely small variations in surface of the holographic screen, its volume, and entropy. The corresponding infinitesimal quantities in this case must be replaced by the "minimal variations possible"finite quantities dependent on the existent energies. As a result, the initial lowenergy theory (quantum theory or general relativity) inevitably must be replaced by a minimal length theory that gives very close results but operates with absolutely other mathematical apparatus. At the present time all highenergy generalizations (limits) of the basic components in fundamental physics (quantum theory [1] and gravity [2]) of necessity lead to a minimal length on the order of the Planck length min ∝ . This follows from string theory But it is clear that provided a minimal length exists, it is existent at all the energy scales and not at high (Planck's) scales only. What is inferred on this basis for real physics? At least, it is suggested that the use of infinitesimal quantities in a mathematical apparatus of both quantum theory and gravity is incorrect, despite the fact that both these theories give the results correlating well with the experiment (e.g., Indeed, in all cases, the infinitesimal quantities bring about an infinitely small length [2] (1) that is inexistent because of min . The same is true for any function Υ dependent only on different parameters whose dimensions of length of the exponents are equal to or greater than 1 ] ≥ 1: Obviously, the infinitely small variation Υ of Υ is senseless as, according to (2), we have But, because of min , the infinitesimal quantities make no sense and hence Υ makes no sense too. Instead of these infinitesimal quantities, it seems reasonable to denote them as "minimal variations possible" Δ min of the quantity having the dimension of length, that is, the quantity And then However, the "minimal variations possible" of any quantity having the dimensions of length (4) which are equal to min ∝ require, according to the Heisenberg Uncertainty Principle (HUP) But at low energies (far from the Planck energy) there are no such quantities and hence in essence Δ min = min ∝ (4) corresponds to the highenergy (Planck's) case only.
The Minimal Length and the Shannon Entropic Uncertainty Relation
"... In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relation [ , ] = ℏ(1 + 2 ), where is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, BialynickiBirul ..."
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In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relation [ , ] = ℏ(1 + 2 ), where is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, BialynickiBirula, and Mycielski (BBM) depends on both the algebra and the used representation, we show that using the formally selfadjoint representation, that is, = and = tan(√ )/√ , where [ , ] = ℏ, the BBM inequality is still valid in the form + ≥ 1 + ln as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.
Effect of Generalized Uncertainty Principle on MainSequence Stars and White Dwarfs
"... This paper addresses the effect of generalized uncertainty principle, emerged from different approaches of quantum gravity within Planck scale, on thermodynamic properties of photon, nonrelativistic ideal gases, and degenerate fermions. A modification in pressure, particle number, and energy densit ..."
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This paper addresses the effect of generalized uncertainty principle, emerged from different approaches of quantum gravity within Planck scale, on thermodynamic properties of photon, nonrelativistic ideal gases, and degenerate fermions. A modification in pressure, particle number, and energy density are calculated. Astrophysical objects such as mainsequence stars and white dwarfs are examined and discussed as an application. A modification in LaneEmden equation due to a change in a polytropic relation caused by the presence of quantum gravity is investigated. The applicable range of quantum gravity parameters is estimated. The bounds in the perturbed parameters are relatively large but they may be considered reasonable values in the astrophysical regime.
On the Boundary Conditions in Deformed Quantum Mechanics with Minimal Length Uncertainty
"... We find the coordinate space wave functions, maximal localization states, and quasiposition wave functions in a GUP framework that implies a minimal length uncertainty using a formally selfadjoint representation. We show how the boundary conditions in quasiposition space can be exactly determined ..."
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We find the coordinate space wave functions, maximal localization states, and quasiposition wave functions in a GUP framework that implies a minimal length uncertainty using a formally selfadjoint representation. We show how the boundary conditions in quasiposition space can be exactly determined from the boundary conditions in coordinate space.