Planck scale effects on some low energy quantum phenomena

Almost all theories of Quantum Gravity predict modifications of the Heisenberg Uncertainty Principle near the Planck scale to a so-called Generalized Uncertainty Principle (GUP). Recently it was shown that the GUP gives rise to corrections to the Schrodinger and Dirac equations, which in turn affect all non-relativistic and relativistic quantum Hamiltonians. In this paper, we apply it to superconductivity and the quantum Hall effect and compute Planck scale corrections. We also show that Planck scale effects may account for a (small) part of the anomalous magnetic moment of the muon. We obtain (weak) empirical bounds on the undetermined GUP parameter from present-day experiments.Comment: 5 pages. To appear in Physics Letters

Corrections to the black body radiation due to minimum-length deformed quantum mechanics

Planck spectrum of black body radiation is usually derived by considering of quantized free electromagnetic field at a finite temperature. The minimum-length deformed quantization affects field theory both at the first and second quantization levels. Performing an exact calculation to the first order in deformation parameter, both of the corrections turn out to be of the same order. Nevertheless, the correction at the second quantization level has some qualitative difference, that may be interesting for future study to differentiate between these two sorts of corrections. In itself the correction to the black body radiation seems to be innocuous in light of the big-bang nucleosynthesis whenever the minimum length is less or equal to $10^{-19}$cm.Comment: 8 pages, Paper has been substantially revised - version to appear in Phys. Lett.

Discreteness of Space from the Generalized Uncertainty Principle

Various approaches to Quantum Gravity (such as String Theory and Doubly Special Relativity), as well as black hole physics predict a minimum measurable length, or a maximum observable momentum, and related modifications of the Heisenberg Uncertainty Principle to a so-called Generalized Uncertainty Principle (GUP). We propose a GUP consistent with String Theory, Doubly Special Relativity and black hole physics, and show that this modifies all quantum mechanical Hamiltonians. When applied to an elementary particle, it implies that the space which confines it must be quantized. This suggests that space itself is discrete, and that all measurable lengths are quantized in units of a fundamental length (which can be the Planck length). On the one hand, this signals the breakdown of the spacetime continuum picture near that scale, and on the other hand, it can predict an upper bound on the quantum gravity parameter in the GUP, from current observations. Furthermore, such fundamental discreteness of space may have observable consequences at length scales much larger than the Planck scale.Comment: 3 pages, revtex4, no figures, to appear in Phys. Lett.

Effects of the Modified Uncertainty Principle on the Inflation Parameters

In this Letter we study the effects of the Modified Uncertainty Principle as proposed in [8] on the inflationary dynamics of the early universe in both standard and Randall-Sundrum type II scenarios. We find that the quantum gravitational effect increase the amplitude of density fluctuation, which is oscillatory in nature, with an increase in the tensor-to-scalar ratio.Comment: new references adde

Discreteness of Space from GUP II: Relativistic Wave Equations

Various theories of Quantum Gravity predict modifications of the Heisenberg Uncertainty Principle near the Planck scale to a so-called Generalized Uncertainty Principle (GUP). In some recent papers, we showed that the GUP gives rise to corrections to the Schrodinger equation, which in turn affect all quantum mechanical Hamiltonians. In particular, by applying it to a particle in a one dimensional box, we showed that the box length must be quantized in terms of a fundamental length (which could be the Planck length), which we interpreted as a signal of fundamental discreteness of space itself. In this paper, we extend the above results to a relativistic particle in a rectangular as well as a spherical box, by solving the GUP-corrected Klein-Gordon and Dirac equations, and for the latter, to two and three dimensions. We again arrive at quantization of box length, area and volume and an indication of the fundamentally grainy nature of space. We discuss possible implications.Comment: v1:8 pages, revtex4, no figures, to appear in Phys. Lett. B; v2: version to match published version in PLB, corrections in ERRATUM include

Free Motion of a Dirac Particle with a Minimum Uncertainty in Position

In this paper, we present a covariant, relativistic noncommutative algebra which includes two small deformation parameters. Using this algebra, we obtain a generalized uncertainty principle which predicts a minimal observable length in measure of space-time distances. Then, we introduce a new representation for coordinate and momentum operators which leads to a generalized Dirac equation. The solutions of the generalized Dirac equation for a free particle will be explicitly obtained. We also obtain the modified fermionic propagator for a free Dirac particle.Comment: 14 pages . . . accepted for publication in Reports on Mathematical Physics. arXiv admin note: text overlap with arXiv:1103.1015,arXiv:1103.3805 by different author

A Higher Order GUP with Minimal Length Uncertainty and Maximal Momentum

We present a higher order generalized (gravitational) uncertainty principle (GUP) in the form $[X,P]=i\hbar/(1-\beta P^2)$. This form of GUP is consistent with various proposals of quantum gravity such as string theory, loop quantum gravity, doubly special relativity, and predicts both a minimal length uncertainty and a maximal observable momentum. We show that the presence of the maximal momentum results in an upper bound on the energy spectrum of the momentum eigenstates and the harmonic oscillator.Comment: 13 pages, 4 figure

Quantum gravity effects on statistics and compact star configurations

The thermodynamics of classical and quantum ideal gases based on the Generalized uncertainty principle (GUP) are investigated. At low temperatures, we calculate corrections to the energy and entropy. The equations of state receive small modifications. We study a system comprised of a zero temperature ultra-relativistic Fermi gas. It turns out that at low Fermi energy $\varepsilon_F$, the degenerate pressure and energy are lifted. The Chandrasekhar limit receives a small positive correction. We discuss the applications on configurations of compact stars. As $\varepsilon_F$ increases, the radius, total number of fermions and mass first reach their nonvanishing minima and then diverge. Beyond a critical Fermi energy, the radius of a compact star becomes smaller than the Schwarzschild one. The stability of the configurations is also addressed. We find that beyond another critical value of the Fermi energy, the configurations are stable. At large radius, the increment of the degenerate pressure is accelerated at a rate proportional to the radius.Comment: V2. discussions on the stability of star configurations added, 17 pages, 2 figures, typos corrected, version to appear in JHE

Effect of the Generalized Uncertainty Principle on Post-Inflation Preheating

We examine effects of the Generalized Uncertainty Principle, predicted by various theories of quantum gravity to replace the Heisenberg's uncertainty principle near the Planck scale, on post inflation preheating in cosmology, and show that it can predict either an increase or a decrease in parametric resonance and a corresponding change in particle production. Possible implications are considered.Comment: v1: 9 pages, revtex4, no figures, accepted for publication in JCAP; v2: one reference added and various cosmetic (but no physics) changes to match published versio

Do the Modified Uncertainty Principle and Polymer Quantization predict same physics?

In this Letter we study the effects of the Modified Uncertainty Principle as proposed in Ali et al. (2009) [5] in simple quantum mechanical systems and study its thermodynamic properties. We have assumed that the quantum particles follow Maxwell-Boltzmann statistics with no spin. We compare our results with the results found in the GUP and polymer quantum mechanical frameworks. Interestingly we find that the corrected thermodynamic entities are exactly same compared to the polymer results but the length scale considered has a theoretically different origin. Hence we express the need of further study for an investigation whether these two approaches are conceptually connected in the fundamental level.Comment: 8 pages, comments and references added, accepted in Phys. Lett. B. arXiv admin note: text overlap with arXiv:1105.531