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 Schrödinger 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 Letter, 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.     

The shear modulus of the human vocal fold in a transverse direction

The aim of this study was to measure the shear modulus of the vocal fold in a human hemilarynx, such that the data can be related to direction of applied stress and anatomical context. Dynamic spring rate data were collected using a modified linear skin rheometer using human hemilarynges, and converted to estimated shear modulus via application of a simple shear model. The measurement probe was attached to the epithelial layer of the vocal fold cover using suction. A sinusoidal force of 3g was applied to the epithelium, and the resultant displacement logged at a rate of 1kHz. Force measurement accuracy was 20microg and position measurement accuracy was 4microm. The force was applied in a transverse direction at the midmembranous point between the vocal process and the anterior commissure. The shear modulus of the three female vocal folds ranged from 814 to 1232Pa. The shear modulus of the three male vocal folds ranged from 1021 to 1796Pa. These data demonstrate that it is possible to obtain estimates for the shear modulus of the vocal fold while preserving anatomical context. The modulus values reported here are higher than those reported using parallel plate rheometry. This is to be expected as the tissue is attached to surrounding structures, and is under natural tension.     

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.     

Generalized uncertainty principle and self-adjoint operators

In this work we explore the self-adjointness of the GUP-modified momentum and Hamiltonian operators over different domains. In particular, we utilize the theorem by von-Neumann for symmetric operators in order to determine whether the momentum and Hamiltonian operators are self-adjoint or not, or they have self-adjoint extensions over the given domain. In addition, a simple example of the Hamiltonian operator describing a particle in a box is given. The solutions of the boundary conditions that describe the self-adjoint extensions of the specific Hamiltonian operator are obtained.     

UV and MO study on the deprotonation of some 2-aryl-Δ2-1,3,4-oxadiazoline-5-thiones

Substituent effects on the deprotonation processes of a series of 2-aryl-Δ²-1,3,4-oxadiazoline-5-thione (1) derivatives have been studied experimentally as well as theoretically. The acid dissociation constants pK_a have been determined spectrophotometrically in ethanol-water solutions (7.5-92.5%) and vary between 3.76 and 5.80. Semiempirical molecular orbital (MO) calculations (AM1 and PM3) were used for the investigation of the existence of possible tautomeric thione and thiol forms of the studied compounds. Strong correlation between the pK_a values and the deprotonation enthalpies were evaluated.     

Growth of calcium oxalate monohydrate on uric acid crystals at sustained supersaturation

The effect of uric acid seeds on calcium oxalate formation was studied at pH 4.50 and 37 °C using a system providing constant supersaturation with respect to calcium oxalate and saturated in uric acid. In all cases the only solid‐phase forming was identified as calcium oxalate monohydrate (COM). Kinetic analysis of the initial rates showed that they were proportional with the relative supersaturation with respect to calcium oxalate monohydrate. The linear dependence of the rate of precipitation of COM on uric acid suggested that growth is mediated through a surface diffusion controlled mechanism. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dependence of the γ carbon-13 shielding effect on electronegativity

Three γ effects on ¹³C shielding in 3,3-dimethylheteracyclohexanes as a function of the hetero-atom X have been examined. The γ-anti effect on the equatorial 3-methyl group is small in absolute magnitude but strongly dependent on the polar properties of X. The plot of the ¹³C shielding of this carbon vs the electronegativity of X is linear, with a slope of ?5.8 ppm/electronegativity unit. The γ-gauche effects on the axial 3-methyl group and on the 4-carbon are large in absolute magnitude but have quite different dependences on the polar properties of X. Whereas the shielding of the 4-carbon exhibits a linear dependence on electronegativity (slope ?3.5), the axial 3-methyl group shows little dependence (slope crudely ?0.7), even though the geometric relationship between X and either carbon is almost the same. Neither gauche carbon shielding appears to be related to the steric properties of X. The polar component of both the γ-anti effect and the γ-gauche effect is interpreted as arising from overlap of appropriately positioned parallel orbitals. For the anti case, the pathway is the familiar zigzag arrangement of bonds. For the gauche case, the pathway may be either through space (the orbitals would be only on X and C-α; for the 4-carbon, this interaction would be through the center of the ring) or through bonds (there are parallel axial orbitals on all four atoms). The absence of a significant polar effect for the axial 3-methyl group suggests that the gauche interaction requires a rigid pathway. The polar component of the general γ-gauche effect is superimposed upon a larger contribution that is essentially independent of the nature of X and may be associated with the removal of the hydrogen on the β-carbon and replacement with the γ-X group.     

Thermodynamical interpretation of the interacting holographic dark energy model in a non-flat universe

Motivated by the recent work of Wang, Lin, Pavon, and Abdalla [B. Wang, C.Y. Lin, D. Pavon, E. Abdalla, Phys. Lett. B 662 (2008) 1, arXiv: 0711.2214 [hep-th]], we generalize their work to the non-flat case. In particular, we provide a thermodynamical interpretation for the holographic dark energy model in a non-flat universe. For this case, the characteristic length is no more the radius of the event horizon (RE$R_E$) but the event horizon radius as measured from the sphere of the horizon (L  ). Furthermore, when interaction between the dark components of the holographic dark energy model in the non-flat universe is present its thermodynamical interpretation changes by a stable thermal fluctuation. A relation between the interaction term of the dark components and this thermal fluctuation is obtained. In the limiting case of a flat universe, i.e. k=0$k=0$, all results given in [B. Wang, C.Y. Lin, D. Pavon, E. Abdalla, Phys. Lett. B 662 (2008) 1, arXiv: 0711.2214 [hep-th]] are obtained.     

COMPOSITION OF INDOOR AEROSOLS AT EMPEROR QIN''''S TERRA-COTTA MUSEUM, XI''''AN, CHINA, DURING SUMMER, 2004

Particle samples were collected in August 2004 both inside and outside Emperor Qin's Terra-Cotta Museum in Xi'an, China. Mass and chemical composition of total suspended particles (JSP, particles with aerodynamic diameter less than-30μm), PM2.5(particles with aerodynamic diameter <2.5μm) were determined. The average levels of indoor PM2.5 and TSP were 108.4 and 172.4 μg·m-3, respectively, with PM2.5 constituting 62.9% of the TSP mass. Sulfate ((32.4±6.2)%), organics ((27.7±8.0)%), and geological material ((12.5±3.4)%) dominated indoor PM2.5, followed by ammonium ((8.9±2.8)%), nitrate ((7.0±2.9)%), and elemental carbon (EC, (3.9±1.5)%). Particle size distribution varied with the number of tourists in the museum. The size of sulfate, organics, EC, nitrate, and ammonium was found to vary in the range of 0.43 to 3.3 μm in fraction. Ion balance indicated that the aerosol was acidic, with insufficient ammonium ions to neutralize the sulfuric and nitric acids. High concentrations of acidic aerosols will erode the Terra-cotta warriors and horses especially in the summer season with high temperature (30℃) and relative humidity (70%) and undesirable solar radiation inside the museum. More attention should be paid to protecting these precious antiques made 2000 years ago.