Snyder Model Research Articles

Worldline formalism in Snyder spaces

We study the $\phi_{\star}^4$ model for a scalar field in a linearization of the Snyder model, using the methods of the Worldline Formalism. Our main result is a master equation for the 1-loop n-point function. From this we derive the renormalization of the coupling parameters of the theory and observe the appearance of a $\phi^6$ divergent contribution that opens the question of whether this theory is renormalizable or not. Additionally, we observe that some terms in the renormalized action can be interpreted as coming from an effective metric proportional to the square of the field.

Thermal and spectral dimension of (generalized) Snyder noncommutative spacetimes

We report an investigation of the Snyder noncommutative spacetime and of some of its most natural generalizations, also looking at them as a powerful tool for comparing different notions of dimensionality of a quantum spacetime. It is known that (generalized-)Snyder noncommutativity, while having rich off-shell implications (kinematical Hilbert space), does not affect noninteracting on-shell particles (physical Hilbert space), and we argue that physically meaningful notions of dimensionality should describe such spacetimes as trivially four-dimensional, without any running with scales. By studying the thermodynamics of a gas of massless particles living on these spacetimes, we find that indeed the Snyder model and its generalizations have constant thermal dimension of four. We also compute the spectral dimension of the Snyder model and its generalizations, finding that, as a result of its sensitivity to off-shell properties, it runs from the standard value of four in the infrared towards lower values in the ultraviolet limit.

Relative-locality phenomenology on Snyder spacetime

We study the effects of relative locality dynamics in the case of the Snyder model. Several properties of this model differ from those of the widely studied models: for example, in the Snyder case the action of the Lorentz group is preserved, and the composition law of momenta is deformed by terms quadratic in the inverse Planck energy. From the investigation of time delay and dual curvature lensing we deduce that, because of these differences, in the Snyder case the properties of the detector are essential for the observation of relative locality effects. The deviations from special relativity do not depend on the energy of the particles and are much smaller than in the case, so are beyond the reach of present astrophysical experiments. However, these results have a conceptual interest, because they show that relative-locality effects can occur even if the action of the Lorentz group on phase space is not deformed.

Relativistic Oscillators in Generalized Snyder Model

We present an exact solution of one-dimensional Klein–Gordon and Dirac oscillators subjected to the uniform electric field with Snyder–de Sitter model in the momentum space, known in quantum mechanics by the stark effect. The energy eigenvalues and eigenfunctions are determined for both cases. The pure relativistic oscillator is obtained as particular case by taking the limit when the electric field vanishes. We also have determined some formulas of thermodynamics properties for relativistic oscillators within this framework.

Snyder-type space–times, twisted Poincaré algebra and addition of momenta

We discuss a generalization of the Snyder model compatible with undeformed Lorentz symmetries, which we describe in terms of a large class of deformations of the Heisenberg algebra. The corresponding deformed addition of momenta, the twist and the R-matrix are calculated to first order in the deformation parameters for all models. In the particular case of the Snyder realization, an analytic formula for the twist is obtained.

Evaluation of Class-A Pan Coefficient Models for Estimation of Reference Crop Evapotranspiration for Dry Sub-humid Climates

Evaporation and evapotranspiration are important parameters for various agricultural activities, which may be estimated using a pan coefficient value obtained from several models. It is a function of wind speed, temperature, relative humidity and fetch length. For dry sub-humid climate of Varanasi, Snyder model was found to be best for estimating the pan coefficient with an average value of 0.79, which closely agreed with pan coefficient value estimated by the standard FAO-56 model. The maximum and minimum average daily pan evaporation for the region was 8.8 mm.day-1 and 2.5 mm.day-1, respectively, in the month of May and December. The reference evapotranspiration estimated using pan coefficient obtained from Snyder method showed lowest MAE of 0.24, RMSE of 0.30, agreement Index of 0.99, percentage error of estimate of 6.51 % and efficiency of 96 per cent. The Mann-Kendall’s non-parametric test used for identifying trend showed an increasing trend for pan coefficient data series, and a decreasing trend for pan evaporation data series at 5 % significance level. Tempoal variation in pan coefficient may be computed for Varanasi meteorological station to estimate reference evapotranspiration.

Generalized higher-order Snyder model

In this paper we consider the higher-order generalization of the Snyder algebra. We find the modified inner product for this algebra. We also discuss the one-dimensional hydrogen atom eigenvalue problem. We find the invariant phase space volume for the generalized Snyder algebra. Finally we use the density of states in three dimension to discuss the Maxwellian distribution and cosmological constant.

Quantum field theory in generalised Snyder spaces

We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate perturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT.

Study of the slope of the linear relationship between retention and mobile phase composition (Snyder-Soczewiñski model) in normal phase liquid chromatography with bonded and charge-transfer phases

The Snyder model and the Soczewiñski model are compared on classic NPLC bonded phases using literature data, and on the charge transfer 2, 4-dinitroanilinopropyl (DNAP) column using experimentally collected data. Overall, the Snyder model slightly better predicts the n-slope than the Soczewiñski model. However, both models give comparable uncertainty in predicting n-slope for a given compound. The number of aromatic double bonds was the most suitable descriptor for estimating the relative n-slope of PAHs, as it correlated with behavior better than the number of aromatic rings and is simpler to calculate than the solute adsorption area. On the DNAP phase, a modified Soczewiñski model is suggested to allow for the significant contribution of the aromatic rings to the n-slope. For classic NPLC bonded phases and DNAP columns, the contribution of polar group to the n-slope parallels the adsorption energy of each polar group.

STUDI TENTANG MODEL HIDROGRAF SATUAN SINTETIK PADA SUB DAS BAYUR SAMARINDA, KALIMANTAN TIMUR

The present research entitled Study of Synthetic Unit Hydrograph Model of Bayur River Basin at Samarinda, East Kalimantan was aimed to analyze and identify the best-suited synthetic unit hydrograph to apply in Bayur river basin. The research was carried out by developing models, and then those models were compared to main unit hydrograph based on 2004 field measurement. Comparative analysis was performed on two stages. The first was to calculate and compare three-unit hydrograph parameters (Tp, Qp and Tb) of two synthetic unit hydrograph to main unit hydrograph. Second stage was to develop direct runoff hydrograph of two synthetic unit hydrograph models and of the main unit hydrograph using rainfall data of April 9, 2004. Then, deviation percentage averages of the two synthetic direct runoff hydrograph were calculated. Through the two analysis stages, the best-suited synthetic unit hydrograph model to apply in Bayur river basin presumably was obtained by identifying smallest deviation percentage average on main unit hydrograph. By calculating deviation averages of direct runoff hydrograph in the two models of synthetic unit hydrograph, it was found that GAMA I model provided smallest deviation average compared Snyder model. Based on the two analysis, it was showed that GAMA I was the closest model to main unit hydrograph compared to Snyder model. Key word: synthetic unit hydrograph, river basin, direct runoff, Snyder model, GAMA I model.

Classical dynamics on Snyder spacetime

We study the classical dynamics of a particle in Snyder spacetime, adopting the formalism of constrained Hamiltonian systems introduced by Dirac. We show that the motion of a particle in a scalar potential is deformed with respect to special relativity by terms of order βE2. A remarkable result is that in the relativistic Snyder model a consistent choice of the time variable must necessarily depend on the dynamics. This is a consequence of the nontrivial mixing between position and momentum coordinates intrinsic to the Snyder model.

Collapse and dispersal of a homogeneous spin fluid in Einstein–Cartan theory

In the present work, we revisit the process of gravitational collapse of a spherically symmetric homogeneous dust fluid which is described by the Oppenheimer–Snyder (OS) model (Oppenheimer and Snyder in Phys Rev D 56:455, 1939). We show that such a scenario would not end in a spacetime singularity when the spin degrees of freedom of fermionic particles within the collapsing cloud are taken into account. To this purpose, we take the matter content of the stellar object as a homogeneous Weyssenhoff fluid. Employing the homogeneous and isotropic FLRW metric for the interior spacetime setup, it is shown that the spin of matter, in the context of a negative pressure, acts against the pull of gravity and decelerates the dynamical evolution of the collapse in its later stages. Our results show a picture of gravitational collapse in which the collapse process halts at a finite radius, whose value depends on the initial configuration. We thus show that the spacetime singularity that occurs in the OS model is replaced by a non-singular bounce beyond which the collapsing cloud re-expands to infinity. Depending on the model parameters, one can find a minimum value for the boundary of the collapsing cloud or correspondingly a threshold value for the mass content below which the horizon formation can be avoided. Our results are supported by a thorough numerical analysis.

Usefulness of class A Pan coefficient models for computation of reference evapotranspiration for a semi-arid region

The reliability of estimates of reference evapotranspiration (ET0) using pan evaporation (Epan) depends on the accurate determination of pan coefficients (Kpan). Six ET0 models were evaluated for their usefulness using 33-year climatological dataset of a semi-arid region of the Gujarat state of India. The equations compared include Cuenca (1989), Allen and Pruitt (1991), Snyder (1992), Modified Snyder (Grismer et al., 2002), Orang (1998), and Pereira et al. (1995). The ET0 data, calculated using daily Kpan values from these equations, were compared to the Food and Agricultural Organization (FAO)-Penman-Monteith (FAO56-PM) method as a reference. Based on the visual comparison as well as from the statistical criteria, ET0 values computed using Modified Snyder and Orang model have very close agreement with the FAO56-PM method for daily, monthly, and annual estimates as compared to other approaches. The sequential performances of the explored models was found as: Modified Snyder (Eqn. 5) > Orang (Eqn. 6) > Cuenca Eqn. (2) > Allen & Pruitt (Eqn. 3) > Snyder (Eqn. 4) > Pereira et al. (Eqn. 7) model. Therefore, the Modified Snyder model (Grismer et al., 2002) could be recommended as the best model for ET0 computations under these prevailing climatic conditions for a semi arid region.

Snyder dynamics in a Schwarzschild spacetime

We calculate the orbits of a particle in Schwarzschild spacetime, assuming that the dynamics is governed by a Snyder symplectic structure. With this assumption, the perihelion shift of the planets acquires an additional contribution with respect to the one predicted by general relativity. Moreover, the equivalence principle is violated. If one assumes that Snyder mechanics is valid also for macroscopic systems, these results impose strong constraints on the value of the coupling parameter of the Snyder model.

Classical and quantum mechanics of the nonrelativistic Snyder model in curved space

The Snyder–de Sitter (SdS) model is a generalization of the Snyder model to a spacetime background of constant curvature. It is an example of noncommutative spacetime admitting two fundamental scales besides the speed of light, and is invariant under the action of the de Sitter group. Here, we consider its nonrelativistic counterpart, i.e. the Snyder model restricted to a three-dimensional sphere, and the related model obtained by considering the anti-Snyder model on a pseudosphere, that we call anti-Snyder–de Sitter (aSdS). By means of a nonlinear transformation relating the SdS phase-space variables to canonical ones, we are able to investigate the classical and the quantum mechanics of a free particle and of an oscillator in this framework. In their flat space limit, the SdS and aSdS models exhibit rather different properties. In the SdS case, a lower bound on the localization in position and momentum spaces arises, which is not present in the aSdS model. In the aSdS case, instead, a specific combination of position and momentum coordinates cannot exceed a constant value. We explicitly solve the classical and the quantum equations for the motion of the free particle and of the harmonic oscillator. In both the SdS and aSdS cases, the frequency of the harmonic oscillator acquires a dependence on the energy. Moreover, in the aSdS model only a finite number of states is present.

Hansen Solubility Parameters for the Solid Surface of Cellulose Acetate Propionate by Inverse Gas Chromatography

The specific retention volumes, V 0 g , for adsorption of 21 solute probes on the solid surface of cellulose acetate propionate (CAP) were determined in the temperature range 343.15–403.15 K by inverse gas chromatography. In the temperature range studied the retention diagrams drawn between vs. 1/T were found to be linear for all the solutes. Hansen solubility parameters (HSPs) were calculated by relating with the cohesive energy of adsorption of the solutes on the surface of CAP following the Snyder and Karger adsorption model. The three components of HSP decreased linearly with increase of temperature. The HSP of CAP were compared with the corresponding values determined on cellulose acetate butyrate (CAB) and discussed in the light of the chemical nature of the two solid surfaces. Enthalpies of adsorption were found to be more negative for CAP than for CAB.

Classical and quantum mechanics in the Snyder space

The Snyder model is an example of noncommutative spacetime admitting a fundamental length scale and invariant under Lorentz transformations. Here, we consider its nonrelativistic counterpart, i.e. the Snyder model restricted to three-dimensional Euclidean space. We discuss the classical and the quantum mechanics of a free particle in this framework, and show that they strongly depend on the sign of a coupling constant λ, appearing in the fundamental commutators. If λ is negative, momenta are bounded, while for λ > 0 a minimal localization length arises. We also give the exact solution of the harmonic oscillator equations both in the classical and the quantum case, and show that its frequency is energy dependent.

Groseclose and Snyder in finite legislatures

The Groseclose and Snyder (1996) model is one of the best-known models of vote buying in legislatures. Although the logic of the model is compelling, it is not clear that its key propositions, derived in a continuous set-up, hold in finite legislatures. This is an important issue because many real-world legislatures are small and should be modeled as finite in order to make predictions on coalition formation in them. This paper makes two contributions. The main one is to show with full generality that the key propositions in the Groseclose and Snyder model do carry through into finite legislatures. Secondly, it clarifies the role that parameter restrictions played in previous work on this question by Banks (2000) which was not fully general.

Classical and quantum mechanics of the nonrelativistic Snyder model

The Snyder model is an example of noncommutative spacetime admitting a fundamental length scale $\beta$ and invariant under Lorentz transformations, that can be interpreted as a realization of the doubly special relativity axioms. Here, we consider its nonrelativistic counterpart, i.e. the Snyder model restricted to three-dimensional Euclidean space. We discuss the classical and the quantum mechanics of a free particle in this framework, and show that they strongly depend on the sign of a coupling constant $\lambda$, appearing in the fundamental commutators and proportional to $\beta^2$. For example, if $\lambda$ is negative, momenta are bounded. On the contrary, for positive $\lambda$, positions and areas are quantized. We also give the exact solution of the harmonic oscillator equations both in the classical and the quantum case, and show that its frequency is energy dependent.

Snyder–de Sitter model from two-time physics

We show that the symplectic structure of the Snyder model on a de Sitter background can be derived from two-time physics in seven dimensions and propose a Hamiltonian for a free particle consistent with the symmetries of the model.