Higher Spatial Derivative Terms Research Articles

Spontaneous generation of a crystalline ground state in a higher derivative theory

The possibility of Spontaneous Symmetry Breaking in momentum space in a generic Lifshitz scalar model–a non-relativistic scalar field theory with higher spatial derivative terms–has been studied. We show that the minimum energy state, the ground state, has a lattice structure, where the translation invariance of the continuum theory is reduced to a discrete translation symmetry. The scale of translation symmetry breaking (or induced lattice spacing) is proportional to the inverse of the momentum of the condensate particle. The crystalline ground state is stable under excitations below a certain critical velocity. The small fluctuations above the ground state can have a phonon like dispersion under suitable choice of parameters.At the beginning we have discussed the effects of next to nearest neighbor interaction terms in a model of linear triatomic molecule depicted by a linear system of three particles of same mass connected by identical springs. This model is relevant since in the continuum limit the next to nearest neighbor interaction terms generate higher (spatial) derivative wave equation, the main topic of this paper.

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms II: Improving the Third Derivative Spatial Discretization Using the LDG2 Method

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms II: Improving the Third Derivative Spatial Discretization Using the LDG2 Method

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms: Formulation and Analysis for Third Derivative Spatial Terms Using the LDG Discretization Procedure

AbstractIn this paper, we develop a formulation for solving equations containing higher spatial derivative terms in a spectral volume (SV) context; more specifically the emphasis is on handling equations containing third derivative terms. This formulation is based on the LDG (Local Discontinuous Galerkin) flux discretization method, originally employed for viscous equations containing second derivatives. A linear Fourier analysis was performed to study the dispersion and the dissipation properties of the new formulation. The Fourier analysis was utilized for two purposes: firstly to eliminate all the unstable SV partitions, secondly to obtain the optimal SV partition. Numerical experiments are performed to illustrate the capability of this formulation. Since this formulation is extremely local, it can be easily parallelized and a h-p adaptation is relatively straightforward to implement. In general, the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries (KdV) type problems.

On the consistency of the Horava Theory

With the goal of giving evidence for the theoretical consistency of the Hořava Theory, we perform a Hamiltonian analysis on a classical model suitable for analyzing its effective dynamics at large distances. The model is the lowest-order truncation of the Hořava Theory with the detailed-balance condition. We consider the pure gravitational theory without matter sources. The model has the same potential term of general relativity, but the kinetic term is modified by the inclusion of an arbitrary coupling constant λ. Since this constant breaks the general covariance under space-time diffeomorphisms, it is believed that arbitrary values of λ deviate the model from general relativity. We show that this model is not a deviation at all, instead it is completely equivalent to general relativity in a particular partial gauge fixing for it. In doing this, we clarify the role of a second-class constraint of the model. There have been a lot of interest about Hořava’s proposal of a new theory of gravity which in principle has a renormalizable quantum version [1] (an important part of the conceptual and technical basis was previously developed in Ref. [2]). To build such a theory, Hořava has proposed to abandon the principle of space-time relativity as a fundamental symmetry of nature, reducing the freedom to perform coordinate transformations to those transformations that preserve some preferred universal time-like foliation. The advantage of this scheme is that one can include higher spatial-derivative terms in the Lagrangian that render the theory renormalizable. According to Hořava’s point of view, jorgebellorin@usb.ve arestu@usb.ve

Hawking spectrum and high frequency dispersion.

We study the spectrum of created particles in two-dimensional black hole geometries for a linear, Hermitian scalar field satisfying a Lorentz noninvariant field equation with higher spatial derivative terms that are suppressed by powers of a fundamental momentum scale ${k}_{0}$. The preferred frame is the "free-fall frame" of the black hole. This model is a variation of Unruh's sonic black hole analogy. We find that there are two qualitatively different types of particle production in this model: a thermal Hawking flux generated by "mode conversion" at the black hole horizon, and a nonthermal spectrum generated via scattering off the background into negative free-fall frequency modes. This second process has nothing to do with black holes and does not occur for the ordinary wave equation because such modes do not propagate outside the horizon with positive Killing frequency. The horizon component of the radiation is astonishingly close to a perfect thermal spectrum: for the smoothest metric studied, with Hawking temperature ${T}_{H}\ensuremath{\simeq}0.0008{k}_{0}$, agreement is of order ${(\frac{{T}_{H}}{{k}_{0}})}^{3}$ at frequency $\ensuremath{\omega}={T}_{H}$, and agreement to order $\frac{{T}_{H}}{{k}_{0}}$ persists out to $\frac{\ensuremath{\omega}}{{T}_{H}}\ensuremath{\simeq}45$ where the thermal number flux is \ensuremath{\sim}${10}^{\ensuremath{-}20}$. The flux from scattering dominates at large $\ensuremath{\omega}$ and becomes many orders of magnitude larger than the horizon component for metrics with a "kink," i.e., a region of high curvature localized on a static world line outside the horizon. This nonthermal flux amounts to roughly 10% of the total luminosity for the kinkier metrics considered. The flux exhibits oscillations as a function of frequency which can be explained by interference between the various contributions to the flux.