Negative Energy Modes Research Articles

Theory and phenomenology of stressed wave-dark-matter soliton

Soliton in the hostile turbulent wave dark matter ($\mathrm{\ensuremath{\Psi}}\mathrm{DM}$) halo of a galaxy agitates with various kinds of excitation, and the soliton even breathes heavily under great stress. A theory of collective excitation for a $\mathrm{\ensuremath{\Psi}}\mathrm{DM}$ soliton is presented. The collective excitation has different degrees of coupling to negative energy modes, where lower-order excitation generally necessitates more negative energy coupling. A constrained variational principle is developed to assess the frequencies and mode structures of small-amplitude perturbations. The predicted frequencies are in good agreement with those found in simulations. Soliton breathing at amplitudes on the verge of breakup is also a highlight of this work. Even in this extreme nonlinear regime, the wave function perturbation amplitudes are moderate. The simulation data show a stable oscillation with frequency weakly dependent on the oscillation amplitude and hint at a self-consistent quasilinear model for the wave function that accounts for modifications in the ground-state wave function and the equilibrium density. The mock solution, constructed from the simulation data, can shed light on the dynamics of the large-amplitude breathing soliton and support the quasilinear model, as evidenced by its ability to predict well the nonlinear eigenfrequency shifts and large-amplitude breathing frequency observed in simulations.

High-Resolution Coherent Probe Spectroscopy of a Polariton Quantum Fluid.

Characterizing elementary excitations in quantum fluids is essential to study their collective effects. We present an original angle-resolved coherent probe spectroscopy technique to study the dispersion of these excitation modes in a fluid of polaritons under resonant pumping. Thanks to the unprecedented spectral and spatial resolution, we observe directly the low-energy phononic behavior and detect the negative-energy modes, i.e., the ghost branch, of the dispersion relation. In addition, we reveal narrow spectral features precursory of dynamical instabilities due to the intrinsic out-of-equilibrium nature of the system. This technique provides the missing tool for the quantitative study of quantum hydrodynamics in polariton fluids.

κ -deformed complex scalar field: Conserved charges, symmetries, and their impact on physical observables

In this paper we revisit the model of $\kappa$-deformed complex scalar field. We find that this model possesses ten conserved Noether charges that form, under commutators, a representation of (undeformed) Poincar\'e algebra. It follows that the theory is relativistic and does not break Lorentz invariance. However the spacetime representation of boosts is not standard, and contains a non-local translation, different for positive and negative energy modes. It then follows that although the masses of particles and anti-particles are equal, the theory violates CPT symmetry in a subtle way. We explain why the Jost-Wightman-Greenberg theorem of equivalence of the Poincar\'e symmetry and CPT fails in our case. Finally, we discuss the phenomenological consequences of the theory and its possible observational signatures.

Treating divergence in quark matter by using energy projectors

We calculate the gluon self-energy using quark energy projectors in a general quark-gluon plasma. By separating the quark field into positive- and a negative-energy modes, the quark loop constructed with the same mode is always convergent, and the divergence appears only in the mixed loop with different modes and is medium independent. After removing the divergence in vacuum, we obtain the one-loop gluon self-energy at finite temperature, chemical potential, and quark mass without approximation. With the method of quark-loop resummation, we calculate nonperturbatively the gluon Debye mass and thermodynamic potential. In the limit of small gluon momentum in comparison with temperature, chemical potential, and quark mass, our calculation comes back to the known HTL/HDL results in literature.

Effect of water depth on Kelvin–Helmholtz instability in a shallow water flow

It is well known that the interface between two regions of an incompressible ideal fluid flow moving in a relative motion is necessarily destabilized, regardless of the velocity difference’s strength. This phenomenon is the so-called Kelvin–Helmholtz instability. However, a large number of works demonstrated a surprising result that the instability is suppressed for shallow water flows; the interface is stabilized if the Froude number, defined by the velocity difference’s ratio to the gravity wave’s speed, is sufficiently large. In a limited way, these authors used the shallow water equations without the higher-order effect of the dispersive terms. Thus, this investigation aims at examining these higher-order dispersive effects to analyze the interface stability problem of tangential-velocity discontinuity in shallow water flows. In particular, we use the Green–Naghdi equations to introduce the dispersive terms related to the depth and the depth-averaged horizontal velocities of the fluid. We show that the interface stability depends on the Froude number (i.e., the velocity difference’s strength) and the water depth. A critical value of the Froude number to stabilize the interface is smaller than the case of no dispersive terms, and the flow in a deeper region is more stable than in a shallower one. We also consider the distribution of kinetic and potential energy to clarify a feature characteristic of a large class of instabilities in shallow water flow. The instability of flows is caused by the decrease in the kinetic energy during the perturbation of waves. This phenomenon is known as negative energy modes and plays a vital role in applying the model to industrial equipment. A conclusion is that the equipartition of energies occurs if and only if the velocity difference is zero and the water depth is shallow enough to ignore the dispersive terms.

Holographic entanglement entropy in flat limit of the generalized minimal massive gravity model

Previously we have studied the Generalized Minimal Massive Gravity (GMMG) in asymptotically AdS_3 background, and have shown that the theory is free of negative-energy bulk modes. Also we have shown GMMG avoids the aforementioned “bulk-boundary unitarity clash”. Here instead of AdS_3 space we consider asymptotically flat space, and study this model in the flat limit. The dual field theory of GMMG in the flat limit is a BMS_3 invariant field theory, dubbed (BMSFT) and we have BMS algebra asymptotically instead of Virasoro algebra. In fact here we present an evidence for this claim. Entanglement entropy of GMMG is calculated in the background in the flat null infinity. Our evidence for mentioned claim is the result for entanglement entropy in filed theory side and in the bulk (in the gravity side). At first using Cardy formula and Rindler transformation, we calculate entanglement entropy of BMSFT in three different cases. Zero temperature on the plane and on the cylinder, and non-zero temperature case. Then we obtain the entanglement entropy in the bulk. Our results in gravity side are exactly in agreement with field theory calculations.

Hawking radiation in a non-covariant frame: the Jacobi metric approach

The present paper deals with a reformulation of the derivation of Hawking temperature for static and stationary black holes. In contrast to the conventional approach, where the covariant form of the metrics is used, we use the manifestly non-covariant Jacobi metric for the black holes in question. In the latter, a restricted form of Hamilton–Jacobi variational principle is exploited where the energy of the particle (pertaining to Hawking radiation) appears explicitly in the metric as a constant parameter. Our analysis shows that, as far as computation of Hawking temperature (for stationary black holes) is concerned, the Jacobi metric framework is more streamlined and yields the result with less amount of complications (as for example, considerations of positive and negative energy modes and signature change of the metric across horizon do not play any direct role in the present analysis).

Unitarity problems in 3D gravity theories

We revisit the problem of the bulk-boundary unitarity clash in 2 + 1 dimensional gravity theories, which has been an obstacle in providing a viable dual two-dimensional conformal field theory for bulk gravity in anti-de Sitter (AdS) spacetime. Chiral gravity, which is a particular limit of cosmological topologically massive gravity (TMG), suffers from pertur- bative log-modes with negative energies inducing a non-unitary logarithmic boundary field theory. We show here that any f(R) extension of TMG does not improve the situation. We also study the perturbative modes in the metric formulation of minimal massive gravity- originally constructed in a first-order formulation-and find that the massive mode has again negative energy except in the chiral limit. We comment on this issue and also discuss a possible solution to the problem of negative energy modes. In any of these theories, the infinitesimal dangerous deformations might not be integrable to full solutions; this suggests a linearization instability of AdS spacetime in the direction of the perturbative log-modes.

Analysis of self-consistent nonlinear wave-particle interactions of whistler waves in laboratory and space plasmas

Whistler mode chorus is one of the most important emissions affecting the energization of the radiation belts. Recent laboratory experiments that inject energetic electron beams into a cold plasma have revealed several spectral features in the nonlinear evolution of these instabilities that have also been observed in high-time resolution in situ wave-form data. These features include (1) a sub-element structure which consists of an amplitude modulation on time-scales slower than the bounce time, (2) closely spaced discrete frequency hopping that results in a faster apparent frequency chirp rate, (3) fast frequency changes near the sub-element boundaries, and (4) harmonic generation. In this paper, we develop a finite dimensional self-consistent Hamiltonian model for the evolution of the resonant beam of electrons. We analyze a single wave case and demonstrate that the instability occurs due to a Krein collision, which manifests as a coupling between a negative and positive energy mode. This analysis revealed that the nonlinear evolution of the spectrally stable fixed-points of the self-consistent Hamiltonian develop a sub-packet structure similar to that of space observations. We then analyze the case of two whistler waves to show that the model reproduces the nonlinear harmonic generation and leads to a hypothesis for the closely spaced frequency hopping observed in laboratory experiments and space data.

On the nonlinear trapping nature of undamped, coherent structures in collisionless plasmas and its impact on stability

An outstanding notion for collisionless plasmas is the essential nonlinear character of their coherent structures, which in the stationary, weak amplitude limit are described by a continuum of cnoidal electron and ion hole modes governed by a multiparametric nonlinear dispersion relation. The well-known discrete structure of undamped linear plasma modes is seamlessly embedded in this nonlinear continuum as the microscopic texture of plasma begins to reveal itself in the high temperature collisionless plasma limit. This transforms the linear-threshold-based operating mechanism of plasma turbulence into a fundamental nonlinear, multifaceted one. Based on a comprehensive three-level description of increasing profundity, a proof of this novel dictum is presented, which makes use of the joint properties of such structures, their coherency and stationarity, and uses in succession a fluid, linear Vlasov and a full Vlasov description. It unifies discrete and continuum limits by resolving the inevitable resonant region and shows that coherent electrostatic equilibria are generally controlled by kinetic particle trapping and are hence fundamentally nonlinear. By forging a link between damped and growing wave solutions, these modes render plasma stability complex and difficult to evaluate due to the entangled pattern of the stability boundary in function and parameter space, respectively. A direct consequence is the existence of negative energy modes of arbitrarily small amplitudes in the subcritical region of the two-stream instability as well as the failure of linear Landau (Vlasov, van Kampen) theory, whenever resonant particles are involved, in addressing the onset of instability in a current-carrying plasma. Responsible for this subtle phase space behavior is hence the thresholdless omnipresence of the trapping nonlinearity originating from coherency. A high resolution, exact-mass-ratio, multispecies, and collisionless plasma simulation is employed to illustrate exemplarily how tiny seed fluctuations in phase-space can act as a triggering agent for a subcritical plasma excitation verifying an access to these modes in the noisy, collisionless plasma limit.

On the generalized minimal massive gravity

In this paper we study the Generalized Minimal Massive Gravity (GMMG) in asymptotically AdS3 background. The generalized minimal massive gravity theory is realized by adding the CS deformation term, the higher derivative deformation term, and an extra term to pure Einstein gravity with a negative cosmological constant. We study the linearized excitations around the AdS3 background and find that at special point (tricritical) in parameter space the two massive graviton solutions become massless and they are replaced by two solutions with logarithmic and logarithmic-squared boundary behavior. So it is natural to propose that GMMG model could also provide a holographic description for a 3-rank Logarithmic Conformal Field Theory (LCFT). We calculate the energy of the linearized gravitons in AdS3 background, and show that the theory is free of negative-energy bulk modes. Then we obtain the central charges of the CFT dual explicitly and show GMMG also avoids the aforementioned “bulk-boundary unitarity clash”. After that we show that General Zwei–Dreibein Gravity (GZDG) model can reduce to GMMG model. Finally by a Hamiltonian analysis we show that the GMMG model has no Boulware–Deser ghosts and this model propagates only two physical modes.

Ion sound instability driven by the ion flows

Ion sound instabilities driven by the ion flow in a system of a finite length are considered by analytical and numerical methods. The ion sound waves are modified by the presence of stationary ion flow resulting in negative and positive energy modes. The instability develops due to coupling of negative and positive energy modes mediated by reflections from the boundary. It is shown that the wave dispersion due to deviation from quasineutrality is crucial for the stability. In finite length system, the dispersion is characterized by the length of the system measured in units of the Debye length. The instability is studied analytically and the results are compared with direct, initial value numerical simulations.

Horizons of semiclassical black holes are cold

We calculate, using our recently proposed semiclassical framework, the quantum state of the Hawking pairs that are produced during the evaporation of a black hole (BH). Our framework adheres to the standard rules of quantum mechanics and incorporates the quantum fluctuations of the collapsing shell spacetime in Hawking's original calculation, while accounting for back-reaction effects. We argue that the negative-energy Hawking modes need to be regularly integrated out; and so these are effectively subsumed by the BH and, as a result, the number of coherent negative-energy modes $N_{coh}$ at any given time is parametrically smaller than the total number of the Hawking particles $N_{total}$ emitted during the lifetime of the BH. We find that $N_{coh}$ is determined by the width of the BH wavefunction and scales as the square root of the BH entropy. We also find that the coherent negative-energy modes are strongly entangled with their positive-energy partners. Previously, we have found that $N_{coh}$ is also the number of coherent outgoing particles and that information can be continually transferred to the outgoing radiation at a rate set by $N_{coh}$. Our current results show that, while the BH is semiclassical, information can be released without jeopardizing the nearly maximal inside-out entanglement and imply that the state of matter near the horizon is approximately the vacuum. The BH firewall proposal, on the other hand, is that the state of matter near the horizon deviates substantially from the vacuum, starting at the Page time. We find that, under the usual assumptions for justifying the formation of a firewall, one does indeed form at the Page time. However, the possible loophole lies in the implicit assumption that the number of strongly entangled pairs can be of the same order of $N_{total}$.

Zitterbewegung in the honeycomb photonic lattice

The propagation of a wave packet in a honeycomb photonic lattice has been studied using the time-dependent wave packet dynamics. It is found that the wave packet, superposed from the positive and negative energy modes at the vicinity of the two inequivalent Dirac points, can transform into a double-ring structure, which is caused by the interference between the two positive and negative energy modes around the Dirac points and is closely related to the Zitterbewegung (ZB). Also, a possible way to detect the ZB effect is proposed in the honeycomb photonic lattice.

Hawking radiation and the boomerang behavior of massive modes near a horizon

We discuss the behaviour of massive modes near a horizon based on a study of the dispersion relation and wave packet simulations of the Klein-Gordon equation. We point out an apparent paradox between two (in principle equivalent) pictures of black hole evaporation through Hawking radiation. In the picture in which the evaporation is due to the emission of positive-energy modes, one immediately obtains a threshold for the emission of massive particles. In the picture in which the evaporation is due to the absorption of negative-energy modes, such a threshold apparently does not exist. We resolve this paradox by tracing the evolution of the positive-energy massive modes with an energy below the threshold. These are seen to be emitted and move away from the black hole horizon, but they bounce back at a "red horizon" and are re-absorbed by the black hole, thus compensating exactly for the difference between the two pictures. For astrophysical black holes, the consequences are curious but do not affect the terrestrial constraints on observing Hawking radiation. For analogue gravity systems with massive modes, however, the consequences are crucial and rather surprising.

Mode signature and stability for a Hamiltonian model of electron temperature gradient turbulence

Stability properties and mode signature for equilibria of a model of electron temperature gradient (ETG) driven turbulence are investigated by Hamiltonian techniques. After deriving new infinite families of Casimir invariants, associated with the noncanonical Poisson bracket of the model, a sufficient condition for stability is obtained by means of the Energy-Casimir method. Mode signature is then investigated for linear motions about homogeneous equilibria. Depending on the sign of the equilibrium “translated” pressure gradient, stable equilibria can either be energy stable, i.e., possess definite linearized perturbation energy (Hamiltonian), or spectrally stable with the existence of negative energy modes. The ETG instability is then shown to arise through a Kreĭn-type bifurcation, due to the merging of a positive and a negative energy mode, corresponding to two modified drift waves admitted by the system. The Hamiltonian of the linearized system is then explicitly transformed into normal form, which unambiguously defines mode signature. In particular, the fast mode turns out to always be a positive energy mode, whereas the energy of the slow mode can have either positive or negative sign. A reduced model with stable equilibria shear flow that possess a continuous spectrum is also analyzed and brought to normal form by a special integral transform. In this way it is seen how continuous spectra can have signature as well.

Stability and dynamics of matter-wave vortices in the presence of collisional inhomogeneities and dissipative perturbations

In this work, the spectral properties of a singly charged vortex in a Bose–Einstein condensate confined in a highly anisotropic (disc-shaped) harmonic trap are investigated. Special emphasis is placed on the analysis of the so-called anomalous (negative energy) mode of the Bogoliubov spectrum. We use analytical and numerical techniques to illustrate the connection of the anomalous mode to the precession dynamics of the vortex in the trap. Effects due to inhomogeneous interatomic interactions and dissipative perturbations motivated by finite-temperature considerations are explored. We find that both of these effects may give rise to oscillatory instabilities of the vortex, which are suitably diagnosed through the perturbation-induced evolution of the anomalous mode, and monitored by direct numerical simulations.

Lorentz invariant field theory on κ–Minkowski space

It is by now well established that the momentum space associated with the non-commutative κ-Minkowski space is a submanifold of de Sitter space. It has been noted recently that field theories built on such momentum space suffer from a subtle form of Lorentz symmetry breaking. Namely, for any negative energy mode the allowed range of rapidities is bounded above. In this paper we construct a complex scalar field theory with a modified action of Lorentz generators which avoids this problem. For such theory we derive conserved charges corresponding to translational and U(1) symmetries. We also discuss in some detail the inner product and Hilbert space structure of the κ-deformed complex quantum field.

Stability of warpedAdS3vacua of topologically massive gravity

AdS3 vacua of topologically massive gravity (TMG) have been shown to be perturbatively unstable for all values of the coupling constant except the chiral point \mu l=1. We study the possibility that the warped vacua of TMG, which exist for all values of \mu, are stable under linearized perturbations. In this paper, we show that spacelike warped AdS3 vacua with Compere-Detournay boundary conditions are indeed stable in the range \mu l > 3. This is precisely the range in which black hole solutions arise as discrete identifications of the warped AdS3 vacuum. The situation somewhat resembles chiral gravity: although negative energy modes do exist, they are all excluded by the boundary conditions, and the perturbative spectrum solely consists of boundary (pure large gauge) gravitons.

Topologically massive gravity and the AdS/CFT correspondence

We set up the AdS/CFT correspondence for topologically massive gravity (TMG) in three dimensions. The first step in this procedure is to determine the appropriate fall off conditions at infinity. These cannot be fixed a priori as they depend on the bulk theory under consideration and are derived by solving asymptotically the non-linear field equations. We discuss in detail the asymptotic structure of the field equations for TMG, showing that it contains leading and subleading logarithms, determine the map between bulk fields and CFT operators, obtain the appropriate counterterms needed for holographic renormalization and compute holographically one- and two-point functions at and away from the ``chiral point'' (μ = 1). The 2-point functions at the chiral point are those of a logarithmic CFT (LCFT) with cL = 0,cR = 3l/GN and b = −3l/GN, where b is a parameter characterizing different c = 0 LCFTs. The bulk correlators away from the chiral point (μ ≠ 1) smoothly limit to the LCFT ones as μ → 1. Away from the chiral point, the CFT contains a state of negative norm and the expectation value of the energy momentum tensor in that state is also negative, reflecting a corresponding bulk instability due to negative energy modes.