Complex Ghost Research Articles

Real-time ray transfer for lens flare rendering using sparse polynomials

AbstractWhile lens flares are often undesired artifacts of the imaging process, they are also essential for increasing the level of realism in video games and serve as a powerful artistic tool for photography and filmography. For these reasons, computationally reproducing lens flares has always received special attention. Due to the cost of analytical ray tracing, existing solutions are unable to simultaneously achieve the performance needed for real-time environments and retain the ability to simulate arbitrarily complex ghost shapes. Although polynomial optics has been successfully used to increase the efficiency of ray tracing in multiple rendering areas, no complete and validated solution exists that correctly models all aspects of lens flares. This paper presents our polynomial optics-based method for efficiently and accurately ray tracing lens flare ghosts. Our approach successfully models the shape, energy absorption, chromatic effects, and blocking of lens flare rays by partitioning the input domain into local fitting zones. We demonstrate that our model provides a considerable speedup and high accuracy compared to the analytical approach and achieves better fitting speed, output quality, and rendering performance than the naïve application of polynomial optics. The source code for our implementation is available on GitHub.

Anti-Instability of Complex Ghost

Abstract We argue that Lee–Wick’s complex ghost appearing in any higher derivative theory is stable and its asymptotic field exists. It may be more appropriate to call it “anti-unstable”, in the sense that the more the ghost "decays" into lighter ordinary particles, the larger the probability that the ghost remains as itself becomes. This is explicitly shown by analyzing the two-point functions of the ghost Heisenberg field which is obtained as an exact result in the N → ∞ limit in a massive scalar ghost theory with light O(N)-vector scalar matter. The anti-instability is a consequence of the fact that the poles of the complex ghost propagator are located on the physical sheet in the complex plane of four-momentum squared. This should be contrasted with the case of the ordinary unstable particle, whose propagator has no pole on the physical sheet.

Investigating Unitarity Violation of Lee–Wick’s Complex Ghost with Quantum Field Theory

Theories with fourth-order derivatives like Lee–Wick’s quantum electrodynamics model or quadratic gravity result in complex ghosts above a definite energy threshold that violate unitarity.

Unitarity violation in field theories of Lee–Wick’s complex ghost

Abstract Theories with fourth-order derivatives, including the Lee–Wick finite QED model and quadratic gravity, have a better UV behavior, but the presence of negative metric ghost modes endangers unitarity. Noticing that the ghost acquires a complex mass by radiative corrections, Lee and Wick, in particular, claimed that such complex ghosts would never be created by collisions of physical particles because of energy conservation, so that the physical S-matrix unitarity must hold. We investigate the unitarity problem faithfully, working in the operator formalism of quantum field theory. When complex ghosts participate, a complex delta function (a generalization of the Dirac delta function) appears at each interaction vertex, which enforces a specific conservation law of complex energy. Its particular property implies that the naive Feynman rule is wrong if the four-momenta are assigned to the internal lines after taking account of the conservation law in advance. We show that complex ghosts are actually created and unitarity is violated in such fourth-order derivative theories. We also find a definite energy threshold below which ghosts cannot be created: The theories are unitary and renormalizable below the threshold.

Quantum field theory with ghost pairs

We explicitly show that general local higher-derivative theories with only complex conjugate ghosts and normal real particles are unitary at any perturbative order in the loop expansion. The proof presented here relies on integrating the loop energies on complex paths resulting from the deformation of the purely imaginary paths, when the external energies are continued from imaginary to real values. Contrary to the case of nonlocal theories, where the same integration path was first proposed, for the classes of theories studied here the same procedure is not analytic, but the resulting theory is unitary and unique when the complex ghosts are present in pairs. As an explicit application, a special class of higher-derivative super-renormalizable or finite gravitational and gauge theories turns out to be unitary at any perturbative order if we exclude the complex ghosts from the spectrum of the theory, as it is normally accepted for Becchi-Rouet-Stora-Tyutin (BRST) ghosts. Finally, we propose an analogy between confined gluons in quantum Yang-Mills theory and classical complex pairs in local higher-derivative theories. According to such interpretation, complex ghosts will not appear on shell as asymptotic states because confined in what is natural to name “ghostballs.”

Mixed star-plus-wormhole systems with a complex scalar field

We study compact mixed configurations with a nontrivial wormholelike spacetime topology supported by a complex ghost scalar field with a quartic self-interaction and a polytropic fluid. The latter is modeled by a relativistic barotropic equation of state that can approximately describe more or less realistic matter. For such systems, we find regular asymptotically flat equilibrium solutions describing localized configurations in which the fluid is concentrated in a finite-size region. The solutions obtained describe double-throat wormholes which are located outside the fluid (one can say that the fluid is hidden inside the region between the throats). Also, we consider the dependence of the total mass of the system on the central density of the fluid and demonstrate the existence of critical values of the central density at which the mass diverges. In this case all regular solutions possessing finite masses lie in the region between these critical values, and this region also contains a discontinuity in magnitudes of the central density where only physically unacceptable oscillating solutions do exist. Is shown that for some values of the central density of the fluid there can exist solutions describing systems whose fluid density and pressure maxima lie not at the center. This results in the fact that such systems possess two equators (local maxima of the metric function) resided symmetrically with respect to the center.

Animproved ghost cell method for flow simulation involving static and moving boundary

In this paper, an improved ghost cell method is implemented to solve flow problems with static and moving boundary. The two-dimensional unsteady compressible Euler equations are discretized on a fixed Cartesian grid by the finite volume method, and the monotonic upstream-centered scheme for conservation law scheme is adopted to obtain the second-order precision by reconstructing the numerical fluxes computed by the AUSM + (advection upstream splitting method +) scheme. In time discretization, the explicit third-order total-variation-diminishing Runge-Kutta method is considered. To simplify the assignment method for the ghost cells and deal with the slit problem, the cells in the flow field are selected as the mirror points, which avoids complex interpolations. For preventing non-physical solutions when the mirror point is very close to the boundary, the current mirror point will be replaced by another one, which is regarded as the second flow field cell located in the direction away from the boundary. Moreover, properties of a ghost cell along the <i>X</i> and <i>Y</i> direction are computed respectively, and then the final property is obtained by using a weighted average method, where the weight is determined by the distance between the ghost cell and corresponding boundary point. In this method, the selection of mirror points does not change abruptly for adjacent ghost cells, and therefore, no kink occurs. Furthermore, considering the gradient of the variable near the boundary, an extended scheme of the improved ghost cell method is achieved, which can deal with the flow problems with moving boundaries. When computing the property of a ghost cell by using the extended scheme, the result may be wrong if the location of a shock is just between two mirror points. In view of the problem above, a shock monitor is implemented to switch to the appropriate approach, that is, when the monitor detects a shock between mirror points, the extended scheme will be replaced by the original ghost cell method. Two typical test cases are investigated to validate the accuracy of the proposed method. The first test case is the Schardin’s problem, in which a shock impinges on a finite wedge and is reflected and diffracted. The results at different grid sizes are obtained, and good agreement with experiment results as well as the previous numerical results is achieved, which shows that the improved ghost cell method can offer the same precision as the body-fitted grid method. The second test case is the cylinder lift-off problem involving moving boundaries and the slit problem. Good agreement with the previous results of a high-order complex ghost cell method shows that the improved simple ghost cell method can meet the requirement for dealing with flow problems with moving boundaries.

Wormhole solutions with a complex ghost scalar field and their instability

We study compact configurations with a nontrivial wormholelike spacetime topology supported by a complex ghost scalar field with a quartic self-interaction. For this case, we obtain regular asymptotically flat equilibrium solutions possessing reflection symmetry. We then show their instability with respect to linear radial perturbations.

Effective Models of Quantum Gravity Induced by Planck Scale Modifications in the Covariant Quantum Algebra

We introduce modified covariant quantum algebra based on the so-called Quesne-Tkachuk algebra. By means of a deformation procedure we arrive at a class of higher-derivative models of gravity. The study of the particle spectra of these models reveals equivalence with the physical content of the well-known higher-derivative gravities. The particle spectrum exhibits the presence of spurious complex ghosts and, in light of this problem, we suggest an interesting interpretation in the context of minimal length theories. Also, a discussion regarding the nonrelativistic potential energy is proposed.

Suppressing Ghost Targets via Gating and Track History in Y-Shaped Passive Linear Array Sonars

In Y-shaped passive linear array sonar (PLAS) systems composed of 3 sensor legs with consecutively missed detections in a full scan, the complex ghosts that occur due to bearing ambiguity seriously deteriorate the target tracking performance. A solution to the ghost problem is proposed which exploits the track information and geometrical relationship between each PLAS leg and the target. The region of possible ghost targets is first established by making use of the geometrical relationship of the PLAS with respect to the target. The ghost targets are then eliminated by means of the updated track information refined via a combination of 3-D assignment with gating and judicial track management. In the track management and gating procedure, we propose an optimal measurement selection criterion based on maximum joint probability data association (MJPDA), which combines both the soft and hard association techniques. The established tracks are updated by the Kalman filter with MJPDA and are terminated with a logic-based procedure. Representative simulations demonstrate the effectiveness of the proposed approach.

Ghost orbits in the diamagnetic hydrogen spectrum using harmonic inversion

The harmonic inversion method is applied in the case of the hydrogen atom in a magnetic field to extract classical information from the quantum photo-ionization cross section. The study is made close to a saddle-node bifurcation for which the usual semiclassical formulas give diverging contributions. All quantities (actions, stabilities, and Maslov indices) for real orbits above the bifurcation and for complex ghost orbits below the bifurcation, are found to be in excellent agreement with the modified semiclassical predictions based on a normal form approach.

Hydrogen atom in a magnetic field: Ghost orbits, catastrophes, and uniform semiclassical approximations

Applying closed-orbit theory to the recurrence spectra of the hydrogen atom in a magnetic field, one can interpret most, but not all, structures semiclassically in terms of closed classical orbits. In particular, conventional closed-orbit theory fails near bifurcations of orbits where semiclassical amplitudes exhibit unphysical divergences. Here we analyze the role of ghost orbits living in complex phase space. The ghosts can explain resonance structures in the spectra of the hydrogen atom in a magnetic field at positions where no real orbits exist. For three different types of catastrophes, viz. fold, cusp, and butterfly catastrophes, we construct uniform semiclassical approximations and demonstrate that these solutions are completely determined by classical parameters of the real orbits and complex ghosts.

Semiclassical spectra without periodic orbits for a kicked top.

We present an $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$ approximation for the quasienergy spectrum of a periodically kicked top, valid under conditions of both regular and chaotic classical motion. In contrast to conventional periodic-orbit theory we implement the semiclassical limit for each matrix element of the Floquet operator rather than for the trace of the propagator. Even though a classical looking action is involved, the approximate matrix elements are specified in terms of complex ghost trajectories instead of real classical orbits. Our mean error for the quasienergies is a surprisingly small 3% of the mean spacing.

The Higgs model with a complex ghost pair

A higher derivative term is introduced in the kinetic energy of the Higgs lagrangian in the minimal Standard Model. A logically consistent and finite field theory is obtained when some excitations of the Higgs field are quantized with indefinite metric in the Hilbert space. The Landau ghost phenomenon of the ponventional triviality problem is replaced by the state vectors of a complex ghost pair at a finite mass scale with observable physical consequences. It is shown that the ghost states exhibit unusual resonance properties and correspond to a complex conjugate pair of Pauli-Villars regulator masses in the euclidean path integral formulation of the theory. An argument is given that microscopic acausality effects associated with the ghost pair remain undetectable in scattering processes with realistic wave packets, and the S-matrix should exhibit unitarity in the observable sector of the Hilbert space.

Strongly interacting Higgs sector in the minimal Standard Model?

The triviality Higgs mass bound is studied with a higher derivative regulating in the spontaneously broken phase of the four dimensional O(4) symmetric scalar fiel-theory with quartic self-interaction. The phase diagram of the O(4) model is determined in a Monte Carlo simulation which interpolates between the hypercubic lattice regulator and the higher derivative regulator in continuum space-time. The same method can be used to calculate the Higgs mass bound in continuum space-time. In a large- N analysis, when compared with a hypercubic lattice, we find a relative increase in the triviality bound of the higher derivative regulator suggesting a strongly interacting Higgs sector in the TeV region with negligible dependence on regulator parameters. When the higher derivative regulator mass is brought close to the Higgs mass the model requires a more elaborate analysis of complex ghost states in scattering amplitudes.

Observability of Complex Ghosts and Tachyons

nor electron-electron ~;cattenn,g i~ observed at a suitab1y high ener~y, then a peak \Yill bt: found in the domain -1 <co,; O<l, where it is assumed that the exchanged photon accompanies a tachyon.

Complex poles, cuts, indefinite metric and unitarity

In connection withLee—Wick’s electrodynamics, unitarity problems are discussed on exactly soluble models with complex ghosts.

Completeness and the Lee model with indefinite-metric theories

Indefinite-metric theories with complex ghosts and dipole ghosts are studied in the N0 sector and the NNθ sector of the Lee model with regard to completeness. It is shown that, in the Nθ sector, the eigen-states of the pseudo-Hermitian Hamiltonian form a complete set in the complex-ghost case whereas in the dipole-ghost case they do not. In the NNθ sector it is shown that the energy eigenstates do not form a complete set in both cases.

Connection between Marchenko Formalism and N/D Equations: Regular Interactions. II

In a previous paper we have shown, for S-waves, that the resulting integral equations of the f/f equations (equivalent to the N/D approach) can be obtained from the Marchenko formalism. The potential V(μ, r) reconstructed from the discontinuity μΔ(x) is −2(d/dr)[(dD/dr)/D], where D(μ,r) is the Fredholm denominator of the Jost solution and D(μ,0) that of the resulting integral equation. For ``regular discontinuities'' we find different classes of V(μ, r). First, if D(μ,0)=0, then V(μ, r) is not ``regular at the origin'' [in general, we find that V(μ, r) becomes marginally singular: repulsive and singular like r−2]. Secondly, if D(μ,0)≠0, then V(μ, r) is ``regular'' at the origin and we obtain the following: (i) If |μ| is less than the smallest modulus root of D(μ,0), then V(μ, r) has no poles for r ≥ 0. This range of |μ|-values where the iteration series of the resulting integral equations converge is limited by the smallest |μ| value where a real or complex ghost can appear or where a bound state can appear at zero energy. (ii) For |μ| larger than this smallest modulus root but μ inside the interval given by the first positive and negative roots, V(μ, r) has no second-order poles for r ≥ 0. These results (i) and (ii) are obtained with the restriction that in the considered interval there do not exist (μ, r)-values such that D(±μ,r)=0, and from our study we cannot conclude that this is always true. (iii) For μ outside the above interval, V(μ, r) has poles of the second order for r &amp;gt; 0, the ``bound states'' being, in general, real or complex ghosts, or ``bound states'' corresponding to badly behaved potentials. We find also that the Jost solutions for energy equal to zero are D(−μ,r)/D(μ,r). This gives the connection between ghosts and, in general, possible bound states appearing at zero energy; this gives also the relations between poles of V(μ, r) corresponding to opposite μ values. These results for the Jost function correspond to a normalization at infinity, so we have considered the problem of subtractions with normalization at an arbitrary point. Then the new Fredholm determinant is the product of the old one by the value of the Jost function at the subtracted point. It follows that, if the first μ-greater-than-zero and the first μ-lessthan-zero roots of D(μ,r) are not opposite (r ≥ 0), the |μ| interval of convergence of the iteration series is enlarged for the subtracted equation.