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Ghost modules were introduced in [I3] without definitions or proofs. We also introduced stability diagrams or "relative pictures" for torsion classes and torsion-free classes for representations of Dynkin quivers. Modules which were not in the chosen class reappeared as "ghosts", in fact one missing module produced two ghosts. In this short paper, we give a precise definition of ghost modules. We give several examples and prove basic properties of ghosts and pictures for torsion and torsion-free classes. We also introduce a third kind of ghost which we call "extension ghosts". In the next paper we will explain how these new ghosts can be used to visualize the computation of other invariants of $K_3$ of group rings.

General Relativity (GR) is an effective field theory valid in the infrared regime. Quadratic curvature extensions intended to probe ultraviolet physics generically propagate a massive spin-$2$ ghost and are therefore non-unitary. One route to remove ghost is by enlarging the geometric sector (torsion, non-metricity). We investigate the infrared phenomenology of both the standard (ghostful) and ghost-free fourth-order gravity theories by computing Gravitational Wave (GW) emission and confronting the results with observations such as the orbital-period decay of quasi-stable binaries such as PSR B1913+16 and PSR J1738+0333 and the chirp-mass evolution of GW170817. In the ghostful theory, besides the theoretical inconsistency due to non-unitarity, there are also phenomenological problems: the massless spin-$2$ GW flux cancels the combined GW fluxes of the massive spin-$2$ ghost and massive spin-$0$ scalar in the vanishing-mass limit, so the GR quadrupole formula is not recovered at the leading order. As a result, we obtain the GW constraint on the ghostful theory as $m\gtrsim 10^{-11}~\mathrm{eV}$, where $m$ is the mass of the massive modes. By contrast, the ghost-free theory smoothly

In this paper we study various aspects of ghost resonances: the resummation that leads to the dressed propagator, the poles locations, the analytic continuation into the second Riemann sheet and the spectral representations in both first and second sheets. In particular, we show that for real masses above the multiparticle threshold the ghost propagator has a pair of complex conjugate poles in the first sheet, unlike the case of an ordinary unstable resonance which has no pole in the first sheet but a complex conjugate pair in the second sheet. Mathematical and physical implications of this feature are discussed. We also clarify an important point regarding the two absorptive contributions of a ghost propagator in the narrow-width approximation. Furthermore, we argue that finite-time quantum field theories are needed to consistently derive the dressed ghost propagator and capture the true physical properties of ghost resonances. Throughout the work, different prescriptions to define the ghost propagator on the real axis are considered: Feynman, anti-Feynman and fakeon prescriptions.

Recently, Pan and Yu showed that Lascoux polynomials can be defined in terms of certain collections of diagrams consisting of unit cells arranged in the first quadrant. Starting from certain initial diagrams, one forms a finite set of diagrams by applying two types of moves: Kohnert and ghost moves. Both moves cause at most one cell to move to a lower row with ghost moves leaving a new "ghost cell" in its place. Each diagram formed in this way defines a monomial in the associated Lascoux polynomial. Restricting attention to diagrams formed by applying sequences of only Kohnert moves in the definition of Lascoux polynomials, one obtains the family of key polynomials. Recent articles have considered a poset structure on the collections of diagrams formed when one uses only Kohnert moves. In general, these posets are not "well-behaved," not usually having desirable poset properties. Here, as an intermediate step to studying the analogous posets associated with Lascoux polynomials, we consider the posets formed by restricting attention to those diagrams formed by using only ghost moves. Unlike in the case of Kohnert posets, we show that such "ghost Kohnert posets" are always ranked joi

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 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 $N\rightarrow\infty$ 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 to the case of the ordinary unstable particle, whose propagator has no pole on the physical sheet.

To investigate the possibility of a ghost-antighost condensate the coupled Dyson--Schwinger equations for the gluon and ghost propagators in Yang--Mills theories are derived in general covariant gauges, including ghost-antighost symmetric gauges. The infrared behaviour of these two-point functions is studied in a bare-vertex truncation scheme which has proven to be successful in Landau gauge. In all linear covariant gauges the same infrared behaviour as in Landau gauge is found: The gluon propagator is infrared suppressed whereas the ghost propagator is infrared enhanced. This infrared singular behaviour provides indication against a ghost-antighost condensate. In the ghost-antighost symmetric gauges we find that the infrared behaviour of the gluon and ghost propagators cannot be determined when replacing all dressed vertices by bare ones. The question of a BRST invariant dimension two condensate remains to be further studied.

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis---the statement that ghosts between finite-dimensional G-representations factor through a projective---we define the ghost number of kG to be the smallest integer l such that the composition of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C_2 as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.

We demonstrate that Cherenkov radiation can be interpreted as ghost instability of a certain type. Solutions of modified gravity theories often contain ghost instabilities. One type of such ghost instability is associated with existence of different types of species with causal cones that do not share common time, which leads to vacuum decay via creation of particles with positive and negative energies. We show that this ghost instability can be seen as Cherenkov radiation and vice versa.

We derive an approximate dynamical equation for the form-factor of the ghost-gluon vertex that contributes to the Schwinger-Dyson equation of the ghost dressing function in the Landau gauge. In particular, we consider the "one-loop dressed" approximation of the corresponding equation governing the evolution of the ghost-gluon vertex, using fully dressed propagators and tree-level vertices in the relevant diagrams. Within this approximation, we then compute the aforementioned form factor for two special kinematic configurations, namely the soft gluon limit, in which the momentum carried by the gluon leg is zero, and the soft ghost limit, where the momentum of the anti-ghost leg vanishes. The results obtained display a considerable departure from the tree-level value, and are in rather good agreement with available lattice data. We next solve numerically the coupled system formed by the equation of the ghost dressing function and that of the the vertex form factor, in the soft ghost limit. Our results demonstrate clearly that the nonperturbative contribution from the ghost-gluon vertex accounts for the missing strength in the kernel of the ghost equation, and allows for an impressive

We introduce the ghost algebra, a two-boundary generalisation of the Temperley-Lieb (TL) algebra, using a diagrammatic presentation. The existing two-boundary TL algebra has a basis of string diagrams with two boundaries, and the number of strings connected to each boundary must be even; in the ghost algebra, this number may be odd. To preserve associativity while allowing boundary-to-boundary strings to have distinct parameters according to the parity of their endpoints, as seen in the one-boundary TL algebra, we decorate the boundaries with bookkeeping dots called ghosts. We also introduce the dilute ghost algebra, an analogous two-boundary generalisation of the dilute TL algebra. We then present loop models associated with these algebras, and classify solutions to their boundary Yang-Baxter equations, given existing solutions to the Yang-Baxter equations for the TL and dilute TL models. This facilitates the construction of a one-parameter family of commuting transfer tangles, making these models Yang-Baxter integrable.

The recently proposed infrared modification of gravity through the introduction of a ghost scalar field results in a number of interesting cosmological and phenomenological implications. In this paper, we derive the exact cosmological solutions for a number of scenarios where at late stages, the ghost behaves like dark matter, or dark energy. The full solutions give valuable information about the non-linear regime beyond the asymptotic first order analysis presented in the literature. The generic feature is that these ghost cosmologies give rise to smooth transitions between radiation dominated phases (or more general power-law expansions) at early epochs and ghost dark matter resp. ghost dark energy dominated late epochs. The current age of our universe places us right at the non-linear transition phase. By studying the evolution backwards in time, we find that the dominance of the ghost over ordinary baryonic matter and radiative contributions persists back to the earliest times such that the Friedmann-Robertson-Walker geometry is dictated to a good approximation by the ghost alone. We also find that the Jeans instability occurs in the ghost dark energy scenario at late times, wh

This paper investigates the brane-localized interactions, including DGP gravity and higher derivative (HD) gravity localized on the brane. We derive the effective action on the brane, which suggests the brane-localized HD gravity suffers the ghost problem generally. Besides, we obtain novel algebraic identities of the mass spectrum, which reveal the global nature and can characterize the phase transformation of the mass spectrum. We get a powerful ghost-free condition from the spectrum identities, which rules out one type of brane-localized HD gravity. We further prove the mass spectrum is real and non-negative $m^2\ge 0$ under the ghost-free condition. Furthermore, we discuss various constraints on parameters of brane-localized gravity in AdS/BCFT and wedge holography, respectively. They include the ghost-free condition of Kaluza-Klein and brane-bending modes, the positive definiteness of boundary central charges, and entanglement entropy. The ghost-free condition imposes strict constraint, which requires non-negative couplings for pure DGP gravity and Gauss-Bonnet gravity on the brane. It also rules out one class of brane-localized HD gravity. Thus, such HD gravity should be unde

In theories with a hidden ghost sector that couples to visible matter through gravity only, empty space can decay into ghosts and ordinary matter by graviton exchange. Perturbatively, such processes can be very slow provided that the gravity sector violates Lorentz invariance above some cut-off scale. Here, we investigate non-perturbative decay processes involving ghosts, such as the spontaneous creation of self-gravitating lumps of ghost matter, as well as pairs of Bondi dipoles (i.e., lumps of ghost matter chasing after positive energy objects). We find the corresponding instantons and calculate their Euclidean action. In some cases, the instantons induce topology change or have negative Euclidean action. To shed some light on the meaning of such peculiarities, we also consider the nucleation of concentrical domain walls of ordinary and ghost matter, where the Euclidean calculation can be compared with the canonical (Lorentzian) description of tunneling. We conclude that non-perturbative ghost nucleation processes can be safely suppressed in phenomenological scenarios.

Dynamical descriptions and modeling of natural systems have generally focused on fixed points, with saddles and saddle-based phase-space objects such as heteroclinic channels/cycles being central concepts behind the emergence of quasi-stable long transients. Reliable and robust transient dynamics observed for real, inherently noisy systems is, however, not met by saddle-based dynamics, as demonstrated here. Generalizing the notion of ghost states, we provide a complementary framework that does not rely on the precise knowledge or existence of (un)stable fixed points, but rather on slow directed flows organized by ghost sets in ghost channels and ghost cycles. Moreover, we show that appearance of these novel objects is an emergent property of a broad class of models, typically used for description of natural systems.

Dubovsky and Sibiryakov recently proposed a scenario in which particles of different species propagate with different speeds due to their direct couplings to ghost condensate. It was argued that this extended version of ghost condensate allows a gedanken experiment leading to violation of the generalized second law. However, in the original ghost condensate scenario, difference in propagation speeds is suppressed by M^2/M_{Pl}^2, where M is the order parameter of spontaneous Lorentz breaking and M_{Pl} is the Planck scale. In this case the energy transfer necessary for the gedanken experiment is so slow that the timescale of decrease of entropy, if any, is always longer than the Jeans timescale of ghost condensate. Hence the generalized second law is not violated by the gedanken experiment in the original ghost condensate scenario. This conclusion trivially extends to gauged ghost condensation by taking into account accretion of gauged ghost condensate into a black hole.

The braneworld model of Dvali-Gabadadze-Porrati realizes the self-accelerating universe. However, it is known that this cosmological solution contains a spin-2 ghost. We study the possibility of avoiding the appearance of the ghost by slightly modifying the model, introducing the second brane. First we consider a simple model without stabilization of the separation of the brane. By changing the separation between the branes, we find we can erase the spin-2 ghost. However, this can be done only at the expense of the appearance of a spin-0 ghost instead. We discuss why these two different types of ghosts are correlated. Then, we examine a model with stabilization of the brane separation. Even in this case, we find that the correlation between spin-0 and spin-2 ghosts remains. As a result we find we cannot avoid the appearance of ghost by two-branes model.

Ghost imaging is demonstrated using a poly-energetic reactor source of thermal neutrons. The method presented enables position resolution to be incorporated, into a variety of neutron instruments that are not position resolving. In an imaging context, ghost imaging can be beneficial for dose reduction and resolution enhancement. We also demonstrate a super-resolution variant of the method, namely a parallel form of neutron ghost imaging, with the ability to significantly increase the spatial resolution of a pixelated detector such as a CCD or CMOS camera. Extensions of our neutron ghost-imaging protocol are discussed in detail and include neutron ghost tomography, neutron ghost microscopy, dark-field neutron ghost imaging, and isotope-resolved color neutron ghost imaging via prompt gamma-ray bucket detection.

We study the phase space asymptotics of the so called Veneziano ghost dark energy models. Models where the ghost field's energy density: i) $ρ_{ghost}\propto H$, and ii) $ρ_{ghost}\propto H+H^2$, are investigated. Both, cases with and without additional non-gravitational interaction between cold dark matter and ghost dark energy, are subject to scrutiny. We pay special attention to the choice of phase space variables leading to bounded and compact phase space so that no critical point of physical interest is missing. A rich asymptotic structure is revealed: depending on the kind of non-minimal coupling critical points associated with radiation dominance, matter dominance, cold dark matter/ghost dark energy scaling, and ghost dark energy dominance, are found. Past and future attractors, as well as saddle equilibrium points, are identified in the corresponding phase spaces.

One of the most surprising consequences of quantum mechanics is the nonlocal multi-particle interference observable in joint-detection of distant particle-detectors. Ghost imaging is one of such phenomena. Two types of ghost imaging have been experimentally demonstrated since 1995. Type-one ghost imaging uses entangled photon pairs as the light source. The nonlocal point-to-point image-forming correlation is the result of a constructive-destructive superposition among a large number of biphoton amplitudes, a nonclassical entity corresponding to different yet indistinguishable alternative ways of producing a joint-detction event between distant photodetectors. Type-two ghost imaging uses chaotic-thermal light. The type-two image-forming correlation is the result of interferences between paired two-photon amplitudes, corresponding to two different yet indistinguishable alternative ways of triggering a join-detection event by two independent photons. A great deal of confusion about ghost imaging comes from "ghost shadow". Similar to x-ray photography, a ghost shadow can be made in coincidences by "blocking-partial blocking-unblocking" of either co-rotating laser beams or classically c