We present a new understanding of the unstable ghost-like resonance which appears in theories such as quadratic gravity and Lee-Wick type theories. Quantum corrections make this resonance unstable, such that it does not appear in the asymptotic spectrum. We prove that these theories are unitary to all orders. Unitarity is satisfied by the inclusion of only cuts from stable states in the unitarity sum. This removes the need to consider this as a ghost state in the unitarity sum. However, we often use a narrow-width approximation where we do include cuts through unstable states, and ignore cuts through the stable decay products. If we do this with the unstable ghost resonance at one loop, we get the correct answer only by using a contour which was originally defined by Lee and Wick. The quantum effects also provide damping in both the Feynman and the retarded propagators, leading to stability under perturbations.
We discuss aspects of non-perturbative unitarity in quantum field theory. The additional ghost degrees of freedom arising in "truncations" of an effective action at a finite order in derivatives could be fictitious degrees of freedom. Their contributions to the fully-dressed propagator -- the residues of the corresponding ghost-like poles -- vanish once all operators compatible with the symmetry of the theory are included in the effective action. These "fake ghosts" do not indicate a violation of unitarity.
The unitarity of a Lorentz-invariance violating QED model with higher-order Myers and Pospelov photons coupled to standard fermions is studied. As expected, we find ghost states associated to the higher-order terms that may lead to the loss of unitarity. An explicit calculation to check perturbative unitarity in the process of electron-positron scattering is performed and it is found to be possible to be preserved.
The unitarity of the NS supersymmetric coset SL(2,R)/U(1) is studied for the discrete representations. The results are applied to the proof of the no-ghost theorem for fermionic strings in AdS(3) x N in the NS sector. A no-ghost theorem is proved for states in flowed discrete representations.
We give a simple discussion of ghosts, unitarity violation, negative norm states and quantum vs classical behavior in the simplest model with four derivative action - the Pais-Uhlenbeck oscillator. We also point out that the normalizable "vacuum state" (in the sense defined below) of this model can be understood as spontaneous breaking of the emergent conformal symmetry. We provide an example of an interacting system that couples the "particle" and "ghost" degrees of freedom and nevertheless remains unitary on both classical and quantum level.
We analyze the unitarity properties of higher derivative quantum field theories which are free of ghosts and ultraviolet singularities. We point out that in spite of the absence of ghosts most of these theories are not unitary. This result confirms the difficulties of finding a consistent quantum field theory of quantum gravity.
Scalars in AdS${}_{d+1}$ with squared masses in the Breitenlohner-Freedman window $-d^2/4 \le m^2 < -d^2/4 +1$ (in units with the AdS scale $\ell$ set to 1) are known to enjoy a variety of boundary conditions. For larger masses $m^2 > -d^2/4 +1$, unitarity bounds in possible dual CFTs suggest that such general boundary conditions should lead to ghosts. We show that this is not always the case as, for conformally-invariant boundary conditions in Poincaré AdS that would naively violate unitarity bounds, the system is generically ghost-free. Conflicts with unitarity bounds are avoided due to the …
For a non-equilibrium physical system defined along a closed time path (CTP), a key constraint is the so-called largest time equation, which is a consequence of unitarity and implies causality. In this paper, we present a simple proof that if the propagators of a non-equilibrium effective action have the proper pole structure, the largest time equation is obeyed to all loop orders. Ghost fields and BRST symmetry are not needed. In particular, the arguments for the proof can also be used to show that if ghost fields are introduced, their contributions vanish.
In theories with unstable particles, unitarity is satisfied by the inclusion of only stable states in unitarity sums. Hence unitarity cuts are not to be taken through unstable particles. This raises a challenge to the generalized unitarity method, whose aim is to reconstruct amplitudes by analyzing sets of unitarity cuts. Nevertheless, under some general physical conditions, and perhaps some methodological modifications, we prove that the method is still reliable for one-loop amplitudes containing resonances. We discuss some simple examples which illustrate these features.
We revisit the previously unsolved problems of ensuring Lorentz invariance and non-perturbative unitarity in Lee-Wick theories. We base our discussion on an ultraviolet completion of QED by Lee-Wick ghost fields, which is argued to be asymptotically safe. We argue that as long as the state space is based upon a suitable choice of distributions of a type invented by Gel'fand and Shilov, the Lee-Wick ghosts can be eliminated while preserving Lorentz invariance to produce a unitary theory. The method for eliminating ghosts is in principle non-perturbatively well-defined, in contrast with some pre…
With fourth-order derivative theories leading to propagators of the generic ghost-like $1/(k^2-M_1^2)-1/(k^2-M_2^2)$ form, it would appear that such theories have negative norm ghost states and are not unitary. However on constructing the associated quantum Hilbert space for the free theory that would produce such a propagator, Bender and Mannheim found that the Hamiltonian of the free theory is not Hermitian but is instead $PT$ symmetric, and that there are in fact no negative norm ghost states, with all Hilbert space norms being both positive and preserved in time. Even though perturbative r…
We analyze the most general case of third-order Chern-Simons-like theories of massive 3D gravity. Results show the conditions for finding the unitary regions on the parameter space. There exists $(n-1)$th order theories on the boundary of a unitary $n$th order model on parameter space under certain conditions, as the first example recently demonstrated from the bimetric generalization of exotic massive gravity. We investigated the mechanism that causes this type of transition for third-order models. Hamiltonian analysis of the theory also presents that ghost and no-ghost regions can be separat…
The effective field theory of massive gravity had long been formulated in a generally covariant way arXiv:hep-th/0210184. Using this formalism, it has been found recently that there exists a class of massive nonlinear theories that are free of the Boulware-Deser ghosts, at least in the decoupling limit arXiv:1007.0443. In this work we study other recently proposed models that go under the name of "gravitational Higgs theories" arXiv:1002.3877, arXiv:1008.5132. We show that these models, although seemingly different from the effective field theories of massive gravity, are in fact equivalent to…
Ghosts versus Unstable Particles in Quantum Field Theory
Departamento de Física de Partículas, Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学加利西亚高能物理研究所粒子物理系）
We present an example of the quantum system with higher derivatives in the Lagrangian, which is ghost-free: the spectrum of the Hamiltonian is bounded from below and unitarity is preserved.
We investigate the instability of the ghost dark energy model against perturbations in different cases. To this goal we use the squared sound speed $v_s^2$ whose sign determines the stability of the model. When $v_s^2<0$ the model is unstable against perturbation. At first we discuss the noninteracting ghost dark energy model in a flat FRW universe and find out that such a model is unstable due to the negativity of the $v_s^2$ in all epoches. The interacting ghost dark energy model in both flat and non-flat universe are studied in the next parts and in both cases we find that the squared sound…
We review the history of the ghost problem in quantum field theory from the Pauli-Villars regulator theory to currently popular fourth-order derivative quantum gravity theories. While these theories all appear to have unitarity-violating ghost states with negative norm, we show that in fact these ghost states only appear because the theories are being formulated in the wrong Hilbert space. In these theories the Hamiltonians are not Hermitian but instead possess an antilinear symmetry. Consequently, one cannot use inner products that are built out of states and their Hermitian conjugates. Rathe…
Numerous claims in the literature suggest that gravity induced on a higher co-dimensional surface violates unitarity in the weak coupling regime. However, it remained unclear, why a conserved source localized on this surface and giving rise to an induced gravity term at low energies would absorb and emit the associated ghost, given a consistent source-free theory. In this article it is shown that the appearance of the induced Einstein Hilbert term does not threaten the unitarity of the theory. The physics arguments behind this statement are presented in a semi-covariant language, but the detai…
Here we prove unitarity of the recently found fourth-order self-dual model of spin-2 by investigating the analytic structure of its propagator. The model describes massive particles of helicity +2 (or -2) in $D=2+1$ and corresponds to the quadratic truncation of a higher derivative topologically massive gravity about a flat background. It is an intriguing example of a theory where a term in the propagator of the form $1/\lbrack\Box^2 (\Box - m^2)\rbrack $ does not lead to ghosts. The crucial role of the linearized Weyl symmetry in getting rid of the ghosts is pointed out. We use a peculiar pai…
We investigate unitarity within the Complex-Mass Scheme, a convenient universal scheme for perturbative calculations involving unstable particles in Quantum Field Theory which guarantees exact gauge invariance. Since this scheme requires to introduce complex masses and complex couplings, the Cutkosky cutting rules, which express perturbative unitarity in theories of stable particles, are no longer valid. We derive corresponding rules for scalar theories with unstable particles based on Veltman's Largest-Time Equation and prove unitarity in this framework.
The presence of an unstable massive spin-2 ghost state in the renormalizable theory of massive conformal gravity leads to a pair of complex poles appearing in the first sheet of the energy plane. Here we show that the positions of these poles are gauge dependent, which makes the theory unitary.
We show that in most conformal field theories the condition of the energy flux positivity, proposed by Hofman and Maldacena, is equivalent to the absence of ghosts. At finite temperature and large energy and momenta, the two-point functions of the stress energy tensor develop lightlike poles. The residues of the poles can be computed, as long as the only spin two conserved current, which appears in the stress energy tensor OPE and acquires nonvanishing expectation value at finite temperature, is the stress energy tensor. The condition for the residues to stay positive and the theory to remain …
Higher-order effects of CPT and Lorentz violation within the SME effective framework including Myers-Pospelov dimension-five operator terms are studied. The model is canonically quantized by giving special attention to the arising of indefinite-metric states or ghosts in an indefinite Fock space. As is well-known, without a perturbative treatment that avoids the propagation of ghost modes or any other approximation, one has to face the question of whether unitarity and microcausality are preserved. In this work, we study both possible issues. We found that microcausality is preserved due to th…
It is conventionally taken for granted that the unitary gauge formulation of quantum gauge field theory has the advantage of preservation unitarity because only physical fields are involved but has the disadvantage of losing renormalizability because of severe ultraviolet divergences due to vector meson propagators. In this paper, we show how to handle the ultraviolet divergent loops so that the physical amplitudes remain gauge invariant. One of the consequences we arrive at is that ghost loops are needed to cancel the divergences due to vector mesons and to give gauge invariant physical ampli…
In this paper we investigate wormhole and spherically symmetric solutions in 4D gravity plus a matter source consisting of a ghost scalar field with a sine-Gordon potential. For the wormhole solutions we also include the possibility of electric and/or magnetic charges. For both types of solutions we perform a linear stability analysis and show that the wormhole solutions are stable and that when one turns on the electric and/or magnetic field the solution remains stable. The linear stability analysis of the spherically symmetric solutions indicates that they can be stable or unstable depending…
In this work we explore general leading singularities of one-loop amplitudes in higher-derivative Yang-Mills and quadratic gravity. These theories are known to possess propagators which contain quadratic and quartic momentum dependence, which leads to the presence of an unstable ghostlike resonance. However, unitarity cuts are not to be taken through unstable particles and therefore unitarity is still satisfied. On the other hand, this could engender issues when calculating leading singularities which are generalizations of unitarity cuts. Nevertheless, we will show with explicit examples how …
Investigations for decay of unstable D-brane and rolling of accelerated D-brane dynamics have revealed that various proposed prescriptions give different result for spectral amplitudes and observables. Here, we study them with particular attention to unitarity and open-closed channel duality. From "ab initio" derivation in the open string channel, both in Euclidean and Lorentzian worldsheet approaches, we find heretofore overlooked contribution to the spectral amplitudes and obervables. The contribution is fortuitously absent for decay of unstable D-brane, but is present for rolling of acceler…
We review some of the recent results which can be useful for better understanding of the problem of stability of vacuum and in general classical solutions in higher derivative quantum gravity. The fourth derivative terms in the purely gravitational vacuum sector are requested by renormalizability already in both semiclassical and complete quantum gravity theories. However, because of these terms the spectrum of the theory has unphysical ghost states which jeopardize the stability of classical solutions. At the quantum level ghosts violate unitarity, and and thus ghosts look incompatible with t…
The unitarity problem in curved spacetime is solved for the string described by the SL(2,R) WZW model. The spectrum is computed exactly and demonstratedto be ghost-free. The new features include (i) SL(2,R) left/right symmetrycurrents that have logarithmic cuts on the world sheet but that satisfy theusual local operator products or commutation rules, (ii) physical statesconsistent with the monodromy condition of closed strings despite thelogarithmic singularity in the currents, and (iii) a new free boson realization for these currents which render the SL(2,R) WZW model completely solvable.
In this paper we present an iterative method to generate an infinite class of new nonlocal field theories whose propagators are ghost-free. We first examine the scalar field case and show that the pole structure of such generalized propagators possesses the standard two derivative pole and in addition can contain complex conjugate poles which, however, do not spoil at least tree level unitarity as the optical theorem is still satisfied. Subsequently, we define analogous propagators for the fermionic sector which is also devoid of unhealthy degrees of freedom. As a third case, we apply the same…
The Lee model was introduced in the 1950s as an elementary quantum field theory in which mass, wave function, and charge renormalization could be carried out exactly. In early studies of this model it was found that there is a critical value of g^2, the square of the renormalized coupling constant, above which g_0^2, the square of the unrenormalized coupling constant, is negative. Thus, for g^2 larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. It was also discovered that in this non-Hermitian regime a new state appears whose norm is negative. This state i…
We construct a class of theories which extend New Massive Gravity to higher orders in curvature in any dimension. The lagrangians arise as limits of a new class of bimetric theories of Lovelock gravity, which are unitary theories free from the Boulware-Deser ghost. These Lovelock bigravity models represent the most general non-chiral ghost-free theories of an interacting massless and massive spin-two field in any dimension. The scaling limit is taken in such a way that unitarity is explicitly broken, but the Boulware-Deser ghost remains absent. This automatically implies the existence of a hol…
We investigate the holographic description of CFTs defined on the cylinder and on AdS, which include an operator saturating the unitarity bound. The standard Klein-Gordon field with the corresponding mass and boundary conditions on global AdS_(d+1) and on an AdS_(d+1) geometry with AdS_d conformal boundary contains ghosts. We identify a limit in which the singleton field theory is obtained from the bulk theory with standard renormalized inner product, showing that a unitary bulk theory corresponding to an operator which saturates the unitarity bound can be formulated and that this yields a fre…
We prove a no-ghost theorem for a bosonic string propagating in Nappi-Witten spacetime. This is achieved in two steps. We first demonstrate unitarity for a class of NW/U(1) modules: the norm of any state which is primary with respect to a chosen timelike U(1) is non-negative. We then show that physical states - states satisfying the Virasoro constraints - in a class of modules of an affinisation of the Nappi-Witten algebra are contained in the NW/U(1) modules. Similar to the case of strings on $AdS_3$, in order to saturate the spectrum obtained in light-cone quantization we are led to include …
We discuss corrections to the ratio of shear viscosity to entropy density $η/s$ in higher-derivative gravity theories. Generically, these theories contain ghost modes with Planck-scale masses. Motivated by general considerations about unitarity, we propose new boundary conditions for the equations of motion of the graviton perturbations that force the amplitude of the ghosts modes to vanish. We analyze explicitly four-derivative perturbative corrections to Einstein gravity which generically lead to four-derivative equations of motion, compare our choice of boundary conditions to previous propo…
We explore a dark energy model with a ghost scalar field in the context of the runaway dilaton scenario in low-energy effective string theory. We address the problem of vacuum stability by implementing higher-order derivative terms and show that a cosmologically viable model of ``phantomized'' dark energy can be constructed without violating the stability of quantum fluctuations. We also analytically derive the condition under which cosmological scaling solutions exist starting from a general Lagrangian including the phantom type scalar field. We apply this method to the case where the dilaton…
In a recently proposed model of ``ghost condensation'', spatially homogeneous states may mix, via tunneling, with inhomogeneous states which are somewhat similar to bubbles in the theory of false vacuum decay, the corresponding bubble nucleation rate being exponentially sensitive to the ultraviolet completion of the model. The conservation of energy and charge requires that the energy density is negative and the field is strongly unstable in a part of the nucleated bubble. Unlike in the theory of false vacuum decay, this region does not expand during subsequent real-time evolution. In the oute…
We investigate the stability against inhomogeneous perturbations and the appearance of ghost modes in Gauss-Bonnet gravitational theories with a non-minimally coupled scalar field, which can be regarded as either the dilaton or a compactification modulus in the context of string theory. Through cosmological linear perturbations we extract four no-ghost and two sub-luminal constraint equations, written in terms of background quantities, which must be satisfied for consistency. We also argue that, for a general action with quadratic Riemann invariants, homogeneous and inhomogeneous perturbations…
Recently modified gravitational theories which mimic the behaviour of dark matter, the so-called "Mimetic Dark Matter", have been proposed. We study the consistency of such theories with respect to the absence of ghost instability and propose a new tensor-vector-scalar theory of gravity, which is a generalization of the previous models of mimetic dark matter with additional desirable features. The original model proposed by Chamseddine and Mukhanov [JHEP 1311 (2013) 135, arXiv:1308.5410] is concluded to describe a regular pressureless dust, presuming that we consider only those configurations …
Following the same steps made for a scalar field in a parallel publication, we propose a class of perturbative theories of quantum gravity based on fractional operators, where the kinetic operator of the graviton is either made of fractional derivatives or a covariant fractional d'Alembertian. The classical action for each theory is constructed and the equations of motion are derived. Unitarity and renormalizability of theories with a fractional d'Alembertian are also considered. We argue that unitarity and power-counting renormalizability never coexist, although in some cases one-loop unitary…
Stable states (particles), ghosts and unstables states (particles) are discussed with respect to the time representations involved, their unitary groups and the induced Hilbert spaces. Unstable particles with their decay channels are treated as higher dimensional probability collectives with nonabelian probability groups U(n) generalizing the individual abelian U(1)-normalization for stable states (particles).
We improve the study of the lack of perturbative unitarity of noncommutative space-time quantum field theories derived from open string theory in electric backgrounds, enforcing the universality of the mechanism by which a tachyonic branch cut appears when the Seiberg-Witten limit freezes the string in an unstable vacuum. The main example is realized in the context of the on-shell four-tachyon amplitude of the bosonic string, and the dependence of the phenomenon on the brane-worldvolume dimension is analysed. We discuss the possibility of a proof in superstring theory, and finally mention the …
We present a detailed investigation of scattering processes in $W_3$ string theory. We discover further physical states with continuous momentum, which involve excitations of the ghosts as well as the matter, and use them to gain a better understanding of the interacting theory. The scattering amplitudes display factorisation properties, with states from the different sectors of the theory being exchanged in the various intermediate channels. We find strong evidence for the unitarity of the theory, despite the unusual ghost structure of some of the physical states. Finally, we show that by per…
The recent Planck data of Cosmic Microwave Background (CMB) temperature anisotropies support the Starobinsky theory in which the quadratic Ricci scalar drives cosmic inflation. We build up a multi-dimensional quantum consisted ultraviolet completion of the model in a phenomenological "bottom-up approach". We present the maximal class of theories compatible with unitarity and (super-)renormalizability or finiteness which reduces to the Starobinsky theory in the low-energy limit. The outcome is a maximal extension of the Krasnikov-Tomboulis-Modesto theory including an extra scalar degree of free…
Poincaré gauge theories provide an approach to gravity based on the gauging of the Poincaré group, whose homogeneous part generates curvature while the translational sector gives rise to torsion. In this note we revisit the stability of the widely studied quadratic theories within this framework. We analyse the presence of ghosts without fixing any background by obtaining the relevant interactions in an exact post-Riemannian expansion. We find that the axial sector of the theory exhibits ghostly couplings to the graviton sector that render the theory unstable. Remarkably, imposing the absence …
Most of the massive states in superstring theory are expected to undergo mass renormalization at one loop order. Typically these corrections should contain imaginary parts, indicating that the states are unstable against decay into lighter particles. However in such cases, direct computation of the renormalized mass using superstring perturbation theory yields divergent result. Previous approaches to this problem involve various analytic continuation techniques, or deforming the integral over the moduli space of the torus with two punctures into the complexified moduli space near the boundary.…
