We consider the Lagrangian of gravity covariantly amended by the mass and polynomial interaction terms with arbitrary coefficients, and reinvestigate the consistency of such a theory in the decoupling limit, up to the fifth order in the nonlinearities. We calculate explicitly the self-interactions of the helicity-0 mode, as well as the nonlinear mixing between the helicity-0 and -2 modes. We show that ghost-like pathologies in these interactions disappear for special choices of the polynomial interactions, and argue that this result remains true to all orders in the decoupling limit. Moreover, we show that the linear, and some of the nonlinear mixing terms between the helicity-0 and -2 modes can be absorbed by a local change of variables, which then naturally generates the cubic, quartic, and quintic Galileon interactions, introduced in a different context. We also point out that the mixing between the helicity-0 and 2 modes can be at most quartic in the decoupling limit. Finally, we discuss the implications of our findings for the consistency of the effective field theory away from the decoupling limit, and for the Boulware-Deser problem.
The non linear aspects of a recently proposed model of massive spin-2 particles with propagating torsion are studied. We obtain a nonlinear equation which reduces at linear order to a generalized Fierz-Pauli equation in any background space-time with or without a vanishing torsion. We contrast those results with properties of a class of bigravity theories in an arbitrary background Einstein manifold. It is known that the non perturbative spectrum of the bigravity model has 8 propagating physical degrees of freedom. This is identical to the physical propagating degrees of freedom of the massive…
We expose a new symmetry for linear perturbations around a solution of non-linear Fierz-Pauli massive gravity plus a bare cosmological constant. The cosmological constant is chosen such that the background metric is flat while the Stuckelberg fields have a non-trivial profile. Around this background, at linear order the new symmetry reduces the propagating degrees of freedom to those of General Relativity, namely the massless helicity 2 modes only. We discuss the physical consequences and possible applications of these findings.
Starting with a 5D physical vacuum described by a 5D Ricci-flat background metric, we study the emergence of gravitational waves (GW) from the Induce Matter (IM) theory of gravity. We obtain the equation of motion for GW on an 4D curved spacetime which has the form of a Fierz-Pauli one. In our model the mass of gravitons $m_g$ is induced by a static foliation on the noncompact space-like extra dimension and the source-term is originated in the interaction of the GW with the induced connections of the background 5D metric. Here, relies the main difference of this formalism with the original Fie…
We show that all non-linear completions of the Fierz-Pauli massive gravity are ruled out, because all theories which might be ghost-free have superluminal modes.
We study the propagation of helicity-1 gravitons in the Fierz--Pauli massive gravity in nearly Minkowski backgrounds. We show that, generically, there exist backgrounds consistent with field equations, in which the propagation is superluminal. The relevant distances are much longer than the ultraviolet cutoff length inherent in the Fierz--Pauli gravity, so superluminality occurs within the domain of validity of the effective low energy theory. There remains a possibility that one may get rid of this property by imposing fine tuning relations between the coefficients in the non-linear generaliz…
Modifying gravity at large distances by means of a massive graviton may explain the observed acceleration of the Universe without Dark Energy. The standard paradigm for Massive Gravity is the Fierz-Pauli theory, which, nonetheless, displays well known flaws in its massless limit. The most serious one is represented by the vDVZ discontinuity, which consists in a disagreement between the massless limit of the Fierz-Pauli theory and General Relativity. Our approach is based on a field theoretical treatment of Massive Gravity: General Relativity, in the weak field approximation, is treated as a ga…
We show that, in four-dimensional spacetimes with an arbitrary Einstein metric, with and without a cosmological constant, perturbative dynamical degrees of freedom in generic quadratic-curvature gravity can be decoupled into massless and massive parts. The massive part has the structure identical to, modulo the over-all sign, the non-Fierz-Pauli-type massive gravity, and a further decomposition into the spin-2 and spin-0 sectors can be done. The equivalence at the level of equations of motion allows us to translate various observational bounds on the mass of extra fields into constraints on th…
The asymptotic structure of the Pauli-Fierz theory at spatial infinity is investigated in four spacetime dimensions. Boundary conditions on the massless spin-$2$ field that are invariant under an infinite-dimensional group of non-trivial ``improper'' gauge symmetries are given. The compatibility of these boundary conditions with invariance of the theory under Lorentz boosts is a subtle issue which is investigated in depth and leads to the identification of the improper gauge symmetries with the pure BMS supertranslations. It is also shown how rigid Poincaré transformations and improper gauge s…
We investigate the non-Pauli-Fierz(nPF) theory, a linearized massive gravity with a generic graviton mass term, which has been ignored due to a ghost in its spectrum and the resultant loss of unitarity. We first show that it is possible to use the Lee-Wick mechanism, a unitarization through the decay of a ghost, in order to handle the sixth mode ghost of nPF, and then check for the quantum consistency. Once proven to be consistent, nPF could become a viable candidate for a large distance modification of gravity, because it naturally solves the intrinsic problems that most dark energy/modified …
We utilise a quotient of the universal enveloping algebra of the Poincaré algebra in three spacetime dimensions, on which we formulate a covariant constancy condition. The equations so obtained contain the Fierz-Pauli equations for non-interacting, massive higher-spin fields, and can thus be regarded as an unfolding of the Fierz-Pauli system. All fundamental fields completely decouple from each other. In the non-truncated case, the field content includes infinitely many copies of each field at fixed spin.
We study general Lorentz invariant theories of massive gravitons. We show that, contrary to the standard lore, there exist consistent theories where the graviton mass term violates Pauli-Fierz structure. For theories where the graviton is a resonance this does not imply the existence of a scalar ghost if the deviation from Pauli-Fierz becomes sufficiently small at high energies. These types of mass terms are required by any consistent realization of the DGP model in higher dimension.
We consider an $O(d,d;\mathbb{Z})$ invariant massive deformation of double field theory at the level of free theory. We study Kaluza-Klein reduction on $R^{1,n-1} \times T^{d}$ and derive the diagonalized second order action for each helicity mode. Imposing the absence of ghosts and tachyons, we obtain a class of consistency conditions which include the well known weak constraint in double field theory as a special case. Consequently, we find two-parameter sets of $O(d,d;\mathbb{Z})$ invariant Fierz-Pauli massive gravity theories.
Izawa's gauge-fixing procedure based on BRS symmetry is applied twice to the massive tensor field theory of Fierz-Pauli type. It is shown the second application can remove massless singularities which remain after the first application. Massless limit of the theory is discussed.
A possible nonlinear completion of massive gravity of the Fierz-Pauli type is proposed. The theory describes a system consisting of a massive tensor field of the Fierz-Pauli type and an additional massive vector field. Massless limit as well as flat-spacetime limit can be taken smoothly. Constructing a nonlinear version of the physical-state condition which drives an extra scalar ghost from physical states is still unsettled.
We study the constraint structure of Fierz -Pauli action in both flat and curved space in the framework of Hamiltonian formalism. We observe an abrupt change in the constraint algebra and the characteristics of the constraints when the mass term is turned off. As is well-known, for de Sitter background with a special tuning of the mass, we will have a gauge symmetry and one degree of freedom less. We will show how this abrupt change in the behavior of the system is reflected on the properties of the constraint algebra. We also obtain the generating functional of gauge transformations both for …
In this paper we consider the issue of massive gravity from a pure field theoretical point of view, as the massive deformation of the gauge theory for a symmetric rank-2 tensor field. We look for the most general massive theory with well defined propagators, imposing the absence of unphysical poles (ghosts). We find several possibilities, depending on the choice of the gauge fixing term. {Amongst these, two solutions with good massless limit are found.} The request of the absence of ghosts, alone, does not isolate the Fierz-Pauli case: several examples of massive theories, which may include or…
In theories of massive gravity with Fierz-Pauli mass term at the linearized level, perturbative radially symmetric asymptotic solutions are singular in the zero mass limit, hence van Dam-Veltman-Zakharov (vDVZ) discontinuity. In this note, in the context of gravitational Higgs mechanism, we argue that in non-Fierz-Pauli theories, which non-perturbatively are unitary, perturbative radially symmetric asymptotic solutions have a smooth massless limit, hence no vDVZ discontinuity. Perturbative vDVZ discontinuity as an artifact of the Fierz-Pauli mass term becomes evident in the language of constra…
We study SO(4) BF theory plus a general quadratic potential, which describes a bi- metric theory of gravity. We identify the profile of the potential leading to a Pauli-Fierz mass term for the massive graviton, thereby avoiding the linearized ghost. We include the Immirzi parameter in our analysis, and find that the mass of the second graviton depends on it. At the non-perturbative level, we find a situation similar to genuine bi-gravities: even choosing the Pauli-Fierz mass term, the ghost mode propagates through the interactions. We present some simple potentials leading to two and three deg…
Here we obtain alternative descriptions of massive spin-2 particles by an embedding procedure of the Fierz-Pauli equations of motion. All models are free of ghosts at quadratic level although most of them are of higher order in derivatives. The models that we obtain can be nonlinearly completed in terms of a dynamic and a fixed metric. They include some $f(R)$ massive gravities recently considered in the literature. In some cases there is an infrared (no derivative) modification of the Fierz-Pauli mass term altogether with higher order terms in derivatives. The analytic structure of the propag…
We derive the Hamiltonian describing Pauli-Fierz massive gravitons on a flat Friedmann-Robertson-Walker (FRW) cosmology in a particular, non-generic effective field theory. The cosmological evolution is driven by a scalar field Phi with an arbitrary potential V(Phi). The model contains two coupled scalar modes, corresponding to the fluctuations of Phi and to the propagating scalar component of the Pauli-Fierz graviton. In order to preserve the full gauge invariance of the massless version of the theory, both modes have to be taken into account. We canonically normalize the Hamiltonian and gene…
Towards the correct prescription of the Problem of divergence of Propagator of Siegal-Zwiebach action from String theory in Fierz-Pauli Gauge in the small mass limit
We consider some flat space theories for spin 2 gravitons, with less invariance than full diffeomorphisms. For the massless case, classical stability and absence of ghosts require invariance under transverse diffeomorphisms (TDiff). Generic TDiff invariant theories contain a propagating scalar, which disappears if the symmetry is enhanced in one of two ways. One possibility is to consider full diffeomorphisms (Diff). The other (which we denote WTDiff) adds a Weyl symmetry, by which the Lagrangian becomes independent of the trace. The first possibility corresponds to General Relativity, whereas…
Linearized gravity is considered as an ordinary gauge field theory. This implies the need for gauge fixing in order to have well defined propagators. Only after having achieved this, the most general mass term is added. The aim of this paper is to study of the degrees of freedom of the gauge fixed theory of linearized gravity with mass term. The main result is that, even outside the usual Fierz-Pauli constraint on the mass term, it is possible to choose a gauge fixing belonging to the Landau class, which leads to a massive theory of gravity with the five degrees of freedom of a spin two massiv…
It is possible to show that there are three independent families of models describing a massive spin-2 particle via a rank-2 tensor. One of them contains the massive Fierz-Pauli model, the only case described by a symmetric tensor. The three families have different local symmetries in the massless limit and can not be interconnected by any local field redefinition. We show here however, that they can be related with the help of a decoupled and non dynamic (spectator) field. The spectator field may be either an antisymmetric tensor $B_{μν}=-B_{νμ}$, a vector $A_μ$ or a scalar field $φ$, corresp…
Cross-couplings between a massless spin-two field (described in the free limit by the Pauli-Fierz action) and an Abelian three-form gauge field in D=11 are investigated in the framework of the deformation theory based on local BRST cohomology. These consistent interactions are obtained on the grounds of smoothness in the coupling constant, locality, Lorentz covariance, Poincare invariance, and the presence of at most two derivatives in the interacting Lagrangian. Our results confirm the uniqueness of the eleven-dimensional interactions between a graviton and a three-form prescribed by General …
By explicitly constructing the Hilbert space, Higuchi showed that there is a lower bound on the mass of a minimally-coupled free spin-2 field on a curved background \cite{HiguchiBound}. Using the vacuum persistence amplitude, we show that this bound is modified by taking into account additional terms not prohibited by symmetry in the case of a maximally symmetric spacetime. This result can further be generalized to the maximally symmetric space case, such as the FRW universe, and its corresponding bound of the deformation parameter is discussed.
Starting with a description of massive spin-2 particles in D=4 in terms of a mixed symmetry tensor $\tmnd $ without totally antisymmetric part ($T_{[[μν]ρ]}=0$) we obtain a dual model in terms of a nonsymmetric tensor $e_{μν}$. The model is of second-order in derivatives and its mass term $(e_{μν}e^{νμ} + c \, e^2)$ contains an arbitrary real parameter $c$. Remarkably, it is free of ghosts for any real value of $c$ and describes a massive spin-2 particle as expected from duality. The antisymmetric part $e_{[μν]}$ plays the role of auxiliary fields, vanishing on shell. In the massless case the …
The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behaviour of the survival and transition amplitudes for th…
We analyze the effect of the Pauli-Fierz mass term on a recently established, new massive gravity theory in three space-time dimensions. We show that the Pauli-Fierz mass term makes the new massive gravity theory non-unitary. Moreover, although we add the gravitational Chern-Simons term to this model, the situation remains unchanged and the theory stays non-unitary in spite that structure of the graviton propagator is greatly changed. Thus, the Pauli-Fierz mass term is not allowed to coexist with mass-generating higher-derivative terms in the new massive gravity.
We consider the effect of quantum interactions on Pauli-Fierz massive gravity. With generic graviton cubic interactions, we observe that the 1-loop counterterms do not conform to the tree level structure of Pauli-Fierz action, resulting in the reappearance of the 6th mode ghost. Then to explore the quantum effects to the full extent, we calculate the resummed graviton propagator with an arbitrary interaction and analyze its complete structure, from which a minimal condition for the absence of the ghost is obtained.
We investigate generally covariant theories which admit a Fierz-Pauli mass term for metric perturbations around an arbitrary curved background. For this we restore the general covariance of the Fierz-Pauli mass term by introducing four scalar fields which preserve a certain internal symmetry in their configuration space. It is then apparent that for each given spacetime metric this construction corresponds to a completely different generally covariant massive gravity theory with different symmetries. The proposed approach is verified by explicit analysis of the physical degrees of freedom of m…
We apply the functional renormalization group equation to a massive Fierz-Pauli action in curved space and find that, even though a massive term is a modification in the infrared sector, the mass term modifies the value of the non-gaussian fixed point in the UV sector.We obtained the beta function for the scale dependent mass parameter and found that the massive Fierz-Pauli case still seems to be an asymptotically safe theory.
We present a new massive spin two field theory which has smooth massless limit based on BRS formalism. Our model contains a parameter a which extendes Fierz-Pauli's mass term. Although redundant scalar ghost exists in our model, physical degree of freedom is still five for the sake of existence of the Deser-Waldron type gauge symmetry. Origin of vDVZ discontiunity can be studied at the Fierz-Pauli limit a -> 1.
The (free) graviton admits, in addition to the standard Pauli-Fierz description by means of a rank-two symmetric tensor, a description in which one dualizes the corresponding (2,2)-curvature tensor on one column to get a (D-2,2)-tensor, where D is the spacetime dimension. This tensor derives from a gauge field with mixed Yound symmetry (D-3,1) called the "dual graviton" field. The dual graviton field is related non-locally to the Pauli-Fierz field (even on-shell), in much the same way as a p-form potential and its dual (D-p-2)-form potential are related in the theory of an abelian p-form. Sinc…
We use Kaluza-Klein reduction to construct flat-space massive spin-2 Lagrangians based on a kinetic term that has local Weyl and transverse-diffeomorphism gauge symmetries in the massless limit. This yields Lagrangians describing a free massive spin 2 which differ from the usual Fierz-Pauli theory, but are physically equivalent to it. These Lagrangians require the use of auxiliary fields, which appear naturally from the higher-dimensional construction. We discuss how these Lagrangians are related to each other and to the Fierz-Pauli theory through Stückelberg transformations, gauge fixings, an…
We introduce a technique for restoring general coordinate invariance into theories where it is explicitly broken. This is the analog for gravity of the Callan-Coleman-Wess-Zumino formalism for gauge theories. We use this to elucidate the properties of interacting massless and massive gravitons. For a single graviton with a Planck scale Mpl and a mass mg, we find that there is a sensible effective field theory which is valid up to a high-energy cutoff Lambda parametrically above mg. Our methods allow for a transparent understanding of the many peculiarities associated with massive gravitons, am…
Using the parent Lagrangian approach we construct a dual formulation, in the sense originally proposed by Curtright and Freund, of a massive spin two Fierz-Pauli theory in arbitrary dimensions $D$. This is achieved in terms of a mixed symmetry tensor $T_{A[B_{1}B_{2}... B_{D-2}]}$, without the need of auxiliary fields. The relation of this method with an alternative formulation based on a gauge symmetry principle proposed by Zinoviev is elucidated. We show that the latter formulation in four dimensions, with a given gauge fixing together with a definite sequence of auxiliary fields elimination…
30-component, of the first order, equation for a spin 2 particle, equivalent to the second order Pauli-Fierz one, is generalized to presence of an external electromagnetic field as well as a curved background space-time geometry. The essential property of the generally covariant wave equation obtained is that here from the very beginning, in accordance with requirement of the Pauli-Fierz approach, a set of additional relations on 30-component wave function for eliminating complementary spin 0 and spin 1 fields is present at the starting equation.
A general approach to description of multigravity models in D-dimensional space-time is presented. Different possibilities of generalization of the invariant volume are given. Then a most general form of the interaction potential is constructed, which for bigravity coincides with the Pauli-Fierz model. A thorough analysis of the model along the 3+1 expansion formalism is done. It is shown that the absence of ghosts the considered bigravity model is equivalent in the weak field limit to the massive gravity (the Pauli-Fierz model). Thus, on the concrete example it is shown, that the interaction …
The cross-couplings among several massless spin-two fields (described in the free limit by a sum of Pauli-Fierz actions) in the presence of a Dirac field are investigated in the framework of the deformation theory based on local BRST cohomology. Under the hypotheses of locality, smoothness of the interactions in the coupling constant, Poincare invariance, (background) Lorentz invariance and the preservation of the number of derivatives on each field, we prove that there are no consistent cross-interactions among different gravitons in the presence of a Dirac field. The basic features of the co…
Massive spin-2 particles has been a subject of great interest in current research. If the graviton has a small mass, the gravitational force at large distances decreases more rapidly, which could contribute to explain the accelerated expansion of the universe. The massive spin-2 particles are commonly described by the known Fierz-Pauli action which is formulated in terms of a symmetric tensor $h_{μν}=h_{νμ}$. However, the Fierz-Pauli theory is not the only possible description of massive spin-2 particles via a rank-2 tensor. There are other two families of models $\mathcal{L}(a_1)$ and $\mathc…
The so called unimodular theory of gravitation is compared with general relativity in the quadratic (Fierz-Pauli) regime, using a quite broad framework, and it is argued that quantum effects allow in principle to discriminate between both theories.
We review the non-trivial issue of the relativistic description of a quantum mechanical system that, contrary to a common belief, kept theoreticians busy from the end of 1920s to (at least) mid 1940s. Starting by the well-known works by Klein-Gordon and Dirac, we then give an account of the main results achieved by a variety of different authors, ranging from de Broglie to Proca, Majorana, Fierz-Pauli, Kemmer, Rarita-Schwinger and many others. A particular interest comes out for the general problem of the description of particles with \textit{arbitrary} spin, introduced (and solved) by Majoran…
Filling a much-needed gap, we exhibit the D=4 Fierz-Pauli (FP) massive s=2 action, and its--manifestly positive--energy, in terms of its 2s+1=5 unconstrained helicity (2,1,0) excitations, after reducing and diagonalizing the troublesome helicity-0 sector.
The cross-couplings among several massless spin-two fields (described in the free limit by a sum of Pauli-Fierz actions) in the presence of a massive Rarita-Schwinger field are investigated in the framework of the deformation theory based on local BRST cohomology. Under the hypotheses of locality, smoothness of the interactions in the coupling constant, Poincare invariance, Lorentz covariance, and the preservation of the number of derivatives on each field, we prove that there are no consistent cross-interactions among different gravitons with a positively defined metric in internal space in t…
The couplings that can be introduced between a massless Rarita-Schwinger field, a Pauli-Fierz field and an Abelian three-form gauge field in eleven spacetime dimensions are analyzed in the context of the deformation of the solution of the master equation.
Generalized structure of the interaction term of multigravity is analyzed in detail. The coincidence limit of any multigravity theory is defined and the compatibility equation for the interaction potential is derived which is studied in the weak perturbation limit of metric. The most general properties of the invariant volume and the scalar potential of multigravity are investigated. The general formula for multigravity invariant volume using three means (arithmetic, geometric and harmonic) is derived. The Pauli-Fierz mass term for bigravity in the weak field limit is obtained.
The Fierz-Pauli (FP) free field theory for massive spin 2 particles can be extended, in a spacetime of (1+2) dimensions (3D), to a generally covariant parity-preserving interacting field theory, in at least two ways. One is "new massive gravity" (NMG), with an action that involves curvature-squared terms. Another is 3D "bigravity", which involves non-linear couplings of the FP tensor field to 3D Einstein-Hilbert gravity. We review the proof of the linearized equivalence of both "massive 3D gravity" theories to FP theory, and we comment on their similarities and differences.
Generalized Pauli's theorem, proved by D. S. Shirokov for two sets of anticommuting elements of a real or complexified Clifford algebra of dimension $2^n$, is extended to the case, when both sets of elements depend smoothly on points of Euclidian space of dimension $r$. We prove that in the case of even $n$ there exists a smooth function such that two sets of Clifford algebra elements are connected by a similarity transformation. All cases of connection between two sets are considered in the case of odd $n$. Using the equation for the spin connection of general form, it is shown that the probl…
