The ultraviolet completion of gauge theories by higher derivative terms can dramatically change their behavior at high energies. The requirement of asymptotic freedom imposes very stringent constraints that are only satisfied by a small family of higher derivative theories. If the number of derivatives is large enough ($n > 4$) the theory is strongly interacting both at extreme infrared and ultraviolet regimes whereas it remains asymptotically free for a low number of extra derivatives ($n \leqslant 4$). In all cases the theory improves its ultraviolet behavior leading in some cases to ultraviolet finite theories with vanishing $β$-function. The usual consistency problems associated to the presence of extra ghosts in higher derivative theories may not harm asymptotically free theories because in that case the effective masses of such ghosts are running to infinity in the ultraviolet limit.
As an example of what happens with physically relevant theories like effective gravity, we consider the covariant relativistic theory of a scalar field of arbitrarily higher differential order. A procedure based on the Legendre transformation and suitable field redefinitions allows to recast it as a theory of second order with one explicit independent field for each degree of freedom. The physical and ghost fields are then apparent. The full (classical) equivalence of both Higher and Lower Derivative versions is shown. An artifact of the method is the appearance of irrelevant spurious fields w…
Typically higher-derivative theories are unstable. Instabilities manifest themselves from extra propagating degrees of freedom, which are unphysical. In this paper, we will investigate an infinite derivative field theory and study its true dynamical degrees of freedom via Hamiltonian analysis. In particular, we will show that if the infinite derivatives can be captured by a Gaussian kinetic term, i.e. exponential of entire function, then it is possible to prove that there are only finite number of dynamical degrees of freedom. We will further extend our investigation into infinite derivative t…
We show that some higher derivative theories have a BRST symmetry. This symmetry is due to the higher derivative structure and is not associated to any gauge invariance. If physical states are defined as those in the BRST cohomology then the only physical state is the vacuum. All negative norm states, characteristic of higher derivative theories, are removed from the physical sector. As a consequence, unitarity is recovered but the S-matrix is trivial. We show that a class of higher derivative quantum gravity theories have this BRST symmetry so that they are consistent as quantum field theorie…
The article is devoted to the investigation of the Noether currents and integrals of motion in the special subclass of theories with higher field derivatives -- theories under differential field transformations in action (DFTA). Under fairly general assertions, we derive a simple representation for the integrals as sums of "old" integrals of motion, terms, that vanishes on the "old" equations of motion and surface integrals. We show that for some cosmological theories of that kind with high order derivatives the last contribution can violate gauge invariance, specifically, diffeomorphisms. The…
Infinite derivative theory of gravity is a modification to the general theory of relativity. Such modification maintains the massless graviton as the only true physical degree of freedom and avoids ghosts. Moreover, this class of modified gravity can address classical singularities. In this thesis some essential aspects of an infinite derivative theory of gravity are studied. Namely, we considered the Hamiltonian formalism, where the true physical degrees of freedom for infinite derivative scalar models and infinite derivative gravity are obtained. Furthermore, the Gibbons-Hawking-York boundar…
We construct no-ghost theories of analytic mechanics involving arbitrary higher-order derivatives in Lagrangian. It has been known that for theories involving at most second-order time derivatives in the Lagrangian, eliminating linear dependence of canonical momenta in the Hamiltonian is necessary and sufficient condition to eliminate Ostrogradsky ghost. In the previous work we showed for the specific quadratic model involving third-order derivatives that the condition is necessary but not sufficient, and linear dependence of canonical coordinates corresponding to higher time-derivatives also …
We review a perturbative approach to deal with Lagrangians with higher or infinite order time derivatives. It enables us to construct a consistent Poisson structure and Hamiltonian with only first time derivatives order by order in coupling. To the lowest order, the Hamiltonian is bounded from below whenever the potential is. We consider spacetime noncommutative field theory as an example.
In general coordinate invariant gravity theories whose Lagrangians contain arbitrarily high order derivative fields, the Noether currents for the global translation and for the Nakanishi's IOSp(8|8) choral symmetry containing the BRS symmetry as its member, are constructed. We generally show that for each of those Noether currents a suitable linear combination of equations of motion can be brought into the form of Maxwell-type field equation possessing the Noether current as its source term.
In this paper we will consider the ultraviolet (UV) finiteness of the most general one-particle irreducible ($1$PI) Feynman diagrams within the context of ghost-free, infinite-derivative scalar toy model, which is inspired from ghost free and singularity-free infinite-derivative theory of gravity. We will show that by using dressed vertices and dressed propagators, $n$-loop, $N$-point diagrams constructed out of lower-loop $2$- & $3$-point and, in general, $N_i$-point diagrams are UV finite with respect to internal and external loop momentum. Moreover, we will demonstrate that the external mom…
In this paper we present a class of higher derivative theories of gravity which admit Birkhoff's theorem. In particular, we explicitly show that in this class of theories, although generically the field equations are of fourth order, under spherical (plane or hyperbolic) symmetry, all the field equations reduce to second order and have exactly the same or similar structure to those of Lovelock theories, depending on the spacetime dimensions and the order of the Lagrangian.
Within the general framework of spatially covariant theories of gravity, we study the conditions for having only the two tensorial degrees of freedom. Generally, there are three degrees of freedom propagating in the theory, of which two are tensorial and one is of the scalar type. Through a detailed Hamiltonian analysis, we find two necessary and sufficient conditions to evade the scalar type degree of freedom. The first condition implies that the lapse-extrinsic curvature sector must be degenerate. The second condition ensures that the dimension of the phase space at each spacetime point is e…
We consider diffeomorphism invariant theories of gravity with arbitrary higher derivative terms in the Lagrangian as corrections to the leading two derivative theory of Einstein's general relativity. We construct a proof of the zeroth law of black hole thermodynamics in such theories. We assume that a stationary black hole solution in an arbitrary higher derivative theory can be obtained by starting with the corresponding stationary solution in general relativity and correcting it order by order in a perturbative expansion in the coupling constants of the higher derivative Lagrangian. We prove…
We study the gravitational production of heavy X-particles of mass of the order of the inflaton mass, produced after the end of inflation. We find that, in the presence of a derivative coupling of the inflaton field or of the X-field to the Einstein tensor, the number of gravitationally produced particles is suppressed as the strength of the coupling is increased.
We study the Hawking-Turok (HT) instanton solutions which have been employed to describe the creation of an open inflationary universe, in the context of higher derivative theories. We consider the effects of adding quadratic and cubic terms of the forms $αR^{2}$ and $βR^{3}$ to the gravitational action. Using a conformal transformation to convert the higher derivative theories into theories of self interacting scalar fields minimally coupled to Einstein gravity, we argue that the cubic term represents a generic perturbation of the polynomial type to the action and obtain the conditions on the…
We study a novel class of higher derivative theories for interacting massless gravitons in Minkowski spacetime. These theories were first discussed by Wald decades ago, and are characterized by scattering amplitudes essentially different from general relativity and many of its modifications. We discuss various aspects of these higher derivative theories, including the Lagrangian construction, violation of asymptotic causality, scattering amplitudes, non-renormalization, and possible implications on emergent gravitons from condensed matter systems.
We propose an entropy current for dynamical black holes in a theory with arbitrary four derivative corrections to Einstein's gravity, linearized around a stationary black hole. The Einstein-Gauss-Bonnet theory is a special case of the class of theories that we consider. Within our approximation, our construction allows us to write down a completely local version of the second law of black hole thermodynamics, in the presence of the higher derivative corrections considered here. This ultra-local, stronger form of the second law is a generalization of a weaker form, applicable to the total entro…
A discussion of the number of degrees of freedom, and their dynamical properties, in higher-derivative gravitational theories is presented. The complete non-linear sigma model for these degrees of freedom is exhibited using the method of auxiliary fields. As a by-product we present a consistent non-linear coupling of a spin-2 tensor to gravitation. It is shown that non-vanishing $(C_{μναβ})^{2}$ terms arise in $N=1$, $D=4$ superstring Lagrangians due to one-loop radiative corrections with light field internal lines.
As an alternative to the covariant Ostrogradski method, we show that higher-derivative relativistic Lagrangian field theories can be reduced to second differential-order by writing them directly as covariant two-derivative theories involving Lagrange multipliers and new fields. Notwithstanding the intrinsic non-covariance of the Dirac's procedure used to deal with the constraints, the Lorentz invariance is recovered at the end. We develope this new setting for a simple scalar model and then its applications to generalized electrodynamics and higher-derivative gravity are outlined. This method …
We study metric transformations which depend on a scalar field $ϕ$ and its first derivatives and confirm that the number of physical degrees of freedom does not change under such transformations, as long as they are not singular. We perform a Hamiltonian analysis of a simple model in the gauge $ϕ= t$. In addition, we explicitly show that the transformation and the gauge fixing do commute in transforming the action. We then extend the analysis to more general gravitational theories and transformations in general gauges. We verify that the set of all constraints and the constraint algebra are le…
As an alternative to the covariant Ostrogradski method, we show that higher-derivative relativistic Lagrangian field theories can be reduced to second differential-order by writing them directly as covariant two-derivative theories involving Lagrange multipliers and new fields. Despite the intrinsic non-covariance of the Dirac's procedure used to deal with the constraints, the explicit Lorentz invariance is recovered at the end. We develop this new setting on the grounds of a simple scalar model and then its applications to generalized electrodynamics and higher-derivative gravity are worked o…
We study the information-geometric properties of the Deser-Sarioglu-Tekin black hole, which is a higher derivative gravity solution with contributions from a non-polynomial term of the Weyl tensor to the Einstein-Hilbert Lagrangian. Our investigation is focused on deriving the relevant information metrics and their scalar curvatures on the space of equilibrium states. The analysis is conducted within the framework of thermodynamic information geometry and shows highly non-trivial statistical behavior. Furthermore, the quasilocal formalism, developed by Brown and York, was successfully implemen…
In String Theory there often appears a rather interesting class of higher derivative theories containing an infinite set of derivatives in the form of an exponential. These theories may provide a way to tame ultraviolet divergences without introducing ghost-like states. In this invited article we provide a brief overview on the progress that has been made over the last decade to construct such infinite derivative theories of gravity which may be able to address the singularity problems in gravity. In the process we will also be able to present some general results that applies to covariant tor…
In this work we consider gravitational theories in which the effect of coupling characteristic classes, appropriately introduced as operators in the Einstein-Hilbert action, has been taken into account. As it is well known, this approach strays from the framework of general relativity since it results in theories in which torsion can be present. We consider here all the characteristic classes that are consistent with a four-dimensional space-time manifold. Then, we present explicit expressions for the contortion $1$-form and torsion $2$-form that can be obtained under general conditions in var…
We construct a proof of the second law of thermodynamics in an arbitrary diffeomorphism invariant theory of gravity working within the approximation of linearized dynamical fluctuations around stationary black holes. We achieve this by establishing the existence of an entropy current defined on the horizon of the dynamically perturbed black hole in such theories. By construction, this entropy current has non-negative divergence, suggestive of a mechanism for the dynamical black hole to approach a final equilibrium configuration via entropy production as well as the spatial flow of it on the nu…
Recent work has shown that non-local modifications of the Einstein equations can have interesting cosmological consequences and can provide a dynamical origin for dark energy, consistent with existing data. At first sight these theories are plagued by ghosts. We show that these apparent ghost-like instabilities do not describe actual propagating degrees of freedom, and there is no issue of ghost-induced quantum vacuum decay.
As a continuation of a previous work, here we examine the admittance of Birkhoff's theorem in a class of higher derivative theories of gravity. This class is contained in a larger class of theories which are characterized by the property that the trace of the field equations are of second order in the metric. The action representing these theories are given by a sum of higher curvature terms. Moreover the terms of a fixed order k in the curvature are constructed by taking a complete contraction of k conformal tensors. The general spherically (hyperbolic or plane) symmetric solution is then giv…
Theories with higher order time derivatives generically suffer from ghost-like instabilities, known as Ostrogradski instabilities. This fate can be avoided by considering "degenerate" Lagrangians, whose kinetic matrix cannot be inverted, thus leading to constraints between canonical variables and a reduced number of physical degrees of freedom. In this work, we derive in a systematic way the degeneracy conditions for scalar-tensor theories that depend quadratically on second order derivatives of a scalar field. We thus obtain a classification of all degenerate theories within this class of sca…
In this paper, we consider an infinite derivative scalar field action with infinite derivative kinetic and interaction terms. We establish that the theory is unitary if the correlation functions are formulated in Euclidean space and then analytically continued in their external momenta to Minkowski space.
We discuss the extent to which it is necessary to include higher-derivative operators in the effective field theory of general scalar-tensor theories. We explore the circumstances under which it is correct to restrict to second-order operators only, and demonstrate this using several different techniques, such as reduction of order and explicit field redefinitions. These methods are applied, in particular, to the much-studied Horndeski theories. The goal is to clarify the application of effective field theory techniques in the context of popular cosmological models, and to explicitly demonstra…
We discuss higher derivative corrections to black hole entropy in theories that allow a near horizon AdS_3 x X geometry. In arbitrary theories with diffeomorphism invariance we show how to obtain the spacetime central charge in a simple way. Black hole entropy then follows from the Euclidean partition function, and we show that this gives agreement with Wald's formula. In string theory there are certain diffeomorphism anomalies that we exploit. We thereby reproduce some recent computations of corrected entropy formulas, and extend them to the nonextremal, nonsupersymetric context. Examples inc…
In this note, we discuss the gravitational memory effect in higher derivative and infinite derivative gravity theories and give the detailed relevant calculations whose results were given in our recent works. We show that the memory effect in higher derivative gravity takes the same form as in pure GR at large distances, whereas at small distances, the results are different. We also demonstrated that, in infinite derivative gravity, the memory is reduced via error function as compared to Einstein's gravity. For the lower bound on the mass scale of non-locality, the memory is essentially reprod…
The canonical structure of higher dimensional pure Chern-Simons theories is analysed. It is shown that these theories have generically a non-vanishing number of local degrees of freedom, even though they are obtained by means of a topological construction. This number of local degrees of freedom is computed as a function of the spacetime dimension and the dimension of the gauge group.
We carry out the extension of the covariant Ostrogradski method to fermionic field theories. Higher-derivative Lagrangians reduce to second order differential ones with one explicit independent field for each degree of freedom.
In this paper we will provide a non-singular rotating space time metric for a ghost free infinite derivative theory of gravity. We will provide the predictions for the Lense-Thirring effect for a slowly rotating system, and how it is compared with that from general relativity.
An alternative version of Hamiltonian formalism for higher-derivative theories is presented. It is related to the standard Ostrogradski approach by a canonical transformation. The advantage of the approach presented is that the Lagrangian is nonsingular and the Legendre transformation is performed in a straightforward way.
In arXiv:1310.5713 and arXiv:1310.6659 a formula was proposed as the entanglement entropy functional for a general higher-derivative theory of gravity, whose lagrangian consists of terms containing contractions of the Riemann tensor. In this paper, we carry out some tests of this proposal. First, we find the surface equation of motion for general four-derivative gravity theory by minimizing the holographic entanglement entropy functional resulting from this proposed formula. Then we calculate the surface equation for the same theory using the generalized gravitational entropy method of arXiv:1…
Higher Derivative (HD) Field Theories can be transformed into second order equivalent theories with a direct particle interpretation. In a simple model involving abelian gauge symmetries we examine the fate of the possible gauge fixings throughout this process. This example is a useful test bed for HD theories of gravity and provides a nice intuitive interpretation of the "third ghost" occurring there and in HD gauge theories when a HD gauge fixing is adopted.
In this paper, we demonstrate that the Wald's entropy for any spherically symmetric blackhole within an infinite derivative theory of gravity is determined solely by the area law. Thus, the infrared behaviour of gravity is captured by the Einstein-Hilbert term, provided that the massless graviton remains the only propagating degree of freedom in the spacetime.
We carry out the extension of the Ostrogradski method to relativistic field theories. Higher-derivative Lagrangians reduce to second differential-order with one explicit independent field for each degree of freedom. We consider a higher-derivative relativistic theory of a scalar field and validate a powerful order-reducing covariant procedure by a rigorous phase-space analysis. The physical and ghost fields appear explicitly. Our results strongly support the formal covariant methods used in higher-derivative gravity.
Higher derivative terms in the effective action of certain Yang-Mills theories can be severely constrained by supersymmetry. We show that requiring sixteen supersymmetries in quantum mechanical gauge theory determines the $v^6$ term in the effective action. Even the numerical coefficient of the $v^6$ term is fixed in terms of lower derivative terms in the effective action.
We study the problem of decoupling of heavy chiral superfields in four-dimensional $N=1$ supersymmetric field theories with Lorentz-invariant and Lorentz-violating higher-derivative terms. We demonstrate that the earlier found effect of large logarithmic quantum corrections, due to heavy chiral superfields, takes place not only if the theory possesses quantum divergences, but also for essentially finite theories involving higher derivative terms, both Lorentz-invariant and Lorentz-breaking ones.
We discuss the contribution of ADHM multi-instantons to the higher-derivative terms in the gradient expansion along the Coulomb branch of N=2 and N=4 supersymmetric SU(2) gauge theories. In particular, using simple scaling arguments, we confirm the Dine-Seiberg nonperturbative nonrenormalization theorems for the 4-derivative/8-fermion term in the two finite theories (N=4, and N=2 with N_F=4).
We consider theories describing the dynamics of a four-dimensional metric, whose Lagrangian is diffeomorphism invariant and depends at most on second derivatives of the metric. Imposing degeneracy conditions we find a set of Lagrangians that, apart form the Einstein-Hilbert one, are either trivial or contain more than two degrees of freedom. Among the partially degenerate theories, we recover Chern-Simons gravity, endowed with constraints whose structure suggests the presence of instabilities. Then, we enlarge the class of parity violating theories of gravity by introducing new "chiral scalar-…
We show that the recently constructed higher-derivative 6D SYM theory involves an internal chiral anomaly breaking gauge invariance. The anomaly is cancelled when adding to the theory an adjoint matter hypermultiplet.
We review some results of applying the higher covariant derivative regularization to the investigation of quantum corrections structure in N=1 supersymmetric theories. In particular, we demonstrate that all integrals, defining the Gell-Mann--Low function in supersymmetric theories, are integrals of total derivatives. As a consequence, there is an identity for Green functions, which does not follow from any known symmetry of the theory, in N=1 supersymmetric theories. We also discuss how to derive the exact $β$-function by methods of the perturbation theory.
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.
We consider warped AdS_3 black holes in generic higher derivatives gravity theories in 2+1 dimensions. The asymptotic symmetry group of the phase space containing these black holes is the semi-direct product of a centrally extended Virasoro algebra and an affine u(1) Kac-Moody algebra. Previous works have shown that in some specific theories, the entropy of these black holes agrees with a Cardy-like entropy formula derived for warped conformal field theories. In this paper, we show that this entropy matching continues to hold for the most general higher derivative theories of gravity. We also …
Interacting theories with higher derivatives involve ghosts. They correspond to instabilities that display themselves at the classical level. We notice that comparatively "benign" mechanical higher-derivative systems exist where the classical vacuum is stable with respect to small perturbations and the problems appear only at the nonperturbative level. We argue that benign higher-derivative field theories exist which are stable with respect to small fluctuations with nonzero momenta. A particular example is the 6D N=2 higher-derivative SYM theory, which is finite and unitary at the perturbativ…
It has frequently been claimed in the literature that the classical physical predictions of scalar tensor theories of gravity depend on the conformal frame in which the theory is formulated. We argue that this claim is false, and that all classical physical predictions are conformal-frame invariants. We also respond to criticisms by Vollick [gr-qc/0312041], in which this issue arises, of our recent analysis of the Palatini form of 1/R gravity.
We provide a full analysis of ghost free higher derivative field theories with coupled degrees of freedom. Assuming the absence of gauge symmetries, we derive the degeneracy conditions in order to evade the Ostrogradsky ghosts, and analyze which (non)trivial classes of solutions this allows for. It is shown explicitly how Lorentz invariance avoids the propagation of "half" degrees of freedom. Moreover, for a large class of theories, we construct the field redefinitions and/or (extended) contact transformations that put the theory in a manifestly first order form. Finally, we identify which cla…
We study the one-loop quantum corrections for higher-derivative superfield theories, generalizing the approach for calculating the superfield effective potential. In particular, we calculate the effective potential for two versions of higher-derivative chiral superfield models. We point out that the equivalence of the higher-derivative theory for the chiral superfield and the one without higher derivatives but with an extended number of chiral superfields occurs only when the mass term is contained in the general Lagrangian. The presence of divergences can be taken as an indication of this equ…
We discuss a formulation of Galileon actions in terms of matrix determinants in four dimensions. This approach allows one to straightforwardly determine derivative couplings between Galileons and scalar or vector degrees of freedom that lead to equations of motion with at most two space-time derivatives. We use this method to easily build generalizations of Galileon set-ups preserving conformal symmetry, finding explicit examples of couplings between Galileons and additional degrees of freedom that preserve the Galileon conformal invariance. We discuss various physical applications of our meth…
We study the classical and quantum dynamics of generally covariant theories with vanishing a Hamiltonian and with a finite number of degrees of freedom. In particular, the geometric meaning of the full solution of the relational evolution of the degrees of freedom is displayed, which means the determination of the total number of evolving constants of motion required. Also a method to find evolving constants is proposed. The generalized Heinsenberg picture needs M time variables, as opposed to the Heisenberg picture of standard quantum mechanics where one time variable t is enough. As an appli…
In this work, we employ renormalization group methods to study the general behavior of field theories possessing anisotropic scaling in the spacetime variables. The Lorentz breaking symmetry that accompanies these models are either soft, if no higher spatial derivative is present, or it may have a more complex structure if higher spatial derivatives are also included. Both situations are discussed in models with only scalar fields and also in models with fermions as a Yukawa like model.
