We analyse the power counting renormalizability of the quantum field theory of Einstein or Einstein-Gauss-Bonnet gravity in D+2 dimensional Lifshitz spacetime. We show the spectral dimension becomes 2+(D/z) at the UV region where z is the critical exponent. Since it is larger than two, the quantum theory of Einstein gravity is not power counting renormalizable. For the pure Einstein-Gauss-Bonnet gravity, where Lifshitz spacetime is allowed only when the parameters are fine tuned, it happens that the graviton modes do not propagate and the quantum field theory is accidentally renormalizable when z>=D. Another method is discretizing the radial coordinate which changes the spectral dimension to 1+(D/z) at the UV region. Since our four dimensional spacetime is continuous, the four dimensional Lorentz symmetry is recovered at the low energy and the power counting renormalizability is still kept for z>=D, if the spacetime near the null singularity in Lifshitz spacetime is modified into AdS spacetime and the discrete radial direction is compactified like a brane world scenario. We also comment on the AdS/CFT correspondence.
In this note, we discuss the renormalizability of Horava-Lifshitz-type gravity theories. Using the fact that Horava-Lifshitz gravity is very closely related to the stochastic quantization of topologically massive gravity, we show that the renormalizability of HL gravity only depends on the renormalizability of topologically massive gravity. This is a consequence of the BRST and time-reversal symmetries pertinent to theories satisfying the detailed balance condition.
We analyse the power counting renormalizability of scalar theory in Lifshitz spacetime in D+2 dimensions. We show the spectral dimension becomes 2+(D/z) (z is the critical exponent) after integrating out the radion field. We comment on the AdS/CFT correspondence, and on how to avoid the Lifshitz singularity by flowing into AdS spacetime in the infrared. We also comment on the quantum gravity in Lifshitz spacetime.
We study unitarity and renormalizability in the Lifshitz scalar field theory, which is characterized by an anisotropic scaling between the space and time directions. Without the Lorentz symmetry, both the unitarity and the renormalizability conditions are modified from those in relativistic theories. We show that for renormalizability, an extended version of the power counting condition is required in addition to the conventional one. The unitarity bound for S-matrix elements also gives stronger constraints on interaction terms because of the reference frame dependence of scattering amplitudes…
We argue that the true nature of the renormalizability of Horava-Lifshitz gravity lies in the presence of higher order spatial derivatives and not in the anisotropic Lifshitz scaling of space and time. We discuss the possibility of constructing a higher order spatial derivatives model that has the same renormalization properties of Horava-Lifshitz gravity but that does not make use of the Lifshitz scaling. In addition, the state-of-the-art of the Lorentz symmetry restoration in Horava-Lifshitz-type theories of gravitation is reviewed.
We study the renormalizability in theories of a self-interacting Lifshitz scalar field. We show that although the statement of power-counting is true at one-loop order, in generic cases where the scalar field is dimensionless, an infinite number of counter terms are involved in the renormalization procedure. This problem can be avoided by imposing symmetries, the shift symmetry in the present paper, which allow only a finite number of counter terms to appear. The symmetry requirements might have important implications for the construction of matter field sectors in the Horava-Lifshitz gravity.
We study tree-unitarity and renormalizability in Lifshitz-scaling theory, which is characterized by an anisotropic scaling between the spacial and time directions. Due to the lack of the Lorentz symmetry, the conditions for both unitarity and renormalizability are modified from those in relativistic theories. For renormalizability, the conventional discussion of the power counting conditions has to be extended. Because of the dependence of $S$-matrix elements on the reference frame, unitarity requires stronger conditions than those in relativistic cases. We show that the conditions for unitari…
In an earlier article [arXiv:0902.0590 [hep-th], Phys. Rev D80 (2009) 025011], I discussed the potential benefits of allowing Lorentz symmetry breaking in quantum field theories. In particular I discussed the perturbative power-counting finiteness of the normal-ordered :P(phi)^{z>=d}_{d+1}: scalar quantum field theories, and sketched the implications for Horava's model of quantum gravity. In the current rather brief addendum, I will tidy up some dangling issues and fill out some of the technical details of the argument indicating the power-counting renormalizability of a z>=d variant of Horava…
We consider the version of Hořava gravity where "detailed balance" is consistently implemented, so as to limit the huge proliferation of couplings in the full theory and obtain healthy dynamics at low energy. Since a superpotential which is third-order in spatial derivatives is not sufficient to guarantee the power-counting renormalizability of the spin-0 graviton, one needs to go an order beyond in derivatives, building a superpotential up to fourth-order spatial derivatives. Here we perturb the action to quadratic order around flat space and show that the power-counting renormalizability of …
It has been argued that Horava gravity needs to be extended to include terms that mix spatial and time derivatives in order avoid unacceptable violations of Lorentz invariance in the matter sector. In an earlier paper we have shown that including such mixed derivative terms generically leads to 4th instead of 6th order dispersion relations and this could be (naively) interpreted as a threat to renormalizability. We have also argued that power-counting renormalizability is not actually compromised, but instead the simplest power-counting renormalizable model is not unitary. In this paper we con…
An anisotropic model describing gravity--vector gauge coupling at all energy scales is presented. The starting point is the 4+1 dimensional non--projectable Hořava--Lifshitz gravity theory subject to a geometrical restriction. Renormalizability arguments require all possible interactions in the potential up to terms with $z=4$ spatial derivatives on the geometrical tensor fields: the Riemann and Weyl tensors. The latter being necessary on a 4+1 dimensional formulation. The dimensional reduction to 3+1 dimensions give rise to a model invariant under {foliation--preserving diffeomorphisms} (FDif…
We consider the relationship between renormalizability and unitarity at a Lifshitz point in d dimensions. We test tree unitarity for theories containing only scalars and fermions, and for pure gauge theory. In both cases, we find the requirement of weighted power-counting renormalizability is equivalent to that of tree unitarity.
We provide Wilsonian proof for renormalizability of four-dimensional quantum field theories with ${\cal N}=1/2$ supersymmetry. We argue that the non-hermiticity inherent to these theories permits assigning noncanonical scaling dimension both for the Grassman coordinates and superfields. This reassignment can be done in such a way that the non(anti)commutativity parameter is dimensionless, and then the rest of the proof ammounts to power counting. The renormalizability is also stable against adding standard four-dimensional soft-breaking terms to the theory. However, with the new scaling dimens…
We initiate the systematic computation of the heat-kernel coefficients for Laplacian operators obeying anisotropic dispersion relations in curved spacetime. Our results correctly reproduce the limit where isotropy is restored and special anisotropic cases considered previously in the literature. Subsequently, the heat kernel is used to derive the scalar-induced one-loop effective action and beta functions of Horava-Lifshitz gravity. We identify the Gaussian fixed point which is supposed to provide the UV completion of the theory. In the present setting, this fixed point acts as an infrared att…
It is well known that General Relativity cannot be considered under the standard of a perturbatively renormalizable quantum field theory, but asymptotic safety is taken into account as a possibility for the formulation of gravity as a non-perturbative renormalizable theory. Recently, the entropy argument has however stepped into the discussion claiming for a "no-go" to the asymptotic safety argument. In this paper, we present simple counter-examples, considering alternative theories of gravity, to the entropy argument as further indications, among others, on the possible flows in the assumptio…
We consider an interacting Lifshitz field with z=3 in a curved spacetime. We analyze the renormalizability of the theory for interactions of the form lambda phi^n, with arbitrary even n. We compute the running of the coupling constants both in the ultraviolet and infrared regimes. We show that the Lorentz violating terms generate couplings to the spacetime metric that are not invariant under general coordinate transformations. These couplings are not suppressed by the scale of Lorentz violation and therefore survive at low energies. We point out that in these theories, unless the effective mas…
We classify the physical observables in spontaneously broken nonlinearly realized gauge theories in the recently proposed loopwise expansion governed by the Weak Power-Counting (WPC) and the Local Functional Equation. The latter controls the non-trivial quantum deformation of the classical nonlinearly realized gauge symmetry, to all orders in the loop expansion. The Batalin-Vilkovisky (BV) formalism is used. We show that the dependence of the vertex functional on the Goldstone fields is obtained via a canonical transformation w.r.t. the BV bracket associated with the BRST symmetry of the model…
We study primordial gravitational waves produced during inflation in quantum gravity at a Lifshitz point proposed by Ho${\rm\check{r}}$ava. Assuming power-counting renormalizability, foliation preserving diffeomorphism invariance, and the condition of detailed balance, we show that primordial gravitational waves are circularly polarized due to parity violation. The chirality of primordial gravitational waves is a quite robust prediction of quantum gravity at a Lifshitz point which can be tested through observations of cosmic microwave background radiation and stochastic gravitational waves.
We investigate some classical and quantum aspects of a general class of higher derivative theories of gravity. We propose a generalized version of the so-called Teyssandier gauge condition and we investigative its implications on the linearized field equations. An exhaustive investigation on the particle spectra is done, including a discussion on the appearance of ghost-like particles. We investigate the UV properties and we determine the power-counting renormalizability of the theory. Finally we probe a conjecture which relates renormalizability with the cancellation of Newtonian singularitie…
In this paper, we discuss questions related to the renormalizability of the classical statistical approximation, an approximation scheme that has been used recently in several studies of out-of-equilibrium problems in Quantum Field Theory. Although the ultraviolet power counting in this approximation scheme is identical to that of the unapproximated quantum field theory, this approximation is not renormalizable. The leading cause of this non-renormalizability is the breakdown of Weinberg's theorem in this approximation. We also discuss some practical implications of this negative result for si…
We analyze the semiclassical Hořava-Lifshitz gravity for quantum scalar fields in 3+1 dimensions. The renormalizability of the theory requires that the action of the scalar field contains terms with six spatial derivatives of the field, i.e. in the UV, the classical action of the scalar field should preserve the anisotropic scaling symmetry ($t \to L^{2z}t,$ $\vec{x} \to L^2 \vec{x}$, with $z=3$) of the gravitational action. We discuss the renormalization procedure based on adiabatic subtraction and dimensional regularization in the weak field approximation. We verify that the divergent terms …
In Hořava-Lifshitz gravity a scaling isotropic in space but anisotropic in spacetime, often called anisotropic scaling with the dynamical critical exponent z=3, lies at the base of its renormalizability. This scaling also leads to a novel mechanism of generating scale-invariant cosmological perturbations, solving the horizon problem without inflation. In this paper we propose a possible solution to the flatness problem, in which we assume that the initial condition of the Universe is set by a small instanton respecting the same scaling. We argue that the mechanism may be more general than the …
In this paper it is studied the propagator for the modified theory of gravity proposed by Ho\vrava. We first calculate the propagator in the $λ=1$ case and show that the main poles that arise correspond to the spin two particle and scalar particle, already known in the literature. The presence of a bad ultraviolet behaving term spoils renormalizability of the theory but it is eliminated by imposing the detailed balance condition, although just a soft version of this condition is actually needed. The problems of a negative mass term and a residue with undefined sign, which is due to the presenc…
For any fundamental quantum field theory, unitarity, renormalizability, and relativistic invariance are considered to be essential properties. Unitarity is inevitably connected to the probabilistic interpretation of the quantum theory, while renormalizability guarantees its completeness. Relativistic invariance, in turn, is a symmetry which derives from the structure of spacetime. So far, the perturbative attempt to formulate a fundamental local quantum field theory of gravity based on the metric field seems to be in conflict with at least one of these properties. In quantum Hořava gravity, a …
We present a 3-dimensional model for massive gravity with masses induced by topological (Chern-Simons) and Proca-like mass terms. Causality and unitarity are discussed at tree-level. Power-counting renormalizability is also contemplated.
In the attempts toward a quantum gravity theory, general relativity faces a serious difficulty since it is non-renormalizable theory. Hořava-Lifshitz gravity offers a framework to circumvent this difficulty, by sacrificing the local Lorentz invariance at ultra-high energy scales in exchange of power-counting renormalizability. The Lorentz symmetry is expected to be recovered at low and medium energy scales. If gravitation is to be described by a Hořava-Lifshitz gravity theory there are a number of issues that ought to be reexamined in its context, including the question as to whether this grav…
Approximately one year ago Horava proposed a power-counting renormalizable theory of gravity which abandons local Lorentz invariance. The proposal has been received with growing interest and resulted in various different versions of Horava-Lifshitz gravity theories, involving a colourful potpourri of new terminology. In this proceedings contribution we first motivate and briefly overview the various different approaches, clarifying their differences and similarities. We then focus on a model referred to as projectable Horava-Lifshitz gravity and summarize the key results regarding its viabilit…
The purpose of this work is to present a model for 3D massive gravity with topological and higher-derivative terms. Causality and unitarity are discussed at tree-level. Power-counting renormalizability is also contemplated.
Non-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation $E\sim k^{z}$ and dynamical exponent $z>1$. The asymptotic growth of the number of states of these theories can be described by an extension of Cardy formula that depends on $z$. We show that this result can be recovered by counting the partitions of an integer into $z$-th powers, as proposed by Hardy and Ramanujan a century ago. This gives a novel relationship between the characteristic energy of the dispersion relation with the cylinder radius and the ground state…
At the present work, it is studied the extension of F (R) gravities to the new recently proposed theory of gravity, the so-called Horava-Lifshitz gravity, which provides a way to make the theory power counting renormalizable by breaking Lorentz invariance. It is showed that dark energy can be well explained in the frame of this extension, just in terms of gravity. It is also explored the possibility to unify inflation and late-time acceleration under the same mechanism, providing a natural explanation the accelerated expansion.
Horava gravity has been constructed so as to exhibit anisotropic scaling in the ultraviolet, as this renders the theory power-counting renormalizable. However, when coupled to matter, the theory has been shown to suffer from quadratic divergences. A way to cure these divergences is to add terms with both time and space derivatives. We consider this extended version of the theory in detail. We perform a perturbative analysis that includes all modes, determine the propagators and discuss how including mixed-derivative terms affects them. We also consider the Lifshitz scalar with mixed-derivative…
In the framework of the power-counting renormalizable theory of gravitation, recently proposed by Hořava, we study the limit $λ\to\infty$, which is arguably the most natural candidate for the ultraviolet fixed point of the renormalization group flow. In the projectable version with dynamical critical exponent $z=3$, we analyze the Friedmann-Robertson-Walker background driven by the so-called "dark matter as integration constant" and perturbations around it. We show that amplitudes of quantum fluctuations for both scalar and tensor gravitons remain finite in the limit and that the theory is wea…
The Horava-Lifshitz gravity, having broken the symmetry of space and time, includes three objects: the spatial metric $g_{ij}$, the lapse variable $N$, and the shift variable $N_{i}$. Each of these objects have their own scaling dimensions. The action of the theory is required to be invariant under this scaling, and spatial diffeomorphism, and the temporal foliation symmetry. Noting that action can non-trivially depend on the lapse variable, we suggest to consider the Horva's approach to quantum gravity in higher dimensions such that a set of extra spacial coordinates possess a scaling dimensi…
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…
We discuss a new covariant scalar-tensor system aimed to realise Hořava proposal for a power-counting renormalizable theory of gravity, with the special feature of not propagating scalar degrees of freedom in an appropriate gauge. The theory is characterized by a new symmetry acting on the metric, that can protect the particular form of its interactions. The set-up spontaneously breaks Lorentz symmetry by means of a time-like scalar field profile. By selecting a unitary gauge for the scalar, we show that this theory describes the dynamics of a spin two degree of freedom, whose equations of mot…
In this paper, we study the effects of high-order operators on the non-relativistic Lifshitz holography in the framework of the Hořava-Lifshitz (HL) theory of gravity, which naturally contains high-order operators in order for the theory to be power-counting renormalizble, and provides an ideal place for such studies. In particular, we show that the Lifshitz space-time is still a solution of the full theory of the HL gravity. The effects of the high-oder operators on the space-time itself is simply to shift the Lifshitz dynamical exponent. However, while in the infrared the asymptotic behavior…
We devise an algorithm which allows one to count the number of Killing vectors for a Lorentzian manifold of dimension 3. Our algorithm relies on the principal traces of powers of the Ricci tensor and branches intricately according to the values of differential invariants arising from the compatibility conditions of the Killing equation. As illustrating examples, we classify the Lifshitz and pp-wave spacetimes into a hierarchy based on their level of symmetry. A complete classification of spacetimes admitting 4 Killing vectors is also presented.
We present a Lorentz gauge theory of gravity in which the metric is not dynamical. Spherically symmetric weak field solutions are studied. We show that this solution contains the Schwarzschild spacetime at least to the first order of perturbation. Next, we present a special case of the theory. It is shown that the Schwarzschild metric is now an exact solution. Moreover, we show that the de Sitter space is an exact vacuum solution and as a result the theory is able to explain the expansion of the universe with no need for a dark energy. Within this special case, quantization of the theory is al…
We study the renormalization of a general field theory on the 2-sphere with tensorial interaction and gauge invariance under the diagonal action of SU(2). We derive the power counting for arbitrary dimension d. For the case d=4, we prove perturbative renormalizability to all orders via multi-scale analysis, study both the renormalised and effective perturbation series, and establish the asymptotic freedom of the model. We also outline a general power counting for the homogeneous space SO(D)/SO(D-1), of direct interest for quantum gravity models in general dimensions, and point out the obstruct…
A proposal for a power-counting renormalizable theory of quantum gravity at a Lifshitz point was recently put forth by Horava (arXiv:0901.3775), and has been since dubbed as Horava-Lifshitz gravity. The theory explicitly breaks Lorentz invariance, which introduces an apparent extra scalar degree of freedom. In this note, we show that the low energy limit of (non-projectible) Horava-Lifshitz gravity is uniquely given by the quadratic cuscuton model: a covariant scalar field theory with an infinite speed of sound and a quadratic potential, which is minimally coupled to Einstein gravity. This imp…
We define a modified dimensional-regularization technique that overcomes several difficulties of the ordinary technique, and is specially designed to work efficiently in chiral and parity violating quantum field theories, in arbitrary dimensions greater than 2. When the dimension of spacetime is continued to complex values, spinors, vectors and tensors keep the components they have in the physical dimension, therefore the $γ$ matrices are the standard ones. Propagators are regularized with the help of evanescent higher-derivative kinetic terms, which are of the Majorana type in the case of chi…
We study the most general case of spherically symmetric vacuum solutions in the framework of the Covariant Horava Lifshitz Gravity, for an action that includes all possible higher order terms in curvature which are compatible with power-counting normalizability requirement. We find that solutions can be separated into two main classes: (i) solutions with nonzero radial shift function, and (ii) solutions with zero radial shift function. In the case (ii), spherically symmetric solutions are consistent with observations if we adopt the view of Horava and Melby-Tomson, according to which the auxil…
We present a candidate quantum field theory of gravity with dynamical critical exponent equal to z=3 in the UV. (As in condensed matter systems, z measures the degree of anisotropy between space and time.) This theory, which at short distances describes interacting nonrelativistic gravitons, is power-counting renormalizable in 3+1 dimensions. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory flows naturally to the relativistic…
We prove the renormalizability of various theories of classical gravity coupled with interacting quantum fields. The models contain vertices with dimensionality greater than four, a finite number of matter operators and a finite or reduced number of independent couplings. An interesting class of models is obtained from ordinary power-counting renormalizable theories, letting the couplings depend on the scalar curvature R of spacetime. The divergences are removed without introducing higher-derivative kinetic terms in the gravitational sector. The metric tensor has a non-trivial running, even if…
A theory of quantum gravity has been recently proposed by means of a novel quantization prescription, which is able to turn the poles of the free propagators that are due to the higher derivatives into fakeons. The classical Lagrangian contains the cosmological term, the Hilbert term, $\sqrt{-g}R_{μν}R^{μν}$ and $\sqrt{-g}R^{2}$. In this paper, we compute the one-loop renormalization of the theory and the absorptive part of the graviton self energy. The results illustrate the mechanism that makes renormalizability compatible with unitarity. The fakeons disentangle the real part of the self ene…
The origin of large-scale magnetic field in the universe is one of the greatest mysteries in modern cosmology. We present a new mechanism for generation of large-scale magnetic field, based on the power-counting renormalizable theory of gravitation recently proposed by Horava. Contrary to the usual case in general relativity, the U(1) gauge symmetry of a Maxwell action in this theory permits terms breaking conformal invariance in the ultraviolet. Moreover, for high frequency modes, the anisotropic scaling intrinsic to the theory inevitably makes the sound horizon far outside the Hubble horizon…
In this thesis we investigate quantum aspects of the Green-Schwarz superstring in various AdS backgrounds relevant for the AdS/CFT correspondence, providing several examples of perturbative computations in the corresponding integrable sigma-models. We start by reviewing in details the supercoset construction of the superstring action in $AdS_5 \times S^5$, pointing out the limits of this procedure for $AdS_4$ and $AdS_3$ backgrounds. For the $AdS_4 \times CP^3$ case we give a thorough derivation of an alternative action, based on the double-dimensional reduction of eleven-dimensional super-mem…
We derive the N=1 supersymmetric extension for a class of weakly nonlocal four dimensional gravitational theories.The construction is explicitly done in the superspace and the tree-level perturbative unitarity is explicitly proved both in the superfield formalism and in field components. For the minimal nonlocal supergravity the spectrum is the same as in the local theory and in particular it is ghost-free. The supersymmetric extension of the super-renormalizable Starobinsky theory and of two alternative massive nonlocal supergravities are found as straightforward applications of the formalism…
