The equivalence principle (EP) and Schiff’s conjecture are discussed en passant, and the connection between the EP and quantum mechanics is then briefly analyzed. Two semiclassical violations of the classical equivalence principle (CEP) but not of the weak one (WEP), i.e., Greenberger gravitational Bohr atom and the tree-level scattering of different quantum particles by an external weak higher-order gravitational field, are thoroughly investigated afterwards. Next, two quantum examples of systems that agree with the WEP but not with the CEP, namely, COW experiment and free fall in a constant gravitational field of a massive object described by its wave-function Ψ, are discussed in detail. Keeping in mind that, among the four examples focused on in this work only COW experiment is based on an experimental test, some important details related to it are presented as well.

In this work we investigate an interesting connection between the absence of Newtonian singularities in the classical nonrelativistic potential and renormalizability properties in higher-derivative models of quantum gravity. In the framework of a large class of D-dimensional higher-derivative models of quantum gravity, we compute the nonrelativistic potential energy associated with two pointlike masses. Investigating its behavior for small distances, we find an algebraic condition which is sufficient for the cancellation of the Newtonian singularity. We verify that the same condition is necessary to ensure power-counting renormalizability and, as a consequence, we conclude that renormalizable higher-derivative models do not exhibit the so-called Newtonian singularity. Finally, we discuss the role of ghosts in the mechanism for the cancellation of Newtonian singularities.

Local gravitational theories with more than four derivatives are superrenormalizable. They also may be unitary in the Lee–Wick sense. Thus it is relevant to study the low-energy properties of these theories, especially to identify observables which might be useful for experimental detection of higher derivatives. Using an analogy with the neutrino physics, we explore the possibility of a gravitational seesaw mechanism in which several dimensional parameters of the same order of magnitude produce a hierarchy in the masses of propagating particles. Such a mechanism could make a relatively light degree of freedom detectable in low-energy laboratory and astrophysical observations, such as torsion-balance experiments and the bending of light. We demonstrate that such a seesaw mechanism in the six- and more-derivative theories is unable to reduce the lightest mass more than in the simplest four-derivative model. Adding more derivatives to the four-derivative action of gravity makes heavier masses even greater, while the lightest massive ghost is not strongly affected. This fact is favorable for protecting the theory from instabilities but makes the experimental detection of higher derivatives more difficult.

As is well known, the deflection of any particle by a gravitational field within the context of Einstein's general relativity - which is a geometrical theory - is, of course, nondispersive. Nevertheless, as we shall show in this paper, the mentioned result will change totally if the bending is analyzed - at the tree level - in the framework of higher-order gravity. Indeed, to first order, the deflection angle corresponding to the scattering of different quantum particles by the gravitational field mentioned above is not only spin dependent, it is also dispersive (energy-dependent). Consequently, it violates the classical equivalence principle (universality of free fall, or equality of inertial and gravitational masses) which is a nonlocal principle. However, contrary to popular belief, it is in agreement with the weak equivalence principle which is nothing but a statement about purely local effects. It is worthy of note that the weak equivalence principle encompasses the classical equivalence principle locally. We also show that the claim that there exists an incompatibility between quantum mechanics and the weak equivalence principle, is incorrect.

A simple expression for calculating the classical potential concerning $D$-dimensional gravitational models is obtained through a method based on the generating functional. The prescription is then used as a mathematical tool to probe the conjecture that renormalizable higher-order gravity models --- which are, of course, nonunitary --- are endowed with a classical potential that is nonsingular at the origin. It is also shown that the converse of this statement is not true, which implies that the finiteness of the classical potential at the origin is a necessary but not a sufficient condition for the renormalizability of the model. The systems we have utilized to verify the conjecture were fourth- and sixth- order gravity models in $D$-dimensions. A discussion about the polemic question related to the renormalizability of new massive gravity, which Oda claimed to be renormalizable in 2009 and three years late was shown to be nonrenormalizable by Muneyuki and Ohta, is considered. We remark that the solution of this issue is straightforward if the aforementioned conjecture is employed. We point out that our analysis is restricted to local models in which the propagator has simple and real poles.

The most simple superrenormalizable model of quantum gravity is based on the general local covariant six-derivative action. In addition to graviton such a theory has massive scalar and tensor modes. It was shown recently that in the case when the massive poles emerge in complex conjugate pairs, the theory has also unitary $S$-matrix and hence can be seen as a candidate to be a consistent quantum gravity theory. In the present work we construct the modified Newton potential and explore the gravitational light bending in a general six-derivative theory, including the most interesting case of complex massive poles. In the case of the light deflection the results are obtained within classical and semiclassical approaches.

Simple prescriptions for computing the D-dimensional classical potential related to electromagnetic and gravitational models, based on the functional generator, are built out. These recipes are employed afterward as a support for probing the premise that renormalizable higher-order systems have a finite classical potential at the origin. It is also shown that the opposite of the conjecture above is not true. In other words, if a higher-order model is renormalizable, it is necessarily endowed with a finite classical potential at the origin, but the reverse of this statement is untrue. The systems used to check the conjecture were D-dimensional fourth-order Lee–Wick electrodynamics, and the D-dimensional fourth- and sixth-order gravity models. A special attention is devoted to New Massive Gravity (NMG) since it was the analysis of this model that inspired our surmise. In particular, we made use of our premise to resolve trivially the issue of the renormalizability of NMG, which was initially considered to be renormalizable, but it was shown some years later to be non-renormalizable. We remark that our analysis is restricted to local models in which the propagator has simple and real poles.

Local gravitational theories with more than four derivatives have remarkable quantum properties, e.g., they are super-renormalizable and may be unitary in the Lee-Wick sense. Therefore, it is important to explore also the IR limit of these theories and identify observable signatures of the higher derivatives. In the present work we study the scattering of a photon by a classical external gravitational field in the sixth-derivative model whose propagator contains only real, simple poles. Also, we discuss the possibility of a gravitational seesaw-like mechanism, which could allow the makeup of a relatively small physical mass from the huge massive parameters of the action. If possible, this mechanism would be a way out of the Planck suppression, affecting the gravitational deflection of low energy photons. It turnsout that the mechanism which actually occurs works only to shift heavier masses to the further UV region. This fact may be favourable for protecting the theory from instabilities, but makes experimental detection of higher derivatives more difficult.

Using a method based on the generating functional plus a kind of "correspondence principle"-which acts as a bridge between the electromagnetic and scalar fields-it is shown that the interparticle potential energy concerning a given D-dimensional electromagnetic model can be obtained in a simple way from that related to the corresponding scalar system. The D-dimensional electromagnetic potential for a general model containing higher derivatives is then found from the corresponding scalar one and the behavior of the former is analyzed at large as well as small distances. In addition, we investigate the presence of ghosts in the four-dimensional version of the potential associated with the model above and analyze the reason why the Coulomb singularity is absent from this system. The no-go theorem by Ostrogradski is demystified as well.

The effect of Lorentz symmetry violation in the phenomenon of photon gravitational bending, is investigated. Using a semiclassical approach, where the photon is described by the Carrol-Field-Jackiw (CFJ) electrodynamics which is responsible for implementing the Lorentz symmetry violation, the gravitational deflection angle related to the CFJ photon is computed. As expected, this bending angle experiences a deviation from the usual Einstein result and the latter is recovered in the appropriate limit. A comparison between the theoretical prediction and the experimental results allows to conclude that no trace of Lorentz symmetry breaking is found provided the components of the background vector field are $\lesssim 10^{-8}$ eV.

The deflection of any particle by an external weak gravitational field in the framework of Einstein's general relativity (which is a geometrical theory) is, of course, non-dispersive. Nonetheless, things are completely different if the deflection is studied within the context of semiclassical gravity (or tree-level gravity). Indeed, to first order the cross sections corresponding to the scattering of different quantum particles by a weak gravity field, treated as an external field, as well as the related bending angles, are spin dependent, while to second order the deflection is dispersive (energy dependent). Interestingly enough, in the framework of semiclassical higher-derivative gravity the bending is already energy dependent to first order. In this paper we discuss these disagreements between quantum mechanics and the equivalence principle.

A straightforward prescription for computing the D-dimensional potential energy of gravitational models, which is strongly based on the Feynman path integral, is built up. Using this method, the static potential energy for the interaction of two masses is found in the context of D-dimensional higher-derivative gravity models, and its behavior is analyzed afterwards in both ultraviolet and infrared regimes. As a consequence, two new gravity systems in which the potential energy is finite at the origin, respectively, in D = 5 and D = 6, are found. Since the aforementioned prescription is equivalent to that based on the marriage between quantum mechanics (to leading order, i.e., in the first Born approximation) and the nonrelativistic limit of quantum field theory, and bearing in mind that the latter relies basically on the calculation of the nonrelativistic Feynman amplitude (M-NR), a trivial expression for computing M-NR is obtained from our prescription as an added bonus.

Among the so-called classical tests of general relativity (GR), light bending has been confirmed with an accuracy that increases as times goes by. Here we study the gravitational deflection of photons within the framework of classical and semiclassical higher-derivative gravity (HDG)-the only version of GR that is known up to now to be renormalizable along with its matter couplings. Since our computations are restricted to scales much below the Planck cutoff we need not be afraid of the massive spin-2 ghost that haunts HDG. An upper bound on the constant related to the R-mu nu(2). sector of the theory is then found by analyzing-from the classical and semiclassical viewpoints-the deflection angle of a photon passing by the Sun. This upper limit greatly improves that available in the literature.

The scattering of a photon by a weak external gravitational field which is solution of the linearized higher-derivative gravity equations sourced by a point-like massive particle located at the origin of the coordinate system, is analyzed. It is shown that the [Formula: see text]-sector of the theory produces dispersive photon propagation. Subsequently, the angle |Δθ|(≡|θ violet -θ red |) at which the visible spectrum would be spread over in the case of a photon passing by the Sun is plotted as a function of the |β|-constant related to the [Formula: see text]-sector. An upper bound on |β| is then found. Interestingly enough, this limit is thirteen orders of magnitude below the accepted upper bound on |β|.

A simple expression for calculating the interparticle potential energy concerning D-dimensional electromagnetic models is obtained via Feynman path integral. This prescription converts the hard task of computing this potential into a trivial algebraic exercise. Since this method is equivalent to that based on the merging of quantum mechanics (to leading order, i.e., in the first Born approximation) with the nonrelativistic limit of quantum field theory, and keeping in mind that the latter relies basically on the computation of the nonrelativistic Feynman amplitude (M-NR), a trivial expression for calculating M-NR is obtained from the alluded prescription as an added bonus. To test the efficacy and simplicity of the method, D-dimensional interparticle potential energy is found for a well-known extension of the standard model in which the massless electrodynamics U(1)(QED) is coupled to a hidden sector U(1)(h), as well as Lee-Wick electrodynamics.

One of the puzzling aspects of N-dimensional Einstein Gravity (NDEG) augmented by curvature-squared terms is why renormalizability and unitarity, two of the most important properties of any physical theory, cannot be reconciled in its framework. Actually, the reason why these properties are mutually incompatible within the context of generic higher-derivative models, not necessarily related to gravity, is one of the unsolved mysteries of physics. Here, a simple solution to the NDEG riddle, based on the analysis of the interparticle gravitational potential, is presented. The main argument used to support our discussion is that tree-level unitarity and the existence of a singularity in the potential are intertwined.

A new set of projection operators for three-dimensional models are constructed. Using these operators, an uncomplicated and easy handling algorithm for analyzing the unitarity of the aforementioned systems is built up. Interestingly enough, this method converts the task of probing the unitarity of a given three-dimensional system, which is in general a time-consuming work, into a straightforward algebraic exercise; besides, it also greatly clarifies the physical interpretation of the propagating modes. To test the efficacy and quickness of the algorithm at hand, the unitarity of some important and timely higher-order electromagnetic (gravitational) systems augmented by both Chern-Simons and higher order Chern-Simons terms are investigated.

We consider the Lee–Wick (LW) electrodynamics, i.e. the U(1) gauge theory where a (gauge-invariant) dimension-6 operator containing higher derivatives is added to the free Lagrangian of the U(1) sector. A quantum bound on the LW heavy particle mass is then estimated by computing the anomalous electron–magnetic moment in the context of the aforementioned model. This limit is not only within the allowed range estimated by LW, it is also of the same order as that considered in early investigations on the possible effects of the LW heavy particle in e-e+ elastic scattering. A comparative study between the LW and the Coulomb potentials is also done.

Some interesting gravitational properties of the Bergshoeff-Hohm-Townsend model (massive 3D gravity), such as the presence of a short-range gravitational force in the nonrelativistic limit and the existence of an impact-parameter-dependent gravitational deflection angle, are studied. Interestingly enough, these phenomena have no counterpart in the usual Einstein 3D gravity. In order to better understand the two aforementioned gravitational properties, they are also analyzed in the framework of 3D higher-derivative gravity with the Einstein-Hilbert term with the "wrong sign."