Advances in High Energy Physics

Volume 2017 (2017), Article ID 4768341, 8 pages

https://doi.org/10.1155/2017/4768341

## Effective Models of Quantum Gravity Induced by Planck Scale Modifications in the Covariant Quantum Algebra

^{1}Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Dr. Xavier Sigaud 150, Urca, 22290-180 Rio de Janeiro, RJ, Brazil^{2}Instituto Federal do Espírito Santo (IFES), Av. Vitória 1729, Jucutuquara, 29040-780 Vitória, ES, Brazil^{3}Departamento de Matemática, Física e Computação, Faculdade de Tecnologia, Universidade do Estado do Rio de Janeiro, Rodovia Presidente Dutra, Km 298, Polo Industrial, 27537-000 Resende, RJ, Brazil

Correspondence should be addressed to G. P. de Brito; moc.liamg@inizzapovatsug

Received 26 May 2017; Accepted 13 August 2017; Published 26 September 2017

Academic Editor: Angel Ballesteros

Copyright © 2017 G. P. de Brito et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We introduce modified covariant quantum algebra based on the so-called Quesne-Tkachuk algebra. By means of a deformation procedure we arrive at a class of higher-derivative models of gravity. The study of the particle spectra of these models reveals equivalence with the physical content of the well-known higher-derivative gravities. The particle spectrum exhibits the presence of spurious complex ghosts and, in light of this problem, we suggest an interesting interpretation in the context of minimal length theories. Also, a discussion regarding the nonrelativistic potential energy is proposed.

#### 1. Introduction

The construction of a quantum theory for gravity consists in one of the most challenging problems of theoretical physics, since there are no experimental hints about what kind of effects we should expect from quantum corrections to the gravitational interaction. However, even without experimental evidences, a great deal of theoretical effort has been done in the last decades. In this vein, several approaches have been proposed as quantum theories of gravity, for instance, string theories, loop quantum gravity, causal dynamical triangulations, causal sets, and induced quantum gravity [1, 2]. Nevertheless, in the present state of art, none of the preceding theories give the final word in quantum gravity.

Rather than considering a full quantum theory of gravity, one can deal with effective theories of quantum gravity, which are promising approaches to implement quantum corrections to GR [3]. However, serious problems must be faced in this approach, mainly the incompatibility between renormalizability and unitarity, which are desired features in the quantum field theory formulation. On one hand, effective theories based on GR are nonrenormalizable by power counting, since the coupling constant associated with gravitation has inverse canonical mass dimension. The issue with renormalizability may be solved by introducing higher-derivative terms into the action, because in such theories the UV behavior of the propagator is more convergent than in the GR case [4–10]. On the other hand, unitarity is usually lost in higher-derivative theories, since it leads to unphysical massive ghosts. The riddle of the incompatibility between renormalizability and unitarity was recently explored in [11], also relating the aforementioned features with the behavior of the nonrelativistic potential energy associated with the gravitational field.

As a consequence of the association between the Planck length and the coupling constant of the gravitational interaction, the Planck scale is the natural regime in which we expect that quantum effects become most relevant to the gravitational interaction. Notwithstanding, gravity does not allow an arbitrarily amount of mass/energy in a very small region of spacetime, since it would collapse into a black hole [12]. This suggests a minimal length hypothesis that is the existence of a fundamental length scale below which we cannot access. The minimal length hypothesis may be achieved by modifications in the quantum algebra of position coordinates and conjugated* momenta*, which was first accomplished in a seminal paper by Snyder in 1947 [13].

The interest on minimal length physics has increased considerably in the last two decades, specially due to certain important results within the context of string theories, loop quantum gravity, and asymptotic safe gravity [14–17]. In special, we mention the noncovariant Kempf algebra [18] which has been vastly explored in the last two decades [19–24]. Its covariant generalization was later introduced by Quesne and Tkachuk [25, 26].

Remarkably the Quesne-Tkachuk algebra leads to a systematic procedure to generate fourth-derivative models; for instance, in the case spin-0 particles, minimal length corrections were implemented in the Klein-Gordon field, leading to a higher-derivative theory for the scalar field [27]; for spin-1/2 fields, the deformation procedure was applied in order to construct a higher-derivative version of the Dirac field [28]; some investigations were also performed in the context of electrostatic [29], magnetostatic [30], electrodynamics with external sources [31], and quantum electrodynamics [32]; finally, in a recent paper, Dias et al. explored the issue of minimal length corrections in the context of Einstein-Hilbert theory [33]. Remarkably, it was recently suggested that this algebra may be viewed as an emergent effect of a supersymmetry breaking of a nonanticommutative superspace [34].

Realizing that there is a clear connection between minimal length deformations and fourth-derivative models, it is natural to think about this kind of deformation as a road to implement quantum corrections in the gravitational interaction. In the last few years there has been an increasing interest in modified theories of gravity including sixth or more derivative terms [9]. In fact, this class of theories is superrenormalizable or even finite at quantum level. Nevertheless, these theories cannot be obtained by means of a deformation procedure in the context of the Quesne-Tkachuk algebra. In this paper, we propose a modification on the Quesne-Tkachuk algebra by introducing higher-order corrections in the deformation parameter in order to include a larger class of higher-derivative effective theories of quantum gravity.

This paper is organized as follows: in Section 2 we introduce and explore some features of the modified Quesne-Tkachuk algebra; in Section 3, we establish a connection between the deformed version of the Einstein-Hilbert Lagrangian and the effective theories of quantum gravity; in Section 4, we explore some features regarding the particle spectra related to the class of effective theories obtained from the minimal length deformation of the GR Lagrangian; in Section 5 we study some low-energy consequences of the deformed gravitational theory; finally in Section 6 we present our conclusions.

Throughout this paper we use , , , , and .

#### 2. Modified Quesne-Tkachuk Algebra

The Quesne-Tkachuk algebra is the simplest possible covariant generalization of the Heisenberg algebra that allows for a minimal length [25]. We may wonder if the Quesne-Tkachuk algebra is just the first-order truncation of a more general covariant algebra. In fact, there are some proposal of higher-order algebras in the literature, for instance, [35, 36]. A commutative spacetime regime of such algebra would also be of special interest. In order to be able to build up a new algebra, we propose the following representations:where the lower-case position and momentum operators satisfy the usual Heisenberg algebra and . For the Heisenberg algebra representations to be recovered as the low-energy limit of the (1), the deformation must satisfy the condition as .

From the considerations above it is possible to set up an algebra. The position and momentum commutators among themselves remain trivially null. It is necessary only to compute the position-momentum commutatorwhere the prime denotes differentiation with respect to . To close algebra this must be expressed fully in terms of the new momentum operator . Thus, the second contribution leads to the differential equationWithout the form of the function we cannot give an explicit solution. We study the simplest possibility where is actually a constant , in which case the solution to (3) is straightforwardInverting the last equation we may express in terms of and, as a consequence, using the result in (2) we arrive at the following algebra:with the representations

The new momentum operator representation in (6), along with the correspondence principle , leads to a deformation of the derivative operator in configuration space

Notice that this deformation is ill-defined only for space-like momenta with . Thus no problem concerning the analyticity of the deformation procedure arises if we exclude the possibility of tachyons.

We emphasize that no approximations were made in the derivation of the algebra (5a), (5b) and representations (6). Only a simplicity assumption and the consistency condition as were needed. In this framework the algebra (5a) and (5b) is exact and holds for arbitrarily high energy. On the other hand interesting physics arises in the low-energy limit, where we can power expand the r.h.s. of (5a) and (6) in the parameter . Each truncation of this series defines different algebras summarized in Table 1.