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Understanding the role of higher derivatives is probably one of the most relevant questions in quantum gravity theory. Already at the semiclassical level, when gravity is a classical background for quantum matter fields, the action of gravity should include fourth derivative terms to provide renormalizability in the vacuum sector. The same situation holds in the quantum theory of metric. At the same time, including the fourth derivative terms means the presence of massive ghosts, which are gauge-independent massive states with negative kinetic energy. At both classical and quantum level such ghosts violate stability and hence the theory becomes inconsistent. Several approaches to solve this contradiction were invented and we are proposing one more, which looks simpler than those what were considered before. We explore the dynamics of the gravitational waves on the background of classical solutions and give certain arguments that massive ghosts produce instability only when they are present as physical particles. At least on the cosmological background one can observe that if the initial frequency of the metric perturbations is much smaller than the mass of the ghost, no instabilities are present.

Recently, a one-parameter extension of the covariant Heisenberg algebra with the extension parameter $l$$(l$ is a non-negative constant parameter which has a dimension of ${[momentum]}^{-1})$ in a $(D+1)$-dimensional globally flat spacetime has been presented which is a covariant generalization of the Kempf–Mangano algebra [see G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior and V. Nikoofard, Adv. High Energy Phys. 2017, 4768341 (2017) and A. Kempf and G. Mangano, Phys. Rev. D 55, 7909 (1997)]. The Abelian Proca model is reformulated from the viewpoint of the above one-parameter extension of the covariant Heisenberg algebra. It is shown that the free space solutions of the above modified Proca model describe two massive vector particles with different effective masses ${{ℳ}}_{\pm }(\mathrm{\Lambda })$ where $\mathrm{\Lambda }=\hslash l$ is the characteristic length scale in our model. In addition, the Feynman propagator in momentum space for the modified Abelian Proca model is calculated analytically. Our numerical estimations show that the maximum value of $\mathrm{\Lambda }$ in a four-dimensional spacetime is near the electroweak length scale, i.e. ${\mathrm{\Lambda }}_{max}\sim l_{\mathrm{\text{electroweak}}}\sim 10^{-18}\mathrm{\text{m}}$. We show that in the infrared/large-distance domain, the modified Proca model behaves like an Abelian massive Lee–Wick model which has been presented by Accioly and his co-workers in A. Accioly, J. Helayel-Neto, G. Correia, G. Brito, J. de Almeida and W. Herdy, Phys. Rev. D 93, 105042 (2016). The short-distance behavior of the modified Proca model is studied in the massless limit and the explicit forms of the inhomogeneous infinite derivative Maxwell equation and the infinite derivative Poisson equation are obtained. Finally, note that in the low-energy limit $(\mathrm{\Lambda }\to 0)$, the results of this paper are compatible with the results of the usual Proca model.

A higher-derivative classical nonrelativistic U(1) × U(1) gauge field model that describes the topologically massive electromagnetic interaction of composite particles in 2+1 dimensions is proposed. This is made by adding a suitable higher-derivative term for the electromagnetic field to the Lagrangian of a model previously proposed. The model contains a Chern–Simons U(1) field and the topologically massive electromagnetic U(1) field, and it uses either a composite boson system or a composite fermion one. The second case is explicitly considered. By following the usual Hamiltonian method for singular higher-derivative systems, the canonical quantization is carried out. By extending the Faddeev–Senjanovic formalism, the path integral quantization is developed. Consequently, the Feynman rules are established and the diagrammatic structure is discussed. The use of the higher-derivative term eliminates in the Landau gauge the ultraviolet divergence of the primitively divergent Feynman diagrams where the electromagnetic field propagator is present. The unitarity problem, related to the possible appearance of states with negative norm, is treated. A generalization of the Becchi–Rouet-Stora–Tyutin algorithm is applied to the model.

A brief review of the physics of systems including higher derivatives in the Lagrangian is given. All such systems involve ghosts, i.e. the spectrum of the Hamiltonian is not bounded from below and the vacuum ground state is absent. Usually, this leads to collapse and loss of unitarity. In certain special cases, this does not happen, however, ghosts are benign.

We speculate that the Theory of Everything is a higher-derivative field theory, characterized by the presence of such benign ghosts and defined in a higher-dimensional bulk. Our Universe then represents a classical solution in this theory, having the form of a 3-brane embedded in the bulk.

Starting from the classical nonrelativistic electrodynamics in 1+1 dimensions, a higher-derivative version is proposed. This is made by adding a suitable higher-derivative term for the electromagnetic field to the Lagrangian of the original electrodynamics, preserving its gauge invariance. By following the usual Hamiltonian method for singular higher-derivative systems, the canonical quantization for the higher-derivative model is developed. By extending the Faddeev–Senjanovic algorithm, the path integral quantization is carried out. Hence, the Feynman rules are established and the diagrammatic structure is analyzed. The use of the higher-derivative term eliminates in the Landau gauge the ultraviolet divergence of the primitively divergent Feynman diagrams of the original model, where the electromagnetic field propagator is present. A generalization of the BRST quantization is also considered.

We perform the Faddeev–Jackiw (FJ) canonical quantization for the Podolsky electrodynamics. To this end, we use an extension of the usual FJ formalism for constrained systems with Grassmann dynamical field variables, proposed by us some time ago. Besides, we compare the obtained results with those corresponding to the implementation of the Dirac formalism to this issue. In this way, we see that the extended FJ and the Dirac formalisms provide the same constraints and generalized brackets, thus suggesting the equivalence between these formalisms, at least for the present case. Furthermore, we find that the extended FJ formalism is more economical than the Dirac one as regards the calculation of both the constraints and the generalized brackets. On the other hand, we also compare the mentioned obtained results with the ones corresponding to the analysis of the issue in study by means of the usual FJ formalism, showing that between the extended and the usual FJ formalisms there are significant differences.

We present a canonical formulation for higher-curvature theories of gravity whose action is a generic function of Riemann tensor. The Arnowitt–Deser–Misner canonical formalism is employed to identify the extra gravitational dynamical degrees of freedom other than metric. We also find a surface term that gives Dirichlet boundary conditions for the dynamical degrees of freedom.

On the basis of the path-integral formulation of the Yang-Mills theory a gauge invariant infrared regularization is introduced. The regularized model includes higher derivatives, but in the limit when the regularization is removed, unphysical excitations decouple.