Programme

Abstract. We present a scenario in which time and Lorentzian signature emerge from a Riemannian theory through a nontrivial scalar-field configuration. At short distances, the system is described by a renormalizable scalar-tensor theory of gravity with a positive-definite metric. At long distances, under certain conditions, it contains a massless spin-2 mode with a Lorentzian dispersion relation. This construction may evade the ghost problem of quadratic gravity. We also briefly discuss applications and the inclusion of the Standard Model.

Abstract. The presence of higher derivatives in the gravitational action is required for renormalizability in semiclassical and quantum gravity. On the other hand, higher derivatives produce unphysical massive ghosts, making classical solutions unstable and quantum theory non-unitary. We discuss different aspects of this problem. In particular, trying to resolve the contradiction, we explore different possibilities related to the natural cut-off on the energy density of gravitons. If this density is far below the Planck order of magnitude, the ghost may be present in the particle spectrum of the theory, but it cannot be generated from the vacuum. The most promising scheme is related to superrenormalizable versions of quantum gravity. In this case, there may be a pair of complex conjugate massive poles and it is possible that the complex ghost-like states form normal bound states. For a while, this mechanism was described only for a simplified toy model that reproduces the general structure of the ghost-like poles. This model is far from the real quantum gravity. However, assuming that the same situation takes place for quantum gravity, such a mechanism may resolve the problem of ghosts.

Abstract. We outline the history of the research of dynamic systems including higher derivatives in the Lagrangian, The story starts in 1744 when Leonhard Euler published a book where the Euler-Lagrange differential equations for a HD system were first written. The next important landmarks were the papers of Ostrogradsky (1850) and of Pais and Uhlenbeck (1950). Many articles written in the second half of the last century were influenced by a misconception that was widespread at the time - it was believed that higher derivatives necessarily bring about a violation of unitarity. We now understand that this is not the case and there are many systems with benign ghosts where HD are present but unitarity is not violated. I will briefly discuss such (quantum) mechanical systems and 2D field theories. Then I present a rather interesting supersymmetric HD gauge theory in 6 dimension. It is rather beautiful: in particular, it is asymptotically free. Unfortunately, it is not unitary - ghosts are not benign there. Finally, I will discuss higher-derivative gravity theories in 4 dimensions. In that case, the question - whether the ghosts there are benign or not - has a special status due to the absence of a nonperturbative Hamiltonian and does not have a definite answer yet.

Abstract. We present the connection between higher-order, infinite-order, and genuinely nonlocal Lagrangian systems. Starting from finite higher-derivative theories, we discuss the formal transition to infinite order and show how this motivates, but does not fully capture, genuine nonlocality. The latter is naturally formulated through functionals of the trajectory rather than infinite derivative expansions. Our presentation follows the framework developed in *Nonlocal Mechanics* by Carlos Heredia and Josep Llosa.