These lecture notes (arXiv:2307.08460) cover materials exposed by Luca Ciambelli at the 59 Winter School of Theoretical Physics and third COST Action CA18108 Training School “Gravity — Classical, Quantum and Phenomenology”, held in Palac Wojanów, Poland, 12-21 Feb 2023.
After introducing the covariant phase space calculus, Noether’s theorems are discussed, with particular emphasis on Noether’s second theorem and the role of gauge symmetries. This is followed by the enunciation of the theory of asymptotic symmetries, and later its application to gravity. Specifically, we review how the BMS group arises as the asymptotic symmetry group of gravity at null infinity. Symmetries are so powerful and constraining that memory effects and soft theorems can be derived from them. The lectures end with more recent developments in the field: the corner proposal as a unified paradigm for symmetries in gravity, the extended phase space as a resolution to the problem of charge integrability, and eventually the implications of the corner proposal on quantum gravity.

In the past decade, significant efforts have been devoted to the study of Relative Locality, which aims to generalize the kinematics of relativistic particles to a nonlocal framework by introducing a nontrivial geometry for momentum space. This paper (arXiv:2306.11451) builds upon a recent proposal to extend the theory to curved spacetimes and investigates the behavior of horizons in certain spacetimes with this nonlocality framework. Specifically, we examine whether nonlocality effects weaken or destroy the notion of horizon in these spacetimes. Our analysis indicates that, in the chosen models, the nonlocality effects do not disrupt the notion of horizon and that it remains as robust as it is in General Relativity.

Recent studies have demonstrated the possibility to uphold classical determinism within gravitational singularities, showcasing the ability to uniquely extend Einstein’s equations across the singularity in certain symmetry-reduced models. This extension can be achieved by allowing the orientation of spatial hypersurfaces to dynamically change. Furthermore, a crucial aspect of the analysis revolves around the formulation of the dynamical equations in terms of physical degrees of freedom, demonstrating their regularity at the singularity. Remarkably, singular behavior is found to be confined solely to the gauge/unphysical degrees of freedom. This paper (arXiv:2306.02941) extends these results to gravity coupled with Abelian and non-Abelian gauge fields in a symmetry-reduced model (homogeneous anisotropic universe). Near the Big Bang, the dynamics of the geometry and the gauge fields is reformulated in a way that shows that determinism is preserved, assuming a change in orientation at the singularity. The gauge fields are demonstrated to maintain their orientation throughout the singularity, indicating that the predicted orientation change of spatial hypersurfaces holds physical significance. This observation suggests that an observer can discern the specific side of the Big Bang they inhabit.

In this paper (arXiv:2304.08843), by using the theory of Lie-Hamilton systems, formal generalized stochastic Hamiltonian systems that enlarge a recently proposed stochastic SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamitonian models based on the book and oscillator algebras, denoted respectively by 𝔟2 and 𝔥4. The last generalization corresponds to a SIS system possessing the so-called two-photon algebra symmetry 𝔥6, according to the embedding chain 𝔟2⊂𝔥4⊂𝔥6, for which an exact solution cannot generally be found, but a nonlinear superposition rule is explicitly given.

In this paper (arXiv:2303.08220), we will discuss the notion of curved momentum space, as it arises in the discussion of noncommutative or doubly special relativity theories. We will illustrate it with two simple examples, the Casimir effect in anti-Snyder space and the introduction of fermions in doubly special relativity. We will point out the existence of intriguing results, which suggest nontrivial connections with spectral geometry and Hopf algebras.

Doubly special relativity considers a deformation of the special relativistic kinematics parametrized by a high-energy scale, in such a way that it preserves a relativity principle. When this deformation is assumed to be applied to any interaction between particles, one faces some inconsistencies. In order to avoid them, in this paper we propose a new perspective where the deformation affects only the interactions between elementary particles. A consequence of this proposal is that the deformation cannot modify the special relativistic energy–momentum relation of a particle.

In the last few years, much attention has been paid to the exotic properties that graphene nanostructures exhibit, especially those emerging upon deforming the material. In this new paper we present a study of the mechanical and electronic properties of bent hexagonal graphene quantum dots employing density functional theory. We explore three different kinds of surfaces with Gaussian curvature exhibiting different shapes—spherical, cylindrical, and one-sheet hyperboloid—used to bend the material, and several boundary conditions regarding what atoms are forced to lay on the chosen surface. In each case, we study the curvature energy and two quantum regeneration times (classic and revival) for different values of the curvature radius. A strong correlation between Gaussian curvature and these regeneration times is found, and a special divergence is observed for the revival time for the hyperboloid case, probably related to the pseudo-magnetic field generated by this curvature being capable of causing a phase transition.

In this paper (arXiv:2212.13575) we introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions, defined as a λ−deformation of the N-dimensional Dunkl oscillator. This deformation can be interpreted either as the introduction of a non-constant curvature related to λ on the underlying space or, equivalently, as a Dunkl oscillator with a position-dependent mass function. This new quantum model is shown to be exactly solvable in arbitrary dimension N, and its eigenvalues and eigenfunctions are explicitly presented. Moreover, it is shown that in the two-dimensional case both the Darboux III and the Dunkl oscillators can be separately coupled with a constant magnetic field, thus giving rise to two new exactly solvable quantum systems in which the effect of a position-dependent mass and the Dunkl derivatives on the structure of the Landau levels can be explicitly studied. Finally, the whole 2D Dunkl-Darboux III oscillator is coupled with the magnetic field and shown to define an exactly solvable Hamiltonian, where the interplay between the λ-deformation and the magnetic field is explicitly illustrated.