These notes (arXiv:2311.01911) cover part of the lectures presented by Andrea Maselli for the 59th Winter School of Theoretical Physics and third COST Action CA18108 Training School ‘Gravity — Classical, Quantum and Phenomenology’. The school took place at Palac Wojanów, Poland, from February 12th to 21st, 2023. The lectures focused on some key aspects of black hole physics, and in particular on the dynamics of particles and on the scattering of waves in the Schwarzschild spacetime. The goal of the course was to introduce the students to the concept of black hole quasi normal modes, to discuss their properties, their connection with the geodesic motion of massless particles, and to provide numerical approaches to compute their actual values.

In this paper (arXiv:2406.17479) a new procedure for the construction of higher-dimensional Lie-Hamilton systems is proposed. This method is based on techniques belonging to the representation theory of Lie algebras and their realization by vector fields. The notion of intrinsic Lie-Hamilton system is defined, and a sufficiency criterion for this property given. Novel four-dimensional Lie-Hamilton systems arising from the fundamental representation of the symplectic Lie algebra 𝔰𝔭(4,ℝ) are obtained and proved to be intrinsic. Two distinguished subalgebras, the two-photon Lie algebra 𝔥6 and the Lorentz Lie algebra 𝔰𝔬(1,3), are also considered in detail. As applications, coupled time-dependent systems which generalize the Bateman oscillator and the one-dimensional Caldirola-Kanai models are constructed, as well as systems depending on a time-dependent electromagnetic field and generalized coupled oscillators. A superposition rule for these systems, exhibiting interesting symmetry properties, is obtained using the coalgebra method.

This paper (arXiv:2404.08729) is the second part of a series that develops the mathematical framework necessary for studying field theories on “T-Minkowski” noncommutative spacetimes. These spacetimes constitute a class of noncommutative geometries, introduced in Part I, that are each invariant under distinct quantum group deformations of the Poincaré group. All these noncommutative geometries possess certain physically desirable characteristics, which allow me to develop all the tools of differential geometry and functional analysis, that are necessary in order to build consistent and T-Poincaré invariant noncommutative classical field theories.

In this paper (arXiv:2404.01073), integrable deformations of a class of Rikitake dynamical systems are constructed by deforming their underlying Lie-Poisson Hamiltonian structures as linearizations of Poisson–Lie structures on certain (dual) Lie groups. By taking into account that there exists a one-to one correspondence between Poisson–Lie groups and Lie bialgebra structures, a number of deformed Poisson coalgebras can be obtained, which allow the construction of integrable deformations of coupled Rikitake systems. Moreover, the integrals of the motion for coupled systems can be explicitly obtained by means of the deformed coproduct map. The same procedure can be also applied when the initial system is bi-Hamiltonian with respect to two different Lie-Poisson algebras. In this case, to preserve a bi-Hamiltonian structure under deformation, a common Lie bialgebra structure for the two Lie-Poisson structures has to be found. Coupled dynamical systems arising from this bi-Hamiltonian deformation scheme are also presented, and the use of collective `cluster variables’, turns out to be enlightening in order to analyse their dynamical behaviour. As a general feature, the approach here presented provides a novel connection between Lie bialgebras and integrable dynamical systems.

In this work (arXiv:2403.19520), we present the technical details of the discussion presented in [J.J. Relancio, L.Santamaría-Sanz (2024) arXiv:2403.18772], where we establish the basis of quantum theories of the free massive scalar, the massive fermionic, and the electromagnetic fields, in a doubly special relativity scenario. This construction is based on a geometrical interpretation of the kinematics of these kind of theories. In order to describe the modified actions, we find that a higher (indeed infinite) derivative field theory is needed, from which the deformed kinematics can be read. From our construction we are able to restrict the possible models of doubly special relativity to particular bases that preserve linear Lorentz invariance. We quantize the theories and also obtain a deformed version of the Maxwell equations. We analyze the electromagnetic vector potential either for an electric point-like source and a magnetic dipole. We observe that the electric and magnetic fields do not diverge at the origin for some models described with an anti de Sitter space but do for the de Sitter one in both problems.

Non-local quantum field theories could be a solution to the inconsistencies arising when quantizing gravity. Doubly special relativity is regarded as a low-energy limit of a quantum gravity theory with testable predictions. In this paper (arXiv:2403.18772) we present a new formulation of quantum field theories in doubly special relativity with non-local behavior. Our construction restricts the models to those showing linear Lorentz invariance. The deformed Klein–Gordon, Dirac, and electromagnetic Lagrangians are derived. The deformed Maxwell equations and the electric potential of a point charge are discussed.

In this short review (arXiv:2403.06652) we present the key definitions, ideas and techniques involved in the study of symmetry resolved entanglement measures, with a focus on the symmetry resolved entanglement entropy. In order to be able to define such entanglement measures, it is essential that the theory under study possess an internal symmetry. Then, symmetry resolved entanglement measures quantify the contribution to a particular entanglement measure that can be associated to a chosen symmetry sector. Our review focuses on conformal (gapless/massless/critical) and integrable (gapped/massive) quantum field theories, where the leading computational technique employs symmetry fields known as (composite) branch point twist fields.

In this paper (arXiv:2311.08435) the one-loop quantum corrections to the internal energy of lattices due to the quantum fluctuations of the scalar field of phonons are studied. The band spectrum of the lattice is characterised in terms of the scattering data, allowing to compute the total Helmholtz free energy and the entropy at finite non zero temperature. Some examples of three dimensional periodic potentials built from the repetition of the same punctual or compact supported potential are addressed: the generalised Dirac comb and the Pöschl-Teller comb, respectively.

In this paper (arXiv:2305.01438) the quantum vacuum interaction energy between a pair of semitransparent two-dimensional plates represented by Dirac delta potentials and its first derivative, embedded in the topological background of a sine-Gordon kink is studied through an extension of the TGTG-formula (firstly discovered by O. Kenneth and I. Klich) to weak curved backgrounds. Quantum vacuum oscillations around the sine-Gordon kink solutions are interpreted as a quantum scalar field theory in the spacetime of a domain wall. Moreover, the relation between the phase shift and the density of states (the well-known Dashen-Hasslacher-Neveu formula) is also exploited to characterize the quantum vacuum energy.

In this paper (arXiv:2312.16586) it is shown that the complex Bernoulli differential equations admitting the supplementary structure of a Lie-Hamilton system related to the book algebra can always be solved by quadratures, providing an explicit solution of the equations. In addition, considering the quantum deformation of Bernoulli equations, their canonical form is obtained and an exact solution by quadratures is deduced as well. It is further shown that the approximations of kth-order in the deformation parameter from the quantum deformation are also integrable by quadratures, although an explicit solution cannot be obtained in general. Finally, the multidimensional quantum deformation of the book Lie-Hamilton systems is studied, showing that, in contrast to the multidimensional analogue of the undeformed system, the resulting system is coupled in a non-trivial form.