Mixed superposition rules are, in short, a method to describe the general solutions of a time-dependent system of first-order differential equations, a so-called Lie system, in terms of particular solutions of other ones. This article (arXiv:2511.01063) is concerned with the theory of mixed superposition rules and their connections with geometric structures. We provide methods to obtain mixed superposition rules for systems admitting an imprimitive finite-dimensional Lie algebra of vector fields or given by a semidirect sum. In particular, we develop a novel mixed coalgebra method for Lie systems that are Hamiltonian relative to a Dirac structure, which is quite general, although we restrict to symplectic and contact manifolds in applications. This provides us with practical methods to derive mixed superposition rules and extends the coalgebra method to a new field of application while solving minor technical issues of the known formalism. Throughout the paper, we apply our results to physical systems including Schrödinger Lie systems, Riccati systems, time-dependent Calogero-Moser systems with external forces, time-dependent harmonic oscillators, and time-dependent thermodynamical systems, where general solutions can be obtained from reduced system solutions. Our results are finally extended to Lie systems of partial differential equations and a new source of such PDE Lie systems, related to the determination of approximate solutions of PDEs, is provided. An example based on the Tzitzéica equation and a related system is given.

The unification of all physical fields into one mathematical object and the derivation of all physical field equations from that object in one framework is a long-lasting endeavor in fundamental physics. We suggest (arXiv:2510.15812) a new approach to achieve this goal by encoding physical fields into the geometry of the 1-particle phase space on spacetime (the cotangent bundle) through Hamilton geometry. The fundamental field, which contains information about all physical fields in spacetime and defines the phase space geometry, is a scalar field in phase space that is interpreted as a point-particle Hamiltonian. We construct an action principle for scalar fields in phase space and derive the corresponding scalar field equation. By choosing a specific scalar field, namely the Hamiltonian describing a charged particle in curved spacetime with an electromagnetic field, we show that this phase-space scalar field equation is equivalent to the coupled Einstein-Maxwell equations in spacetime, thus providing a geometric unification of gravity and electromagnetism. We further discuss how this approach differs from previous unification attempts and its potential for describing further physical fields and their dynamics in a unified manner in terms of phase-space geometry.

The Darboux III oscillator is an exactly solvable N-dimensional nonlinear oscillator defined on a radially symmetric space with non-constant negative curvature. Its one-dimensional version can be seen as a position dependent mass system whose mass function depends on the nonlinearity parameter λ, such that in the limit λ→0 the harmonic oscillator is recovered. In this paper (arXiv:2510.06221), a detailed study of the entropic moments and of the Rényi and Tsallis information entropies for the quantum version of the one-dimensional Darboux III oscillator is presented. In particular, analytical expressions for the aforementioned quantities in position space are obtained. Since the Fourier transform of the Darboux III wave functions does not admit a closed form expression, a numerical analysis of these quantities has been performed. Throughout the paper the interplay between the entropy parameter α and the nonlinearity parameter λ is analysed, and known results for the Shannon entropy of the Darboux III and for the Rényi and Tsallis entropies of the harmonic oscillator are recovered in the limits α→1 and λ→0, respectively. Finally, motivated by the strong non-linear effects arising when large values of λ and/or highly excited states are considered, an approximation to the probability density function valid in those regimes is presented. From it, an analytical approximation to the probability density in momentum space can be obtained, and some of the previously observed effects arising from the interplay between α and λ can be explained.

IR/UV mixing (a mechanism causing ultraviolet quantum-gravity effects to manifest themselves also in a far-infrared regime) is a rare case of feature found in several approaches to the quantum-gravity problem. We here (arXiv:2508.06171) derive the implications for “soft” IR/UV mixing (corrections to the dispersion relation that are linear in momentum) of some recent cold-atom-interferometry measurements. For both signs of the IR/UV-mixing correction term we establish bounds on the characteristic length scale which reach the Planck-length milestone. Intriguingly, for values of the characteristic scale of about half the Planck length we find that IR/UV mixing provides a solution for a puzzling discrepancy between Cesium-based and Rubidium-based atom-interferometric measurements of the fine structure constant.

In this work (arXiv:2508.04527) we develop a quantization scheme for the quantum theory of a real scalar field on a class of non-commutative spacetime models collectively known as T-Minkowski. Requiring the theory to be covariant under T-Poincaré transformations, we find that for a subclass of models the Wightmann functions are equal to their commutative counterparts, and we are able to prove a Wick theorem for Wightmann functions that is structurally equivalent to the one encountered in commutative QFT. For some of these models we further extend the result to Green functions and to N-point functions of interacting QFT, which we also find to be commutative, leaving no space for IR/UV mixing effects advocated in other approaches to noncommutative QFT.

This work (arXiv:2507.03425) aims to bridge the gap between Dunkl superintegrable systems and the coalgebra symmetry approach to superintegrability, and subsequently to recover known models and construct new ones. In particular, an infinite family of N-dimensional quasi-maximally superintegrable quantum systems with reflections, sharing the same set of 2N−3 quantum integrals, is introduced. The result is achieved by introducing a novel differential-difference realization of 𝔰𝔩(2,ℝ) and then applying the coalgebra formalism. Several well-known maximally superintegrable models with reflections appear as particular cases of this general family, among them, the celebrated Dunkl oscillator and the Dunkl-Kepler-Coulomb system. Furthermore, restricting to the case of “hidden” quantum quadratic symmetries, maximally superintegrable curved oscillator and Kepler-Coulomb Hamiltonians of Dunkl type, sharing the same underlying 𝔰𝔩(2,ℝ) coalgebra symmetry, are presented. Namely, the Dunkl oscillator and the Dunkl-Kepler-Coulomb system on the N-sphere and hyperbolic space together with two models which can be interpreted as a one-parameter superintegrable deformation of the Dunkl oscillator and the Dunkl-Kepler-Coulomb system on non-constant curvature spaces. In addition, maximally superintegrable generalizations of these models, involving non-central potentials, are also derived on flat and curved spaces. For all specific systems, at least an additional quantum integral is explicitly provided, which is related to the Dunkl version of a (curved) Demkov-Fradkin tensor or a Laplace-Runge-Lenz vector.

In this work (arXiv:2506.11686) discuss the application of the Jordanian quantum algebra U_h(𝔰𝔩(2,ℝ)), as a Hopf algebra deformation of the Lie algebra 𝔰𝔩(2,ℝ), in the context of entanglement properties of quantum states. For them, several kind of entanglement measures and fidelities are obtained, parametrized by the deformation parameter h. In particular, we construct the associated h-deformed Dicke states on 𝔰𝔩(2,ℝ), comparing them to the q-Dicke states obtained from the quantum deformation of the U_q(𝔰𝔩(2,ℝ)). Moreover, the density matrices of these h-deformed Dicke states are compared to the experimental realizations of those of Dicke states. A similar behavior is observed, pointing out that the h-deformation could be used to describe noise and decoherence effects in experimental settings.

The Lie-Hamilton approach for t-dependent Hamiltonians is extended to cover the so-called nonlinear Lie-Hamilton systems (arXiv:2505.13853), which are no longer related to a linear t-dependent combination of a basis of a finite-dimensional Lie algebra of functions W, but an arbitrary t-dependent function on W. This novel formalism is accomplished through a detailed analysis of related structures, such as momentum maps and generalized distributions, together with the extension of the Poisson coalgebra method to a t-dependent frame, in order to systematize the construction of constants of the motion for nonlinear systems. Several relevant relations between nonlinear Lie-Hamilton systems, Lie-Hamilton systems, and collective Hamiltonians are analyzed. The new notions and tools are illustrated with the study of the harmonic oscillator, Hénon-Heiles systems and Painlevé trascendents within a t-dependent framework. In addition, the formalism is carefully applied to construct oscillators with a t-dependent frequency and Kepler-Coulomb systems with a t-dependent coupling constant on the n-dimensional sphere, Euclidean and hyperbolic spaces, as well as on some spaces of non-constant curvature.

In this work (arXiv:2502.01283) we describe the dynamics of a detector modeled by a harmonic oscillator coupled with an otherwise free quantum field in a curved spacetime in terms of covariant equations of motion leading to local observables. To achieve this, we derive and renormalize the integro-differential equation that governs the detector pointer-variable dynamics, introducing phenomenological parameters such as a dispersion coefficient and a Lamb-shift parameter. Our formal solution, expressed in terms of Green’s functions, allows for the covariant, and causal analysis of induced observables on the field. This formalism can be used for instance to detect non-Gaussianities present in the field’s state.

In this work (arXiv:2504.21833), we introduce a PT-symmetric infinite-dimensional representation of the U_z(sl(2,R)) Hopf algebra, and we analyse a multiparametric family of Hamiltonians constructed from such representation of the generators of this non-standard quantum algebra. It is shown that all these Hamiltonians can be mapped to equivalent systems endowed with a position-dependent mass. From the latter presentation, it is shown how appropriate point canonical transformations can be further defined in order to transform them into Hamiltonians with constant mass over suitable domains. By following this approach, the bound-state spectrum and the corresponding eigenfunctions of the initial PT-symmetric Hamiltonians can be determined. It is worth stressing that a relevant feature of some of the new U_z(sl(2,R)) systems here presented is found to be their connection with double-well and Pöschl-Teller potentials. In fact, as an application we present a particular Hamiltonian that can be expressed as an effective double-well trigonometric potential, which is commonly used to model several relevant systems in molecular physics.