This work (arXiv:2608.31104) introduces the novel Dunkl two-photon and Dunkl-Schrödinger algebras as reflection-extended counterparts of the standard six-dimensional two-photon and Schrödinger algebras. These extensions are realized through the presence of an involutive generator representing a reflection, and they reduce to the undeformed two-photon and Schrödinger algebras in the absence of this additional generator in an appropriate limit. For both algebraic structures, which are shown to be isomorphic, we derive the associated cubic polynomial Casimir invariant. Furthermore, we identify several structurally significant subalgebras and compute their corresponding quadratic Casimir invariants. Representations of the Dunkl two-photon algebra are constructed on the Fock space of the one-dimensional Dunkl harmonic oscillator and subsequently translated into holomorphic realizations within the associated Dunkl-Fock-Bargmann representation. Finally, for the Dunkl-Schrödinger algebra, we present an explicit vector field realization in terms of the Dunkl spatial derivative, while keeping time continuous. When applied to the Casimir invariants of the aforementioned subalgebras, this realization yields a family of six nontrivial Dunkl differential-difference equations in 1+1 dimensions, including, in particular, a Dunkl version of the well-known heat-Schrödinger equation.

In this paper (arXiv:2607.07057) we develop a geometric reduction mechanism generated by systems of particular integrals, namely, families of functions whose time derivatives close linearly on the family. Their common zero set is dynamically invariant. In the Hamiltonian case, under a weak involution condition, the restricted dynamics is presymplectic, and its characteristic quotient carries a reduced Hamiltonian flow. This yields a direct bridge between particular integrals, presymplectic reduction, and lower-dimensional Hamiltonian dynamics, and leads to a Liouville-type notion of particular integrability. We illustrate the framework through mechanical examples and lift constructions, including variants of the Eisenhart lift.

In this paper (arXiv:2606.17964) we study the path integral quantisation of a scalar field on a generic noncommutative deformation of Minkowski space, built as a quantum homogeneous space of a deformed Poincaré group. We show that the procedure depends on the choice of noncommutative functional derivative, and we isolate two natural choices, distinguished by the space in which the Leibniz rule remains undeformed. The first, which carries the undeformed statistics, reproduces the standard scheme and yields n-point functions that break the deformed Poincaré covariance and exhibit the UV/IR mixing of [8]. The second, which adapts the functional calculus to the braided statistics of the fields in the spirit of [20], yields covariant n-point functions free of this mixing. We trace both the covariance breaking and the mixing to a single source, the intertwining of external and loop momenta in the non-planar contributions, and conclude that, in the models considered, the UV/IR mixing of [8] is an artifact of a quantisation that breaks the deformed symmetry rather than a feature of noncommutativity itself. We further disentangle covariance, fixed by the quantisation scheme, from finiteness, fixed independently by the propagator, and illustrate the formalism on the T-Minkowski models, the Euclidean three-dimensional quantum gravity model, and the quantum two-sphere.

This work (arXiv:2606.19462) explores the effects of both the standard quantum q-deformation and the non-standard h-deformation of the Hopf algebra U(𝔰𝔩(2,ℝ)) on multi-qubit systems. By constructing the states of a Hilbert space of N qubits through the Clebsch-Gordan coefficients associated with the deformed algebras, we show that these states naturally coincide with the eigenstates of the Hamiltonian of the q- and h-deformed Kittel-Shore models. We compare the resulting deformed states with those typically targeted in quantum information experiments, providing a bridge between algebraic constructions and experimentally relevant quantum resources. Fidelities with respect to the undeformed states are computed to establish how the quantum correlations are affected, both for few-qubit systems (including Dicke and non-Dicke states), and in the macroscopic limit (N→∞) through closed-form formulas derived for arbitrary Dicke states. The results reveal different behaviors between the two deformations. The q-deformation smoothly modifies the states and maintains a residual overlap with the original configurations, while the h-deformation rapidly makes the states orthogonal to their undeformed counterparts. Both models demand a standard N−1 rescaling to preserve fidelity stability in the macroscopic limit.

The Fock-Darwin (FD) quantum system describes the motion on the plane of a charged particle under the action of an isotropic oscillator potential together with a perpendicular constant magnetic field. When the isotropic oscillator is suppressed, the FD system leads to the Landau Hamiltonian with infinitely degenerate Landau levels. The Fock-Darwin-Darboux (FDD) system is the generalisation of the FD system to a particle moving on the Darboux III space, which is a conformally flat surface with non-constant negative curvature. We present in this paper (arXiv:2604.26466) a systematic study of some information-theoretic entropy and dispersion-like measures for these quantum systems. Since both systems are exactly solvable, analytical expressions for Shannon, Rényi and Tsallis entropies, among others, can be obtained. We show that for the FD system, its information-theoretic measures are formally the same as the ones for the harmonic oscillator, provided a modified effective frequency depending on the magnetic field is introduced. In the FDD case, the nonlinear nature of the underlying manifold precludes the existence of a simple closed form for the wave-function on momentum space, which is numerically analysed. We compare the numerical behaviour of the different entropy measures and we analyse the interplay arising in the FDD system between the curvature parameter and the magnetic field. In particular, it is shown that the Landau system on the Darboux III space has no infinitely degenerate Landau levels.

In this paper (arXiv:2604.01058) universal T-matrices, or Hopf algebra dual forms, for quantum groups are revisited, and their contraction theory is developed. As a first illustrative example, the (1+1) timelike κ-Poincaré T-matrix is explicitly worked out. Afterwards, motivated by recent results on the role of the Hopf algebra dual form of a quantum (1+1) centrally extended Galilei group as the algebraic object underlying non-relativistic quantum reference frame transformations, a new quantum deformation of the (1+1) centrally extended Poincaré Lie algebra is obtained, and its universal T-matrix is presented. Finally, the Hopf algebra dual form contraction is applied to this Poincaré T-matrix, showing that its corresponding non-relativistic counterpart is precisely the Galilei T-matrix associated with quantum reference frames. In this way, the Poincaré Hopf algebra dual form introduced here stands as a natural candidate for describing the symmetry structure of relativistic quantum reference frame transformations. In the appropriate basis, the associated quantum Poincaré group is recognized, remarkably, as a non-trivial central extension of the (1+1) spacelike κ-Poincaré dual Hopf algebra.

The Kittel-Shore Hamiltonian characterizes N spins with identical long-range interactions, and the 𝔰𝔲(2) coalgebra has been proven to be a symmetry of this model, which can be exactly solved. By using quantum groups and, in particular, quantum 𝔰𝔲(2), this Hamiltonian was deformed. In this work (arXiv:2512.15216), we study the thermodynamic properties of this deformed model for spin-1/2 particles. In particular, we discuss how this deformation affects the specific heat, magnetic susceptibility, magnetisation, and phase transitions as a function of the parameter q of the deformation and compare them with those of the undeformed model. Deformation was found to shift the thermodynamic behaviours to higher temperatures and alter the phase transitions. The potential applications of this q-deformed model for describing few-spin quantum systems with non-identical couplings are discussed.

A superintegrable generalization of the classical and quantum Zernike systems is reviewed (arXiv:2511.09541). The corresponding Hamiltonians are endowed with higher-order integrals and can be interpreted as higher-order superintegrable perturbations of the 2D spherical (Higgs), hyperbolic, and Euclidean harmonic oscillators. As a new result, the complete polynomial Higgs-type symmetry algebra of the generalized classical system is presented. For the generalized quantum system, the symmetry algebra and the spectra are provided for a representative case.

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.