Quantum groups have been widely explored as a tool to encode possible nontrivial generalisations of reference frame transformations, relevant in quantum gravity. In quantum information, it was found that the reference frames can be associated to quantum particles, leading to quantum reference frames transformations. The connection between these two frameworks is still unexplored, but if clarified it will lead to a more profound understanding of symmetries in quantum mechanics and quantum gravity. In this paper (arXiv:2504.00569), we establish a correspondence between quantum reference frame transformations and transformations generated by a quantum deformation of the Galilei group with commutative time, taken at the first order in the quantum deformation parameter. This is found once the quantum group noncommutative transformation parameters are represented on the phase space of a quantum particle, and upon setting the quantum deformation parameter to be proportional to the inverse of the mass of the particle serving as the quantum reference frame. These results allow us to show that quantum reference frame transformations are physically relevant when the state of the quantum reference frame is in a quantum superposition of semiclassical states. We conjecture that the all-order quantum Galilei group describes quantum reference frame transformations between more general quantum states of the quantum reference frame.
Deformations of the symmetric subspace of qubit chains
The symmetric subspace of multi-qubit systems, that is, the space of states invariant under permutations, is commonly encountered in applications in the context of quantum information and communication theory. It is known that the symmetric subspace can be described in terms of irreducible representations of the group SU(2), whose representation spaces form a basis of symmetric states, the so-called Dicke states. In this work (arXiv:2503.23554), we present deformations of the symmetric subspace as deformations of this group structure, which are promoted to a quantum group U_q(𝔰𝔲(2)). We see that deformations of the symmetric subspace obtained in this manner correspond to local deformations of the inner product of each spin, in such a way that departure from symmetry can be encoded in a position-dependent inner product. The consequences and possible extensions of these results are also discussed.
Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
We propose (arXiv:2503.20558) an adaptation of the notion of scaling symmetries for the case of Lie-Hamilton systems, allowing their subsequent reduction to contact Lie systems. As an illustration of the procedure, time-dependent frequency oscillators and time-dependent thermodynamic systems are analyzed from this point of view. The formalism provides a novel method for constructing contact Lie systems on the three-dimensional sphere, derived from recently established Lie-Hamilton systems arising from the fundamental four-dimensional representation of the symplectic Lie algebra 𝔰𝔭(4,ℝ). It is shown that these systems are a particular case of a larger hierarchy of contact Lie systems on a special class of three-dimensional homogeneous spaces, namely the Cayley-Klein spaces. These include Riemannian spaces (sphere, hyperbolic and Euclidean spaces), pseudo-Riemannian spaces (anti-de Sitter, de Sitter and Minkowski spacetimes), as well as Newtonian or non-relativistic spacetimes. Under certain topological conditions, some of these systems retrieve well-known two-dimensional Lie-Hamilton systems through a curvature-dependent reduction.
Exceptional Eigenvalues and Braak’s Conjecture in the Quantum Rabi Model
The spectrum of the quantum Rabi model can be separated as regular eigenvalues that need to be computed numerically and exceptional eigenvalues, that match the energies of a shifted quantum harmonic oscillator. Exceptional values can be separated further as Juddean, if they occur under certain algebraic conditions, and non-Juddean, if they obey more elusive transcendental conditions. In this paper (arXiv:2503.15572), we show that simple assumptions on these conditions imply and extend Braak’s conjecture on the distribution of the quantum Rabi spectrum.
Quantum group deformation of the Kittel-Shore model
The Kittel-Shore (KS) Hamiltonian describes N spins with long-range interactions that are identically coupled. In this paper (arXiv:2502.20884) , the underlying 𝔰𝔲(2) coalgebra symmetry of the KS model is demonstrated for arbitrary spins, and the quantum deformation of the KS Hamiltonian (q-KS model) is obtained using the corresponding 𝔰𝔲_q(2) quantum group. By construction, the existence of such a symmetry guarantees that all integrability properties of the KS model are preserved under q-deformation. In particular, the q-KS model for spin-1/2 particles is analysed in both ferromagnetic and antiferromagnetic couplings, and the cases with N=2,3, and 4 spins are studied in detail. The higher-spin q-KS models are sketched.
Thermal time of noncommutative Minkowski spacetime
In this paper (arXiv:2502.12750), we study the thermal time hypothesis of arXiv:gr-qc/9406019 in the context of noncommutative deformations of Minkowski. We show that a natural modular group arises from the modular function of the momentum space. In the specific case of κ-Minkowski, we show that this thermal time flow corresponds to the globally defined time coordinate translation. On the other hand, the absence of thermal time for ρ-Minkowski is directly related to the discreteness of its global time. The impact of inner automorphism transformation on the physics and the treatment of unimodular case (unthermalised spacetimes) are discussed. Moreover, a reflection on the use of thermal field theory for quantum gravity phenomenology is put forward, as we just bridged thermal spacetimes with κ-Minkowski, often considered a “flat limit” of a quantum gravity candidate theory.
Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
In this paper (arXiv:2502.02491) we consider the quantum analog of the generalized Zernike systems. This two-dimensional quantum model, besides the conservation of the angular momentum, exhibits higher-order integrals of motion within the enveloping algebra of the Heisenberg algebra 𝔥2. By constructing suitable combinations of these integrals, we uncover a polynomial Higgs-type symmetry algebra that, through an appropriate change of basis, gives rise to a deformed oscillator algebra. The associated structure function Φ is shown to factorize into two commuting components Φ=Φ1.Φ2. This framework enables an algebraic determination of the possible energy spectra of the model for the cases N=2,3,4, the case N=1 being canonically equivalent to the harmonic oscillator. Based on these findings, we propose two conjectures which generalize the results for all N≥2 and any value of the coefficients of the model, and they are explicitly proven for N=5. In addition, all of these results can be interpreted as superintegrable perturbations of the original quantum Zernike system corresponding to N=2 which are also analyzed and applied to the isotropic oscillator on the sphere, hyperbolic and Euclidean spaces.
Doubly quantum mechanics
Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, in this paper (arXiv:2412.05997) we take the first steps towards a theory of ”Doubly” Quantum Mechanics, a modification of Quantum Mechanics in which the geometrical configurations of physical systems, measurement apparata, and reference frame transformations are themselves quantized and described by ”geometry” states in a Hilbert space. We develop the formalism for spin-1/2 measurements by promoting the group of spatial rotations SU(2) to the quantum group SU_q(2) and generalizing the axioms of Quantum Theory in a covariant way. As a consequence of our axioms, the notion of probability becomes a self-adjoint operator acting on the Hilbert space of geometry states, hence acquiring novel non-classical features. After introducing a suitable class of semi-classical geometry states, which describe near-to-classical geometrical configurations of physical systems, we find that probability measurements are affected, in these configurations, by intrinsic uncertainties stemming from the quantum properties of SU_q(2). This feature translates into an unavoidable fuzziness for observers attempting to align their reference frames by exchanging qubits, even when the number of exchanged qubits approaches infinity, contrary to the standard SU(2) case.
Generalized Buchdahl equations as Lie-Hamilton systems: Quantum deformations and their general solution
In this paper (arXiv:2412.06057) we revisit the nonlinear second-order differential equations x´´(t)=a(x)x´(t)^2+b(t)x´(t), where a(x) and b(t) are arbitrary functions on their argument from the perspective of Lie-Hamilton systems. For the particular choice a(x)=3/x and b(t)=1/t, these equations reduce to the Buchdahl equation considered in the context of General Relativity. It is shown that these equations are associated to the ‘book’ Lie algebra 𝔟2, determining a Lie-Hamilton system for which the corresponding t-dependent Hamiltonian and the general solution of the equations are given. The procedure is illustrated considering several particular cases. We also make use of the quantum deformation of 𝔟2 with quantum deformation parameter z (where q=e^z), leading to a deformed generalized Buchdahl equation. Applying the formalism of Poisson-Hopf deformations of Lie-Hamilton systems, we derive the corresponding deformed t-dependent Hamiltonian, as well as its general solution. The presence of the quantum deformation parameter z is interpreted as the introduction of an integrable perturbation of the initial generalized Buchdahl equation, which is described in detail in its linear approximation.
Mechanical presymplectic structures and Marsden-Weinstein reduction of time-dependent Hamiltonian systems
In 1986, Albert proposed a Marsden-Weinstein reduction process for cosymplectic structures. In this paper (arXiv:2411.11997), we present the limitations of this theory in the application of the reduction of symmetric time-dependent Hamiltonian systems. As a consequence, we conclude that cosymplectic geometry is not appropriate for this reduction. Motived for this fact, we replace cosymplectic structures by more general structures: mechanical presymplectic structures. Then, we develop Marsden-Weinstein reduction for this kind of structures and we apply this theory to interesting examples of time-dependent Hamiltonian systems for which Albert’s reduction method doesn’t work.
