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.

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.

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.

In this paper (arXiv:2409.18489), new classes of Lie-Hamilton systems are obtained from the six-dimensional fundamental representation of the symplectic Lie algebra 𝔰𝔭(6,ℝ). The ansatz is based on a recently proposed procedure for constructing higher-dimensional Lie-Hamilton systems through the representation theory of Lie algebras. As applications of the procedure, we study a time-dependent electromagnetic field and several types of coupled oscillators. The irreducible embedding of the special unitary Lie algebra 𝔰𝔲(3) into 𝔰𝔭(6,ℝ) is also considered, yielding Lie-Hamilton systems arising from the sum of the quark and antiquark three-dimensional representations of 𝔰𝔲(3), which are applied in the construction of t-dependent coupled systems. In addition, t-independent constants of the motion are obtained explicitly for all these Lie-Hamilton systems, which allows the derivation of a nonlinear superposition rule

A postdoctoral research position in the Mathematical Physics Group at the University of Burgos (Spain) is open. We welcome all qualified candidates with a strong research record in the field of Theoretical Quantum Information, including applications in quantum communication, quantum computation, quantum foundations, quantum aspects of the gravitational field and information geometry.
The contract should start on September 1st at the latest, and would end in August 2025. The salary will be set based on the experience and research record of the selected candidate. Applicants are requested to send a complete CV, a short research statement and at least one reference letter to Prof. Angel Ballesteros (angelb@ubu.es) by March 19th.

Two three-year research positions starting on November, 2022, are available in our group within the framework of the Project “Quantum reference frames, quantum groups, quantum information, entanglement and decoherence: applications in quantum communications“. The deadline for applications is October 25th. The call for applications can be found here, and interested candidates should contact us urgently at angelb@ubu.es.

An one-year position starting on January, 2017, is available in our group. The deadline for applications will be December 8th. Candidates willing to join any of the research lines of the group and with strong expertise on symbolic and/or numerical computation are encouraged to apply as soon as possible. Candidates have to be registered at the “Registro de Garantía Juvenil” and should contact us urgently at angelb@ubu.es.