Cervantes, D., Fioresi, R., Lledo, M. A., & Nadal, F. A. (2012). Quadratic deformation of Minkowski space. Fortschritte Phys.-Prog. Phys., 60(9-10), 970–976.
Abstract: We present a deformation of the Minkowski space as embedded into the conformal space (in the formalism of twistors) based in the quantum versions of the corresponding kinematic groups. We compute explicitly the star product, whose Poisson bracket is quadratic. We show that the star product although defined on the polynomials can be extended differentiably. Finally we compute the Eucliden and Minkowskian real forms of the deformation.
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Fioresi, R., Lledo, M. A., & Razzaq, J. (2022). N=2 quantum chiral superfields and quantum super bundles. J. Phys. A, 55(38), 384012–19pp.
Abstract: We give the superalgebra of N = 2 chiral (and antichiral) quantum superfields realized as a subalgebra of the quantum supergroup SL q (4|2). The multiplication law in the quantum supergroup induces a coaction on the set of chiral superfields. We also realize the quantum deformation of the chiral Minkowski superspace as a quantum principal bundle.
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Lledo, M. A., & Sommovigo, L. (2010). Torsion formulation of gravity. Class. Quantum Gravity, 27(6), 065014–16pp.
Abstract: We explain precisely what it means to have a connection with torsion as a solution of the Einstein equations. While locally the theory remains the same, the new formulation allows for topologies that would have been excluded in the standard formulation of gravity. In this formulation it is possible to couple arbitrary torsion to gauge fields without breaking the gauge invariance.
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Fioresi, R., Latini, E., Lledo, M. A., & Nadal, F. A. (2018). The Segre embedding of the quantum conformal superspace. Adv. Theor. Math. Phys., 22(8), 1939–2000.
Abstract: In this paper we study the quantum deformation of the superflag Fl(2 vertical bar 0, 2 vertical bar 1, 4 vertical bar 1), and its big cell, describing the complex conformal and Minkowski superspaces respectively. In particular, we realize their projective embedding via a generalization to the super world of the Segre map and we use it to construct a quantum deformation of the super line bundle realizing this embedding. This strategy allows us to obtain a description of the quantum coordinate superring of the superflag that is then naturally equipped with a coaction of the quantum complex conformal supergroup SLq (4 vertical bar 1).
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Aldana, M., & Lledo, M. A. (2023). The Fuzzy Bit. Symmetry-Basel, 15(12), 2103–25pp.
Abstract: In this paper, the formulation of Quantum Mechanics in terms of fuzzy logic and fuzzy sets is explored. A result by Pykacz, which establishes a correspondence between (quantum) logics (lattices with certain properties) and certain families of fuzzy sets, is applied to the Birkhoff-von Neumann logic, the lattice of projectors of a Hilbert space. Three cases are considered: the qubit, two qubits entangled, and a qutrit 'nested' inside the two entangled qubits. The membership functions of the fuzzy sets are explicitly computed and all the connectives of the fuzzy sets are interpreted as operations with these particular membership functions. In this way, a complete picture of the standard quantum logic in terms of fuzzy sets is obtained for the systems considered.
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Fioresi, R., & Lledo, M. A. (2021). Quantum Supertwistors. Symmetry-Basel, 13(7), 1241–16pp.
Abstract: In this paper, we give an explicit expression for a star product on the super-Minkowski space written in the supertwistor formalism. The big cell of the super-Grassmannian Gr(2|0,4|1) is identified with the chiral, super-Minkowski space. The super-Grassmannian is a homogeneous space under the action of the complexification SL(4|1) of SU(2,2|1), the superconformal group in dimension 4, signature (1,3), and supersymmetry N=1. The quantization is done by substituting the groups and homogeneous spaces by their quantum deformed counterparts. The calculations are done in Manin's formalism. When we restrict to the big cell, we can explicitly compute an expression for the super-star product in the Minkowski superspace associated to this deformation and the choice of a certain basis of monomials.
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Lledo, M. A. (2020). Superfields, Nilpotent Superfields and Superschemes dagger. Symmetry-Basel, 12(6), 1024–32pp.
Abstract: We interpret superfields in a functorial formalism that explains the properties that are assumed for them in the physical applications. We study the non-trivial relation of scalar superfields with the defining sheaf of the supermanifold of super spacetime. We also investigate in the present work some constraints that are imposed on the superfields, which allow for non-trivial solutions. They give rise to superschemes that, generically, are not regular, that is they do not define a standard supermanifold.
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Cervantes, D., Fioresi, R., Lledo, M. A., & Nadal, F. A. (2016). Quantum Twistors. P-Adic Num., 8(1), 2–30.
Abstract: We compute explicitly a star product on the Minkowski space whose Poisson bracket is quadratic. This star product corresponds to a deformation of the conformal spacetime, whose big cell is the Minkowski spacetime. The description of Minkowski space is made in the twistor formalism and the quantization follows by substituting the classical conformal group by a quantum group.
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