Nonrelativistic quantum mechanics and conformal quantum mechanics are deformed through a Jordanian twist. The deformed space coordinates satisfy the Snyder noncommutativity. The resulting deformed Hamiltonians are pseudo-Hermitian Hamiltonians of the type discussed by Mostafazadeh.The quantization scheme makes use of the so-called "unfolded formalism" discussed in previous works. A Hopf algebra structure, compatible with the physical interpretation of the coproduct, is introduced for the Universal Enveloping Algebra of a suitably chosen dynamical Lie algebra (the Hamiltonian is contained among its generators).The multi-particle sector, uniquely determined by the deformed 2-particle Hamiltonian, is composed of bosonic particles.
Noncommutative oscillators are first-quantized through an abelian Drinfel'd twist deformation of a Hopf algebra and its action on a module. Several important and subtle issues making possible the quantization are solved. The spectrum of the single-particle Hamiltonians is computed. The multi-particle Hamiltonians are fixed, unambiguously, by the Hopf algebra coproduct. The symmetry under particle exchange is guaranteed. In d = 2 dimensions the rotational invariance is preserved, while in d = 3 the so(3) rotational invariance is broken down to an so(2) invariance.
In this work we investigate the q-deformation of the so(4) dynamical symmetry of the hydrogen atom using the theory of the quantum group su q (2). We derive the energy spectrum in a physically consistent manner and find a degeneracy breaking as well as a smaller Hilbert space. We point out that using the deformed Casimir as was done before leads to inconsistencies in the physical interpretation of the theory.
We consider spin-boson models composed by a single bosonic mode and an ensemble of N identical two-level atoms. The situation where the coupling between the bosonic mode and the atoms generates real and virtual processes is studied, where the whole system is in thermal equilibrium with a reservoir at temperature β −1 . Phase transitions from ordinary fluorescence to superradiant phase in three different models is investigated. First a model where the coupling between the bosonic mode and the j − th atom is via the pseudo-spin operator σ z (j) is studied. Second, we investigate the generalized Dicke model, introducing different coupling constants between the single mode bosonic field and the environment, g 1 and g 2 for rotating and counter-rotating terms, respectively. Finally it is considered a modified version of the generalized Dicke model with intensity-dependent coupling in the rotating terms. In the first model the zero mode contributes to render the canonical entropy a negative quantity for low temperatures. The last two models presents phase transitions, even when only Hamiltonian terms which generates virtual processes are considered.
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