Non-hermitian And Deformed Hamiltonians.

Résumé: Quantum gravity is a physical theory designed to unify general relativity and quantum mechanics, which are inconsistent and incompatible and each one has a different scale from the other, the first is macroscopic while the second is microscopic, thus it would be difficult to make them integrated into a single theory. There are numerous attempts to unify them using three separate roads: super string theory, loop-quantum gravity and black hole approach, and since these three approaches expect a minimal length on either the position x or the momentum p coordinates, or both of them, this necessitates a deformation on the phase space, with a slight deformation coefficient. Certainly; such deformation affects Quantum Mechanics in particular Heisenberg algebra by producing a modifications in the commutation relations. In such circumstances, the resulted Hamiltonians could be non-Hermitian which means they do not respect Dirac Hermiticity condition defined in a Hermitian-inner product, which is necessary and sufficient condition to obtain a real and positive energy spectrum.In fact, it is contrary to what was found; because there are Hamiltonian operators do not respect this condition.In this context it can be categorized them into three classes, PT-Hermitian, pseudo-Hermitian and Quasi-Hermitan hamiltonian, where all they have the property of Hermiticity but in non standard inner product. First, two examples of PT-symmetric Hamiltonians have been studied, Shifted(displace)-harmonic oscillator p2 + x2 + i x and Cubic-anharmonic oscillator p2 + x2 + i x3, then we have deformed the position operator x, where the Hamiltonians H became pseudo-hermitian or more precisely quasi-Hermitian characterized by the similarity transformation which is written in terms of a positive-definite, Hermitian and invertible operator η. We found that the energy spectrum is real and positive and related to a deformation coefficient β is given for small values,and we studied the effect of time evolution on both the space-time and non-Hermiticity of PT-symmetric spin Hamiltonian H2∗2 with assumption that there is no correlation in the momentum. It turns out that it is possible to generate a bipartite spin quantum entanglement quantified in the Von Newman entropy, as well as the possibility of magnified the amount of quantum entanglement and becomes maximal or reduced (decoherence) depending on the various parameters and physical quantities of the system (Hermiticity and metric) Finally we also expect that quantum entanglement depends strongly on the metric space, especially near the horizon (Schwarzchild’s radius). This result can be generalized to multisystems (Bell, Werner,...etc).

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