This article is an introductory presentation of the quantization of the half-plane based on affine coherent states (ACS). The half-plane is viewed as the phase space for the dynamics of a positive physical quantity evolving with time, and its affine symmetry is preserved due to the covariance of this type of quantization. We promote the interest of such a procedure for transforming a classical model into a quantum one, since the singularity at the origin is systematically removed, and the arbitrariness of boundary conditions can be easily overcome. We explain some important mathematical aspects of the method. Three elementary examples of applications are presented, the quantum breathing of a massive sphere, the quantum smooth bouncing of a charged sphere, and a smooth bouncing of "dust" sphere as a simple model of quantum Newtonian cosmology.

We address the issue of fermionic particle creation in cosmological bouncing models governed by General Relativity, but where the bounce itself takes place due to quantum effects. If the energy scale of the bounce is not very close to the Planck energy, the Wheeler-DeWitt approach can be used to furnish sensible singularity-free background models with a contracting phase preceding an expanding phase in which the fermions evolve. The fermionic fields considered are massive, neutral and minimally coupled to gravity. We are particularly interested in neutrinos, neutrons and heavy neutrinos motivated by theories beyond the Standard Model of Particle Physics. We perform a numerical analysis for a bouncing model containing radiation and a pressureless fluid.The results reveal that the fermionic production is very small, with no back-reaction, unless the fermions are very heavy with masses up to 10 9 GeV. Hence, investigations concerning baryogenesis in such bouncing models should either go beyond the minimal coupling between gravity and the fermionic fields considered here, or assume the existence of such heavy fermions as a starting point. * arthur@cbpf.br † lfog@cbpf.br ‡ nelson.

It is widely known that bouncing models with a dust hydrodynamical fluid satisfying quencies. Indeed, we show that the energy density of primordial gravitational waves is proportional to k 2(9λ−1)/(1+3λ) for wavelengths which become bigger than the Hubble radius when this extra fluid dominates the background. Hence, as λ → 1 (an almost stiff matter fluid), the energy density of primordial gravitational waves will increase faster in frequency, turning them potentially detectable at high frequencies. However, there is an extra factor I q (λ) in the amplitude which decreases exponentially with λ. The net effect of these two contributions turns the energy density of primordial gravitational waves not sufficiently big at high frequencies in order to be detected by present day or near future observations for models which satisfy the nucleosynthesis bounds and is symmetric with respect to the bounce. Hence, symmetric bouncing models where the background is dominated by a dust hydrodynamical fluid with small sound velocity, do not present any significant amount of primordial gravitational waves at any frequency range compatible with observations, even if there are other fields present in the model dominating the bounce phase. Any detection of such waves will then rule out this kind of models. * nelsonpn@cbpf.br † arthur@cbpf.br

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