In this paper, we investigate geometric Quantization of massive superparticles in four dimensional space-time, which preserves $frac{1}{4}$ of the target space supersymmetry. Because the black holes (also massive black holes) are strong gravitational system, hence, application of such Quantization to the supersymmetric cosmological black holes would give us information about quantum gravity. As such supermassive black holes are candidate of dark matter hence our calculations are important from cosmological point of view. The world-line action of this model contains a Wess-Zumino term which breaks $d=4$ Lorentz symmetry. We solve Hamiltonian equation and obtain unique solution, which help to calculate preQuantization operator. It yields to corresponding Dirac equation which is important in relativistic quantum mechanics. It will be useful in the second Quantization of the same models. On the other hand superparticles are itself candidate of dark matter and dark matter are important from their gravitation effects, hence their Quantization may yields to quantum gravity theory.

In this paper we try to formulate the Berezin Quantization on projective Hilbert space P(H) and use its geometric structure to construct a correspondence between a given classical theory and a given quantum theory. It will be shown that the star product in Berezin Quantization is equivalent to the Poisson bracket on coherent states manifold M, embedded in P(H), and the Berezin method is used to define a classical limit for geometric quantum mechanics. With this construction to all of the quantum observables are associated their covariant symbols, which form a poisson algebra on P(H) and since the corresponding classical phase space has a natural Poisson structure, the Berezin Quantization is then a systematic procedure to relate these two piosson algebras.

We prove that a dictionary for a Banach space X has the strong coefficient Quantization property if it has the same property when it restricted on the unit ball of X. We also obtain the same result for the strong net Quantization property.

The recent observational data are strongly in favor of a positive acceleration of the present universe. In a first approximation, the background space-time that we live in, might be considered as a de Sitter space-time. If this is so, it is important to understand how quantum gravity and should formulated in de Sitter space. We discusses the some properties of linear quantum gravity necessitated to the indefinite metric Quantization for preserving the de Sitter invariant and eliminating the infrared divergence in the graviton two points function. There appears two different types of negative norm states. The first one is similar to those non physical states dual to gauge states which appear in gauge quantum field theory in Minkowski space. The other is due to the occurrence of negative frequency solutions in order to preserve de Sitter covariance. This is similar to the case of minimally coupled scalar field in dS space. We discuss the possibility of generalization to the Minkowskian limit and their consequences in the renormalization theory.

In this article, the quantum mechanical formulation in extended phase space has been generalized to the quantized many body systems with degrees of freedom limited to the evolution of classical and quantum mechanical fields. This makes the investigation of such fields possible in phase space. As an example, the problem of vibrating string has been solved using this formalism.

In this article, the quantum mechanical formulation in phase space has been used to solve the damped harmonic oscillator. The applied hamiltonian which is obtained from integrating the equations of motion, gives a reasonable interpretation of the concept of phase space extension based on damped oscillator and its mirror image, so that the energy lost by the damped oscillator is absorbed by its mirror image with the same rate of absorption and these two together form a conservative system.