In this project, we review the global (or rigid) supersymmetry algebra (N= 1 and N=2), Lagrangian supersymmetry fields and its simple supersymmetry transformations. Then we also discuss the local supersymmetry or supergravity briefly including field spin-3/2. We study spontaneous N=2->N=1 supersymmetry breaking in supergravity within consideration of Minkowski background (flat space without cosmological constant). We develop new representation of dual supersymmetry from Ferrara Nieuwenhuizen N=1 massive spin-3/2 supersymmetry which has multiplet (3/2, 1, 1, 1/2) by means Poincare duality of massive vector fields. We couple the supergravity coupling upon each of the supersymmetry representations. Non-linear realization is responsible for partial supersymmetry breaking was studied. We used UnHiggs technique upon each representation and find a new N=2 supergravity for Minkowski background appear...

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The first part of this work focuses on the canonical group quantization approach ap­plied to non-commutative spaces, namely plane 1R2 and two-torus T2. Canonical group quantization is a quantization approach that adopts the group structure that respects the global symmetries of the phase space as a main ingredient. This is fol­lowed by finding its unitary irreduciblere presentations. The use of non commutative space is motivated by the idea of quantum sub structure of space leading to nontrivial modification of the quantization. Extending to noncommuting phase space includes noncommuting momenta that arises naturally in magnetic background as in Landau problem. The approach taken is to modify the symplectic structures corresponding to the noncommutative plane, noncomrnutative phase space and noncommutative torus and obtain their canonical groups. In all cases, the canonical group is found to be central extensions of the Heisenberg group. Next to consider is to generalize the approach.....

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