Two Dimensional Plane, Modified Symplectic Structure and Quantization

Noncommutative quantum mechanics on the plane has been widely studied in the literature. Here, we consider the problem using Isham's canonical group quantization scheme for which the primary object is the symmetry group that underlies the phase space. The noncommutativity of the configuration space coordinates requires us to introduce the noncommutative term in the symplectic structure of the system. This modified symplectic structure will modify the group acting on the configuration space from abelian $\mathbb{R}^2$ to a nonabelian one. As a result, the canonical group obtained is a deformed Heisenberg group and the canonical commutation relation (CCR) corresponds to what is usually found in noncommutative quantum mechanics.Comment: 5 pages. Submitted to Jurnal Fizik Malaysi

Two dimensional plane, modified symplectic structure and quantization

Noncommutative quantum mechanics on the plane has been widely studied in the literature. Here, we consider the problem using Isham's canonical group quantization scheme for which the primary object is the symmetry group that underlies the phase space. The noncommutativity of the configuration space coordinates requires us to introduce the noncommutative term in the symplectic structure of the system. This modified symplectic structure will modify the group acting on the configuration space from abelian R^2 to a nonabelian one. As a result, the canonical group obtained is a deformed Heisenberg group and the canonical commutation relation (CCR) corresponds to what is usually found in noncommutative quantum mechanics

Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension

The first part of this work focuses on the canonical group quantization approach applied to non-commutative spaces, namely plane R2 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 irreducible representations. The use of noncommutative space is motivated by the idea of quantum substructure 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, noncommutative 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 to twisted phase spaces where it employs the technique of Drinfeld twist on the Hopf algebra of the system. The result illustrates that a tool from the deformation quantization can be used in canonical group quantization where the deformed Heisenberg group H2 q is obtained and its representation stays consistent with the discussion in the literature. In the second part, the two-parameter deformations of quantum group for Heisenberg group and Euclidean group are studied. Both can be achieved through the contraction procedure on SU(2)q;p quantum group. The study also continues to develop (q; p)-extended Heisenberg quantum group from the previous result. As conclusion, it is shown that the extensions of Heisenberg group arise from quantizing noncommutative plane, noncommutative phase space, noncommutative two-torus, and twisted phase space. The work on two-parameter deformation of quantum group also further shows generalizations of the extension of Heisenberg group