Generalized Projection Operator Method of Constrained Systems

Alternative Approach to Noncommutative Quantum Mechanics on a Curved Space

“…Following our previous works [2,7,8], we here present the brief review of the constraint star-product quantization formalism of the POM together with the explanation of the notations appearing hereafter.…”

“…, the redundant variables [6] associated to the noncommutativities among the resultant canonically conjugate set of the system. Starting with the Lagrangian mentioned above, we shall construct the noncommutative quantum mechanics on the curved space through the constraint star-product quantization formalism of the projection operator method (POM) [7,8]. Imposing the additional constraints to eliminate the redundant degrees of freedom, then, the noncommutative quantum system with the noncommutativity among the coordinates x on the curved space will be constructed exactly and it will be shown in the exact form that the Hamiltonian contains the quantum correction terms due to the noncommutativity associated to the constraint-opetator G(x) and the successive projections of the Hamiltonian.…”

Star-product Description of Quantization in Second-class Constraint Systems

“…From the Darboux's theorem in dynamical systems, we can, in principle, construct the canonically conjugate set in terms of T α (C), which we call the associated canonically conjugate set (ACCS) [11,12,13].…”

“…In the case of second-class constraints, the quantization of the constraint systems in the canonical formalism has usually been accomplished by using the generalized Hamiltonian formalism with the Dirac bracket [3], which is in the approach I. There have been proposed many formalism within approach II; the operator formalism developed by Batalin and Fradkin [4,5,10], the projection operator formalism [11,12,13,14,15,16], the algebraic approach proposed by Ohnuki and Kitakado [17] and the first-order singular Lagrangian formalism proposed by Faddeev and Jackiw [18,19]. Using the projection operator method [11,12] (POM), then, we have shown that the operator-algebra in the approach II contain the quantum corrections caused by the noncommutativity in the re-ordering of constraint-operators in the products of operators, which can never be obatined by the approach I.…”

“…There have been proposed many formalism within approach II; the operator formalism developed by Batalin and Fradkin [4,5,10], the projection operator formalism [11,12,13,14,15,16], the algebraic approach proposed by Ohnuki and Kitakado [17] and the first-order singular Lagrangian formalism proposed by Faddeev and Jackiw [18,19]. Using the projection operator method [11,12] (POM), then, we have shown that the operator-algebra in the approach II contain the quantum corrections caused by the noncommutativity in the re-ordering of constraint-operators in the products of operators, which can never be obatined by the approach I. Alternative formalism of quantization is the Wely-Wigner-Moyal (WWM) quantization [20,21], and has been investigated for the constraint systems [22,23,24,25].…”