Dirac versus reduced quantization and operator ordering.pdf

a r X i v : g r - q c / 9 6 0 1 0 4 2 v 1 2 6 J a n 1 9 9 6 Dirac versus reduced quantization and operator ordering K Shimizu? Center for mathematical sciences, The University of Aizu, Aizu-Wakamatsu City, Fukushima, 965 Japan Abstract We show an equivalence between Dirac quantization and the reduced phase space quantization. The equivalence of the both quantization methods de- termines the operator ordering of the Hamiltonian. Some examples of the operator ordering are shown in simple models. I. INTRODUCTION In recent years some authors [1] are discussing on Dirac quantization and the reduced phase space quantization. Their arguments are that the reduced phase space quantization and Dirac quantization may be different in the constraint system with a non-trivial metric. In order to clarify the problem, let us consider the simplest model, as an example. Lagrangian is given by L = 1 2 x?2 + f(x) 2 (y? ? λ)2 (1) where λ is a Lagrange multiplier. There is a non-trivial metric f(x). This is not a field theory but a quantum mechanics. The Hamiltonian of this system is H = 1 2 p2x + 1 2f(x) p2y + λpy (2) and there are two constraints pλ ≡ π ≈ 0, (3) py ≈ 0. (4) ?e-mail address: shimizu@u-aizu.ac.jp 1 These are first-class constrains. We set py = 0 in the Hamiltonian before the quantization. Then the Hamiltonian reduces to H = 1 2 p2x (5) and the Hamiltonian operator is H? = 1 2 ?2x. (6) This is the reduced phase space quantization. The procedure of the reduced phase space quantization is to reduce first and then quantize. In the case of Dirac quantization, its procedure is to quantize first and then reduce. The Hamiltonian in this model is defined on the two-dimensional space of x and y with- out a constraint term. To ensure the invariance under the coordinate transformation, the Hamiltonian operator is written by H? = 1 2 √ f ?x √ f?x + 1 2 √ f ?y √ f 1 f ?y, (7) where √ f is √ detgμν . The metric gμν is the two-dimensional metric of x-y space. Since p?y = ?y ≈ 0, y