On the Schrödinger Equation for the Minisuperspace Models

We obtain a time-dependent Schrödinger equation for the Friedmann Robertson Walker (FRW) model interacting with a homogeneous scalar matter field. We show that for this purpose it is necesary to include an additional action invariant under the reparametrization of time. The last one does not change the equations of motion of the system, but changes only the constraint which at the quantum level becomes time-dependent Schrödinger equation. The same procedure is applied to the supersymmetric case and the supersymmetric quantum constraints are obtained, one of them is a square root of the Schrödinger operator. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] One of the most important questions in quantum cosmology is that of identifying a suitable time parameter [1] and a time-dependent Wheeler-DeWitt equation [2, 3]. The main peculiarity of the gravity theory is the presence of non-physical variables (gauge variables) and constraints [3, 4, 5, 6]. They arise due to the general coordinate invariance of the theory. The conventional Wheeler-DeWitt formulation gives a time independent quantum theory [7]. The canonical quantization of the minisuperspace approximation [8] has been used to find results in the hope, that they would illustrate the behaviour of general relativity [9]. In the minisuperspace models [2, 7] there is a residual invariance under reparametrization of time (worldline symmetry). Due to this fact the equation that governs the quantum behaviour of these models is the Schrödinger equation for states with zero energy. On the other hand, supersymmetry transformations are more fundamental than time translations (reparametrization of time) in the sense, that these ones may be generated by anticommutators of the supersymmetry generators. The recent introduction of supersymmetric minisuperspace models has led to the square root equations for states with zero energy [10, 11, 12]. The structure of the world-line supersymmetry or the world-line supersymmetry transformations has led to the zero Hamiltonian phenomena [2, 6, 12]. Investigations of the time evolution problem for such quantum systems have been carried in two directions: the cosmological models of gravity have been quantized by reducing the phase space degrees of freedom [13, 14, 15, 16, 17] and with the help of the WKB approach [18, 19]. In this work we obtain a time-dependent Schrödinger equation for the homogeneous cosmological models. In our approach this equation arises due to an additional action invariant under reparametrization. The last one does not change the equations of motion, but the constraint which becomes time-dependent Schrödinger equation. In the case of the supersymmetric minisuperspace model we obtain the supersymmetric constraints, one of them is a square root of time-dependent Schrödinger equation. We begin by considering an homogeneous and isotropic metric defined by ds = −N(t)dt +R(t)dΩ3, (1) where the only dynamical degree of freedom is the scale factor R(t). The lapse function N(t), being a pure gauge variable, is not dynamical. The quantity dΩ3 is the standard line element on the unit three-sphere. We shall set c = h̄ = 1. The pure gravitational action corresponding to the metric (1) is Sg = 6 κ2 ∫ ( − 2 2N + 1 2 kNR )