Quantization of the Schwarzschild black hole: a Noether symmetry approach

We study the canonical formalism of a spherically symmetric space-time. In the context of the 3+1 decomposition with respect to the radial coordinate $r$, we set up an effective Lagrangian in which a couple of metric functions play the role of independent variables. We show that the resulting $r$-Hamiltonian yields the correct classical solutions which can be identified with the Schwarzschild black hole. The Noether symmetry of model is then investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. According to the Noether symmetry approach, we also quantize the model and show that the existence of a Noether symmetry yields a general solution to the Wheeler-DeWitt equation where exhibits a good correlation with the classical regime. We use the resulting wave function in order to (qualitative) investigate the possibility of the avoidance of classical singularities.

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