Black Holes and Quantum Mechanics

The motion of a particle near the RN black hole horizon is described by conformal mechanics. Models of this type have no ground state with vanishing energy. This problem was resolved in past by a redefinition of the Hamiltonian which breaks translational time invariance but gives a normalizable ground state. We show that this change of the Hamiltonian is a quantum mechanical equivalent of the change of coordinates near the black hole horizon removing the singularity. The new Hamiltonian of quantum mechanics is identified as an operator of a rotation between 2 time-like coordinates of the adS hypersurface which translates global time. Therefore conformal quantum mechanics may eventually help to resolve the puzzles of the classical black hole physics. To appear in proceedings of the 22nd Johns Hopkins Workshop, Gothenburg, 1998. 1. Recently a surprising connection [1] between black holes and conformal mechanics of De Alfaro, Fubini and Furlan [2] (DFF) and superconformal [3] mechanics have been established. The dynamics of a (super)particle near the horizon of an extreme Reissner-Nordström black hole was shown to be governed by an action that reduces to a (super)conformal mechanics model in the limit of a large black hole charge. The Hamiltonian, and the rest of the generators of the conformal group were found in [1] to be equal to H = p 2f + g 2x2f , K = f x 2 , D = xp+ px 4 . (1) Here the function f(x, p) in the limit of the large black hole charge Q tends to 1 and we find a complete agreement with the conformal model [2] in this limit. Such models upon quantization give the commutators of the conformal algebra [H,D] = iH , [K,D] = −iK , [H,K] = 2iD . (2) This quantum mechanical model was shown in [2] to a have a continuous spectrum of energy eigenstates with energy eigenvalue E > 0, but there is no ground state at E = 0. In ref. [1] a black hole interpretation of this model was suggested. The classical analog of an eigenstate of H is an orbit of a timelike Killing vector field k, equal to ∂/∂t in the region outside the horizon, and the energy is then the value of k. The absence of a ground state of H at E = 0 can now be interpreted as due to the fact that the orbit of k with k = 0 is a null geodesic generator of the event horizon, which is not covered by the static coordinates adapted to ∂t. The procedure used by DFF to cure the problem of the absence of a ground state was to choose a different combination of conserved charges as the Hamiltonian. From the perspective of the quantum mechanics this shift of the Hamiltonian is unusual and motivated by the fact that when the theory does not have a ground state, an additional prescription may be used such that the new theory has a well defined vacuum. It has been realized that such additional prescription leads to the breakdown of time translational invariance in favor of maintaining a well-defined vacuum and an anti de Sitter group. Various aspects of black holes and ADS2/CFT1 duality were studied in [4].