ct 2 00 5 Hawking radiation and Quasinormal modes

The spectrum of Hawking radiation by quantum fields in the curved spacetime is continuous, so the explanation of Hawking radiation using quasinormal modes can be suspected to be impossible. We find that quasinormal modes do not explain the relation between the state observed in a region far away from a black hole and the short distance behavior of the state on the horizon. Loop quantum gravity [1] is a background independent, non-perturbative approach to unify general relativity and quantum physics. There has been a lot of progress, especially in the resolution of the big-bang singularity [2], even though we have physical states only in the kinematical level in loop quantum gravity and solving the hamiltonian constraint [3] is still a open problem and the description of low energy physics [6] is not fully understood yet. In loop quantum gravity, there is the Immirzi parameter γ [7]. Neither phase space variables nor their Poisson brackets depend on this parameter. Thus the canonical phase space is γ independent. Therefore there is no ambiguity in quantization [9]. However the expression of the geometrical fields -the spatial triad and the extrinsic curvaturein terms of canonical variables depends on it, so to fix the value of γ is important to figure out the correct semiclassical limit. One of the ways to fix the value of the Immirzi parameter was proposed using SO(3) gauge group instead of SU(2) [10]. This was motivated by observation [11] Even though it still remains if physical states constructed with the master constraint operator contain correct semiclassical states, Thiemann showed the existence of the self-adjoint, positive master constraint operator for loop quantum gravity and we have a good chance to solve this open problem [4]. See also [5]. This parameter does not appear in the equations of motion. Recently it was shown that in the presence of fermions, it appears in the equations of motion [8]. on the quasinormal modes of a black hole with using Bohr’s correspondence principle: “Transition frequencies at large quantum numbers should be equal to classical oscillation frequencies.” Even though a fair argument was made to specify the imaginary part of quasinormal mode as a quantum number in term of the relaxation time [11], there is a criticism for using the correspondence principle [12]. If we can find any quantum mechanical role of the quasinormal modes, above observation would be supported without the correspondence principle. However we find that there is no Hawking radiation by quasinormal modes in the framework of quantum field theory in curved spacetime. Fredenhagen and Haag derived Hawking radiation [13] by the local behavior of the correlation functions, < φ(x1) · · ·φ(xn) > , of the quantum field [14]. What they found was that the asymptotic counting rate of the two-point correlation function of the smeared quantum field living on a Schwarzschild black hole background is governed by the short distance behavior of the ground state two-point function near the horizon, so-called Hadamard form [15]. The smeared field