We consider a real, massive scalar field in BTZ spacetime, a 2+1-dimensional black hole solution of the Einstein's field equations with a negative cosmological constant. First, we analyze the space of classical solutions in a mode decomposition and we characterize the collection of all admissible boundary conditions of Robin type which can be imposed at infinity. Secondly, we investigate whether, for a given boundary condition, there exists a ground state by constructing explicitly its two-point function. We demonstrate that for a subclass of the boundary conditions it is possible to construct a ground state that locally satisfies the Hadamard property. In all other cases, we show that bound state mode solutions exist and, therefore, such construction is not possible. * francesco.bussola01@ateneopv.it † claudio.dappiaggi@unipv.it ‡ hugo.ferreira@pv.infn.it § igor.khavkine@unimi.it arXiv:1708.00271v1 [gr-qc] 1 Aug 2017
We analyse the local behaviour of the two-point correlation function of a quantum state for a scalar field in a neighbourhood of a Killing horizon in a 2 + 1-dimensional spacetime, extending the work of Moretti and Pinamonti in a 3 + 1-dimensional scenario. In particular we show that, if the state is of Hadamard form in such neighbourhood, similarly to the 3 + 1-dimensional case, under a suitable scaling limit towards the horizon, the two-point correlation function exhibits a thermal behaviour at the Hawking temperature. Since the whole analysis rests on the assumption that a Hadamard state exists in a neighbourhood of the Killing horizon, we show that this is not an empty condition by verifying it for a massive, real scalar field subject to Robin boundary conditions in the prototypic example of a three dimensional black hole background: the non-extremal, rotating BTZ spacetime. arXiv:1806.00427v1 [gr-qc] 1 Jun 2018The Hadamard condition has been fist introduced as a selection criterion for physically admissible quantum states, since on the one hand it guarantee that the short-distance behaviour of the state coincides with that of the Poincaré vacuum, while, on the other hand, it ensures that the quantum fluctuations of all observables are finite and that there exists a covariant scheme to construct Wick polynomials, [8,9,10,11].Subsequently, in a very influential paper by Haag and Frendehagen [12], it has been shown that the Hadamard condition is deeply connected to the presence of Hawking radiation, although, in this paper, the focus was on the role of radiative modes. Only in [7] the attention has been switched towards the analysis of the interplay between Hadamard states and the thermal properties of field theoretical models in a neighbourhood of a Killing horizon. The key ingredient in this paper has been the integral kernel of a two-point function ω 2 (x, y). In particular, using the structural properties of a Killing horizon [11,13] and provided that ω 2 is of Hadamard form ina neighborhood of the horizon, it has been proven that, independently of the choice of such bi-distribution, ω 2 acquires a thermal spectrum with respect to the notion of time and energy associated with the Killing field, generating the horizon. This result is obtained by smearing ω 2 with two test functions whose support is close to the horizon. More precisely, if the points x and y of the integral kernel of ω 2 lie on the same side of the horizon, the Fourier transform of the twopoint function displays a Bose factor at the Hawking temperature. Conversely, if x and y are kept at the opposite sides of the horizon, the resulting spectrum agrees with the transition probability between two weakly coupled reservoirs which are in thermal equilibrium at the Hawking temperature, in perfect agreement with the predictions of [2].Yet a close scrutiny of [2] unveils that their analysis can be applied to any spacetime with a Killing horizon, regardless of the dimension of the background, while the results of [7] are based on the leading singular ...
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