Bose–Einstein condensate and spontaneous breaking of conformal symmetry on Killing horizons

Local scalar quantum field theory (in Weyl algebraic approach) is constructed on degenerate semi-Riemannian manifolds corresponding to Killing horizons in spacetime. Covariance properties of the $C*$-algebra of observables with respect to the conformal group $PSL(2,R)$ are studied. It is shown that, in addition to the state studied by Guido, Longo, Roberts, and Verch for bifurcated Killing horizons, which is conformally invariant and KMS at Hawking temperature with respect to the Killing flow and defines a conformal net of von Neumann algebras, there is a further wide class of algebraic (coherent) states representing spontaneous breaking of $PSL(2,R)$ symmetry. This class is labeled by functions in a suitable Hilbert space and their GNS representations enjoy remarkable properties. The states are nonequivalent extremal KMS states at Hawking temperature with respect to the residual one-parameter subgroup of $PSL(2,R)$ associated with the Killing flow. The KMS property is valid for the two local subalgebras of observables uniquely determined by covariance and invariance under the residual symmetry unitarily represented. These algebras rely on the physical region of the manifold corresponding to a Killing horizon cleaned up by removing the unphysical points at infinity [necessary to describe the whole $PSL(2,R)$ action]. Each of the found states can be interpreted as a different thermodynamic phase, containing Bose–Einstein condensate, for the considered quantum field. It is finally suggested that the found states could describe different black holes.