Bose-Einstein condensate and Spontaneous Breaking of Conformal Symmetry on Killing Horizons

TL;DR

This paper constructs local scalar quantum field theories on Killing horizons, revealing spontaneous conformal symmetry breaking and Bose-Einstein condensate phases, which may correspond to different black hole states.

Contribution

It introduces a class of algebraic states on Killing horizons exhibiting spontaneous conformal symmetry breaking and Bose-Einstein condensation, expanding understanding of quantum fields in curved spacetime.

Findings

Existence of a wide class of algebraic states breaking PSL(2,R) symmetry.

States are extremal KMS at Hawking temperature with residual symmetry.

States can be interpreted as different thermodynamic phases with Bose-Einstein condensate.

Abstract

Local scalar QFT (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 $PSL(2,\bR)$ 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 $PSL(2,\bR)$ symmetry. This class is labeled by functions in a suitable Hilbert space and their GNS representations enjoy remarkable properties. The states are non equivalent extremal KMS states at Hawking temperature with respect to the residual one-parameter subgroup of $PSL(2,\bR)$ associated with the Killing flow. The KMS property is valid for the two local sub algebras 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 $PSL(2,\bR)$ 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.