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

Abstract

In a previous paper (hep-th/0407256) local scalar QFT (in Weyl algebraic approach) has been constructed on degenerate semi-Riemannian manifolds $\bS^1\times Σ$ corresponding to the extension of Killing horizons by adding points at infinity to the null geodesic forming the horizon. It has been proved that the theory admits a natural representation of $PSL(2,\bR)$ in terms of $\ast$-automorphisms and this representation is unitarily implementable if referring to a certain invariant state $\hat{I}»$. Among other results it has been proved that the theory admits a class of inequivalent algebraic (coherent) states $\{\hat{I}{»}_{\hat{I}}\P \}$, with $\hat{I}\P \in L^2(\hat{I}\pounds )$, which break part of the symmetry, in the sense that each of them is not invariant under the full group $PSL(2,\bR)$ and so there is no unitary representation of whole group $PSL(2,\bR)$ which leaves fixed the cyclic GNS vector. These states, if restricted to suitable portions of $\bM$ are invariant and extremal KMS states with respect a surviving one-parameter group symmetry. In this paper we clarify the nature of symmetry breakdown. We show that, in fact, {\em spontaneous} symmetry breaking occurs in the natural sense of algebraic quantum field theory: if $\hat{I}\P \ne 0$, there is no unitary representation of whole group $PSL(2,\bR)$ which implements the $\ast$-automorphism representation of $PSL(2,\bR)$ itself in the GNS representation of $\hat{I}{»}_{\hat{I}}\P$ (leaving fixed or not the state).

7 pages, title changed, in print in Journal of Mathematical Physics