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

Valter Moretti
November 29, 2005
In a previous paper (hep-th/0407256) local scalar QFT (in Weyl algebraic approach) has been constructed on degenerate semi-Riemannian manifolds ${\mathbb S}^1\times \Sigma$ 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 $PSL(2,{\mathbb R})$ in terms of $*$-automorphisms and this representation is unitarily implementable if referring to a certain invariant state $\lambda$. Among other results it has been proved that the theory admits a class of inequivalent algebraic (coherent) states $\{\lambda_\zeta\}$, with $\zeta\in L^2(\Sigma)$, which break part of the symmetry, in the sense that each of them is not invariant under the full group $PSL(2,{\mathbb R})$ and so there is no unitary representation of whole group $PSL(2,{\mathbb R})$ which leaves fixed the cyclic GNS vector. These states, if restricted to suitable portions of ${\mathbb M}$ 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 $\zeta \neq 0$, there is no unitary representation of whole group $PSL(2,{\mathbb R})$ which implements the $*$-automorphism representation of $PSL(2,{\mathbb R})$ itself in the GNS representation of $\lambda_\zeta$ (leaving fixed or not the state).
open access link J.Math.Phys. 47 (2006) 032302
@article{Moretti:2005fs, author = "Moretti, Valter", title = "{Bose-Einstein condensate and spontaneous breaking of conformal symmetry on Killing horizons. II.}", journal = "J. Math. Phys.", volume = "47", year = "2006", pages = "032302", doi = "10.1063/1.2173174", eprint = "gr-qc/0511155", archivePrefix = "arXiv", primaryClass = "gr-qc", reportNumber = "UTM-686", SLACcitation = "%%CITATION = GR-QC/0511155;%%" }