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

Valter Moretti
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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 $*$-automorphisms and this representation is unitarily implementable if referring to a certain invariant state $λ$. Among other results it has been proved that the theory admits a class of inequivalent algebraic (coherent) states $\{λ_ζ\}$, with $ζ\in L^2(Σ)$, 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 $ζ\neq 0$, there is no unitary representation of whole group $PSL(2,\bR)$ which implements the $*$-automorphism representation of $PSL(2,\bR)$ itself in the GNS representation of $λ_ζ$ (leaving fixed or not the state).

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BibTeXRIS

Valter Moretti. 2006-03-08. Bose-Einstein Condensate and Spontaneous Breaking of Conformal Symmetry on Killing Horizons II. https://doi.org/10.1063/1.2173174

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