Cosmological horizons and reconstruction of quantum field theories

Abstract

As a starting point, we state some relevant geometrical properties enjoyed by the cosmological horizon of a certain class of Friedmann-Robertson-Walker backgrounds. Those properties are generalised to a larger class of expanding spacetimes $M$ admitting a geodesically complete cosmological horizon $\scrim$ common to all co-moving observers. This structure is later exploited in order to recast, in a cosmological background, some recent results for a linear scalar quantum field theory in spacetimes asymptotically flat at null infinity. Under suitable hypotheses on $M$, encompassing both the cosmological de Sitter background and a large class of other FRW spacetimes, the algebra of observables for a Klein-Gordon field is mapped into a subalgebra of the algebra of observables $\cW(\scrim)$ constructed on the cosmological horizon. There is exactly one pure quasifree state $\lambda$ on $\cW(\scrim)$ which fulfils a suitable energy-positivity condition with respect to a generator related with the cosmological time displacements. Furthermore $\lambda$ induces a preferred physically meaningful quantum state ${\lambda }_M$ for the quantum theory in the bulk. If $M$ admits a timelike Killing generator preserving $\scrim$, then the associated self-adjoint generator in the GNS representation of ${\lambda }_M$ has positive spectrum (i.e. energy). Moreover ${\lambda }_M$ turns out to be invariant under every symmetry of the bulk metric which preserves the cosmological horizon. In the case of an expanding de Sitter spacetime, ${\lambda }_M$ coincides with the Euclidean (Bunch-Davies) vacuum state, hence being Hadamard in this case. Remarks on the validity of the Hadamard property for ${\lambda }_M$ in more general spacetimes are presented.

32 pages, 1 figure, to appear on Comm. Math. Phys., dedicated to Professor Klaus Fredenhagen on the occasion of his 60th birthday

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