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Publication# The Källén-Lehmann representation in de Sitter spacetime

Abstract

We study two-point functions of symmetric traceless local operators in the bulk of de Sitter spacetime. We derive the Kallen-Lehmann spectral decomposition for any spin and show that unitarity implies its spectral densities are nonnegative. In addition, we recover the Kallen-Lehmann decomposition in Minkowski space by taking the flat space limit. Using harmonic analysis and the Wick rotation to Euclidean Anti de Sitter, we derive an inversion formula to compute the spectral densities. Using the inversion formula, we relate the analytic structure of the spectral densities to the late-time boundary operator content. We apply our technical tools to study two-point functions of composite operators in free and weakly coupled theories. In the weakly coupled case, we show how the Kallen-Lehmann decomposition is useful to find the anomalous dimensions of the late-time boundary operators. We also derive the Kallen-Lehmann representation of two-point functions of spinning primary operators of a Conformal Field Theory on de Sitter.

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Conformal field theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

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