Kontorovich-Lebedev-Fourier Space for de Sitter Correlators

TL;DR

This paper introduces a new frequency-momentum space for de Sitter correlators using the Kontorovich-Lebedev-Fourier transform, simplifying perturbative calculations and revealing the underlying group-theoretical structure.

Contribution

It constructs a novel KLF space for de Sitter correlators from first principles, connecting representation theory with practical perturbation techniques.

Findings

Reproduces the Källén-Lehmann representation of two-point functions.

Derives Feynman rules in KLF space for in-in perturbation theory.

Shows propagators are simple rational functions and diagrams as spectral integrals.

Abstract

In this work, we build a novel frequency-momentum space for $(d+1)$-dimensional de Sitter (dS) correlators from first principles. This construction follows directly from the decomposition into unitary irreducible representations (UIRs) of the spacetime isometry group $\mathrm{SO}(1,d+1)$. While the spatial momentum space is given by the standard $d$-dimensional Fourier transform, the frequency space arises from diagonalising the quadratic Casimir operator, leading to the $(d+1)$-dimensional Kontorovich-Lebedev-Fourier (KLF) transform. We show that square-integrable functions decompose only along the principal series, whereas more general functions can receive discrete contributions from other UIRs. Applying this framework to the bulk CFT two-point function reproduces its K\"all\'en-Lehmann representation. Using the path integral formulation, we derive the Feynman rules for in-in perturbation theory in KLF space, leading to the introduction of KLF-space correlators, which are simply related to late-time correlation functions through a reduction formula. Furthermore, the KLF-space formulation sheds light on the simple mathematical structure of perturbative computations. In particular, the propagators take the form of simple rational functions, and tree-level diagrams can be written as spectral integrals over known meromorphic functions, as demonstrated in the example of the single-exchange four-point function. At the loop level, we show, through the example of the self-energy correction to the scalar propagator, that the group-theoretical nature of the construction allows the momentum integral to be recast as an orthogonality relation among $\mathrm{SO}(1,d+1)$ Clebsch-Gordan coefficients.