De Sitter Momentum Space

TL;DR

This paper develops a nonperturbative momentum space framework for quantum field theory in de Sitter spacetime, simplifying calculations and revealing structural similarities to flat space.

Contribution

It introduces the Kontorovitch-Lebedev-Fourier (KLF) space, providing a harmonic expansion adapted to de Sitter symmetry and streamlining perturbative computations.

Findings

KLF space shares properties with Minkowski momentum space.

Equations of motion become algebraic in KLF space.

Perturbative in-in correlators are simplified using this framework.

Abstract

We construct a natural and nonperturbative momentum space for quantum field theory on $(d+1)$-dimensional de Sitter (dS) spacetime in the Poincar\'e slicing, adapted to early Universe cosmology. In particular, we identify the dS frequency as the unitary-representation label of the dS isometry group $\mathrm{SO}(1, d+1)$. By diagonalizing the quadratic Casimir together with spatial translations, we provide a harmonic expansion of operators in what we call the Kontorovitch-Lebedev-Fourier (KLF) space. This momentum space shares many structural properties with its Minkowski counterpart, for instance: equations of motion reduce to algebraic equations, and the quadratic dynamics provides a simple propagator analogous to flat space. We reformulate the perturbative computation of in-in correlators in KLF momentum space, showing from first principles how time integrals turn into frequency-space integrals over meromorphic functions. We show how our construction streamlines computations, naturally accommodates the contributions from principal and complementary series in the K\"all\'en-Lehmann spectral decomposition of composite operators, and leads to a group-theoretical method to evaluate loop momentum integrals.