Quantum Gravity, de Sitter Space, and Normalizability

TL;DR

This work addresses the perturbative normalizability of the Lorentzian Chern-Simons-Kodama state in de Sitter space by employing a holomorphic inner product derived from the reality conditions of self-dual Ashtekar variables. It shows that the linearized CSK state is perturbatively normalizable for super-Planckian $\Lambda$, and proposes a phase-space Wick rotation (Thiemann complexifier) that analytically continues divergent modes to render the full perturbative state normalizable across a broad range of $\Lambda$. The results provide a concrete realization of a CPT-breaking yet normalizable gravitational vacuum within a nonperturbative canonical quantum gravity framework and open a path toward a viable quantum cosmology in the Ashtekar formalism. The authors discuss regularization issues, potential global-de Sitter extensions, and how combining phase-space rotations with compact spatial topology could yield a fully well-defined ground state for quantum gravity with $\Lambda>0$.

Abstract

We propose a resolution to the longstanding problem of perturbative normalizability in canonical quantum gravity of the Lorentzian Chern-Simons-Kodama (CSK) state with a positive cosmological constant in four dimensions. While the CSK state is an exact solution to the Hamiltonian constraint in the self-dual formulation and semiclassically describes de Sitter spacetime, its physical viability has been questioned due to apparent nonnormalizability and CPT asymmetry. Starting from a nonperturbative holomorphic inner product derived from the reality conditions of the self-dual Ashtekar variables, we show that the linearization, in terms of gravitons, of the CSK state is perturbatively normalizable for super-Planckian cosmological constant. Furthermore, we demonstrate that a rotation in phase space, a generalization of Thiemann's complexifier, can render the full perturbative state normalizable for all $Λ$ by analytically continuing the non-convergent modes in phase space. This provides the first concrete realization of a CPT-breaking, yet normalizable, gravitational vacuum state rooted in a nonperturbative quantum gravity framework. Our results establish the CSK state-long thought formal-as a viable candidate for the ground state of quantum gravity in de Sitter space.