The Cosmological CPT Theorem

TL;DR

The paper develops a cosmological generalization of the CPT theorem by organizing discrete symmetries into a Z2×Z2 structure generated by Reflection Reality, a 180° rotation, and CRT. It shows how unitarity and scale invariance in cosmology lead to non-perturbative reality constraints on bulk wavefunction coefficients and to a precise phase formula for boundary wavefunction coefficients at future infinity, with corrections for UV/IR divergences. These results extend to de Sitter globally and in the Poincaré patch, and they provide powerful consistency checks for dS/CFT and inflationary observables, linking bulk unitarity to boundary phase structure. The approach relies on local Lagrangian symmetries and analytic continuations in conformal time or momenta, offering a framework that unifies flat-space CPT with cosmological settings and yields practical constraints for wavefunction coefficients and correlators across loop orders. The work also highlights the potential for accidental unitarity from RR, discusses analytic continuation strategies, and suggests future extensions to spinning fields and heavy modes in cosmology.

Abstract

The CPT theorem states that a unitary and Lorentz-invariant theory must also be invariant under a discrete symmetry $\mathbf{CRT}$ which reverses charge, time, and one spatial direction. In this article, we study a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry group, in which two of the nontrivial symmetries (``Reflection Reality'' and a 180 degree rotation) are implied by Unitarity and Lorentz Invariance respectively, while the third is $\mathbf{CRT}$. (In cosmology, Scale Invariance plays the role of Lorentz Invariance.) This naturally leads to converses of the CPT theorem, as any two of the discrete $\mathbb{Z}_2$ symmetries will imply the third one. Furthermore, in many field theories, the Reflection Reality $\mathbb{Z}_2$ symmetry is actually sufficient to imply the theory is fully unitary, over a generic range of couplings. Building upon previous work on the Cosmological Optical Theorem, we derive non-perturbative reality conditions associated with bulk Reflection Reality (in all flat FLRW models) and $\mathbf{CRT}$ (in de Sitter spacetime), in arbitrary dimensions. Remarkably, this $\mathbf{CRT}$ constraint suffices to fix the phase of all wavefunction coefficients at future infinity (up to a real sign) -- without requiring any analytic continuation, or comparison to past infinity -- although extra care is required in cases where the bulk theory has logarithmic UV or IR divergences. This result has significant implications for de Sitter holography, as it allows us to determine the phases of arbitrary $n$-point functions in the dual CFT.

Figures (1)

• Figure 1: The vacuum wavefunction $\Psi[\tau=0]$ is given by a path integral over one half of the Euclidean $\tau$ plane, with the usual conventions integrating over $-\infty < \tau \leq 0$.