De Sitter quantum gravity and the emergence of local algebras

Quantum theories of gravity are generally expected to have some degree of non-locality, with familiar local physics emerging only in a particular limit. Perturbative quantum gravity around backgrounds with isometries and compact Cauchy slices provides an interesting laboratory in which this emergence can be explored. In this context, the remaining isometries are gauge symmetries and, as a result, gauge-invariant observables cannot be localized. Instead, local physics can arise only through certain relational constructions. We explore such issues below for perturbative quantum gravity around de Sitter space. In particular, we describe a class of gauge-invariant observables which, under appropriate conditions, provide good approximations to certain algebras of local fields. Our results suggest that, near any minimal $S^d$ in dS${d+1}$, this approximation can be accurate only over regions in which the corresponding global time coordinate $t$ spans an interval $Δt \lesssim O(\ln G^{-1})$. In contrast, however, we find that the approximation can be accurate over arbitrarily large regions of global dS${d+1}$ so long as those regions are located far to the future or past of such a minimal $S^d$. This in particular includes arbitrarily large parts of any static patch.

💡 Research Summary

The paper investigates how local quantum field theory (QFT) algebras emerge from a fundamentally non‑local quantum theory of gravity when one works perturbatively around global de Sitter (dS) space. Because the background retains the full connected de Sitter isometry group SO₀(D,1), these isometries remain gauge symmetries in the perturbative expansion. Consequently any gauge‑invariant observable must be invariant under the entire group, which precludes the existence of strictly local operators supported at a single spacetime point. The authors therefore adopt a relational approach: they introduce a “reference frame” built from additional matter fields, and use it to define relational positions and directions.

The technical construction proceeds as follows. Two independent quantum fields, ϕ and ψ, are considered on the fixed dS background. The ψ‑sector supplies a reference state |ψ₀⟩ that is highly excited (in the G→0 limit it can carry arbitrarily large energy and momentum) and is chosen so that its overlap with its image under any de Sitter transformation g behaves like a Haar‑delta function, ⟨ψ₀|U(g)|ψ₀⟩≈δ(g). One then defines a non‑local operator A = ˜A(x)⊗|ψ₀⟩⟨ψ₀| , where ˜A(x) is a local operator acting only on the ϕ‑sector at a chosen point x. The fully gauge‑invariant observable is obtained by averaging A over the whole connected de Sitter group with the Haar measure: O = ∫_{SO₀(D,1)} dg U(g) A U(g)^{-1}. (Eq. 1.5)

In the strict G→0 limit the reference state can be taken to have infinite energy, making the group average collapse to a true delta function. In that limit O reduces exactly to ˜A(x), reproducing the usual local operator and its micro‑causal algebra. At finite Newton constant G, however, back‑reaction prevents the reference state from being arbitrarily energetic. The group average then yields a smearing kernel of width ∼ln G⁻¹ rather than a delta function. As a result O is only an approximate copy of the local operator, and the approximation is reliable only in spacetime regions where the reference particles are well‑localized.

A central result concerns the dependence on global time. Near a minimal Sⁿ slice (the “waist” of global dS at t=0) the smearing grows rapidly with the time separation. The authors find that any region containing such a minimal sphere can support a good approximation to the local algebra only for a global time interval Δt≲O(ln G⁻¹). This logarithmic bound is illustrated in Fig. 1(a). By contrast, if one considers regions that lie far to the future (or far to the past) of the minimal sphere, the smearing remains small even for arbitrarily large Δt. Consequently, arbitrarily large portions of a static patch—or indeed of the entire future (or past) wedge—can be described by an algebra that is arbitrarily close to the standard dS QFT algebra, as shown in Fig. 1(b). This asymmetry reflects the fact that the minimal sphere is the most “fragile” slice: it is where the reference frame’s localization is weakest.

The paper works out the construction explicitly in 1+1 dimensions, where Einstein gravity is trivial but the model serves as a useful toy for higher‑dimensional intuition. The authors then discuss the generalization to d>1, addressing technical issues such as the divergent norm of group‑averaged states, the inclusion of multiple reference particles, and the treatment of graviton modes (included as part of the ϕ‑sector). Appendices provide a summary of prior results, an alternative definition of the averaging, and explicit calculations of generator moments in arbitrary dimensions.

In the discussion, the authors emphasize that their relational construction demonstrates a concrete mechanism by which local QFT algebras can emerge from a diffeomorphism‑invariant theory, even when the background retains non‑trivial isometries. The logarithmic time bound near the minimal sphere suggests a universal limitation on how long a local description can be maintained without moving the reference frame far away in time. Conversely, the ability to recover the local algebra over arbitrarily large future or past regions indicates that, for practical purposes (e.g., observers confined to a static patch), the non‑locality of quantum gravity does not obstruct the use of ordinary QFT techniques.

Overall, the work advances the “quantum reference frame” program, provides a clear quantitative estimate of the regime of validity of local QFT in de Sitter space, and sets the stage for extensions to more realistic cosmological scenarios, including inflating backgrounds and string‑theoretic constructions.