De Sitter quantum gravity and the emergence of local algebras

Abstract

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.

Figures (12)

• Figure 1: A sketch of global dS$_{d+1}$ indicating regions (shaded pink) where we construct good approximations to dS QFT at non-zero $G$. (a): Regions that contain a minimal $S^d$ (blue) can span only global time intervals $\Delta t\lesssim O[\ln(G^{-1})]$. (b): Regions far to the future (or past) of a minimal $S^d$ (blue) can span arbitrarily large global time intervals.
• Figure 2: A positive charge (red) sources a flux of electric field (arrows) as shown. However, if the charge lives on a sphere (say, at the north pole), the resulting field lines are forced to cross again at least at one other point (at the south pole in the example shown here). The Gauss law then requires the resulting convergence to coincide with the location of a negative charge (blue). As a result, only configurations of charges with zero net charge can consistently source electric fields on $S^d$.
• Figure 3: Numerical examples illustrating continuity of the real (blue) and imaginary (orange) parts of smeared correlators under changes of the smearing functions. Here we consider a free scalar two-point function and take the smearing function $F_y(x)$ to be of the near-Gaussian form \ref{['eq:GaussianF']}. Plotting $G_{smeared}(x_0, y, \sigma) : = {\cal N}^{-1} \int_{dS_D} d^Dx \sqrt{-g} F_y(x) \bigl[0;\phi| \hat{\phi}_{QFT}(x) \hat{\phi}_{QFT}(x_0) |0;\phi\bigr]$ as a function of the smearing width $\sigma$ gives results consistent with the desired continuity. Results are shown for scalars of mass $M=d/2\ell$ (for which $\mu=0$). The left panel shows $D=2$ while the right panel shows $D=10$. We take the point $y$ to lie at $\theta=0$ and $t=3\ell$, with $x_0$ on the past light cone of $y$ at $t=0$. Numerical integrations were performed along a deformed contour that avoids the singularities (in a regime where the results are stable with respect to such deformations). Blue and orange show the real and the imaginary parts. The normalization $\mathcal{N}$ was arbitrarily chosen to set $G_{smeared}=1$ at $\sigma=0.8$.
• Figure 4: The Killing vector fields $B_1$ (left), $B_2$ (center), and $R$ (right), in dS$_{1+1}$. The figures are drawn using conformal coordinates $T, \theta$ with $-\pi<\theta<\pi$, $-\pi/2<T<\pi/2$.
• Figure 5: The positive real group averaging kernel $\bigl[\psi_0| e^{i\lambda_i B_i}|\psi_0\bigr] = |\bigl[\psi_+| e^{i\lambda_i B_i}|\psi_+\bigr]|^2$ for $\mu=0$ in the 2-particle state $|\psi_0\bigr]$. Panels (a,b,c) show $i=(1,R,2)$ (with $B_R:=R)$. Each panel displays results for $j_{\mathsmaller \star}=10$ (blue), $j_{\mathsmaller \star}=50$ (magenta), and for the Gaussian whose width matches the corresponding term in \ref{['eq:chi_1']}. For $B_1$ or $R$ (top row), the resulting curves largely coincide when both are plotted against $j_{\mathsmaller \star}\lambda_1$, $j_{\mathsmaller \star}\theta$. This approximate symmetry at large $j_{\mathsmaller \star}$ reflects the scale-invariance of the corresponding Minkowski-space problem involving either time translations (analogous to $B_1$) or space translations (analgous to $R$) and massless particles localized at the origin. Due to \ref{['eq:chi_1']}, we instead plot results for $B_2$ against $\sqrt{j_{\mathsmaller \star}}\lambda_2$. While this gives the correct scaling for the peak (see expanded view in panel (c) at right), there appears to be no scale symmetry at large $j_{\mathsmaller \star}$.