Relational de Sitter State Counting with an SU(3) Clock

TL;DR

The paper tackles the puzzle of de Sitter entropy in Euclidean quantum gravity by embedding an observer into the path integral as a massive equatorial worldline carrying a finite SU$(3)$ clock. This setup yields a mode-by-mode cancellation of the troublesome one-loop phase $i^{D+2}$ between CKV-induced directions and worldline fluctuations, and imposes a microcanonical constraint via a Bromwich inverse Laplace transform to produce a real, nonnegative density of states. Three explicit SU$(3)$ clock models are constructed with closed-form partition functions, and the final observer-inclusive density factorizes into a universal geometric/worldline part and a clock-dependent SU$(3)$ weight; calibrating the clock to a vacuum microstructure links the counting to an SU$(3)$ confinement-based picture of dark energy. The positivity statement is conditional on spectral assumptions (A1)–(A3) that ensure a Laplace representation with a nonnegative spectral density in a strip about the de Sitter saddle, and the work emphasizes the need for higher-loop and backreaction analyses to solidify the framework. Collectively, the results provide a consistent, observer-centric route to positive state counting in Euclidean de Sitter space, with a concrete bridge to microscopic SU$(3)$ vacuum structure and a clear set of future directions.

Abstract

Motivated by Maldacena's observer-centric formulation of de~Sitter physics \cite{Maldacena:2024spf}, we develop an observer-dependent state-counting framework in Euclidean de~Sitter space by modeling the observer as a massive equatorial worldline carrying an SU(3) clock. Starting from the gauge-fixed graviton path integral on $S^D$, we trace the one-loop phase $\ii^{D+2}$ to a finite set of scalar and conformal Killing modes and show that, once the worldline is included, the $(D-1)$ transverse negative modes cancel the corresponding $(D-1)$ conformal Killing directions mode by mode. The residual fixed-$β$ phase from the global conformal factor and reparametrizations is removed by imposing the Hamiltonian constraint $H_{\text{patch}} - H_{\text{clock}} - ν= 0$ via a Bromwich inverse Laplace transform, which under explicit complete-monotonicity assumptions yields a real and positive microcanonical density. We stress that this positivity statement is conditional on Assumptions (A1)--(A3) and is established at one loop about the round $S^D$ saddle in the probe regime $G E_{\rm clock}/R\ll 1$; a self-consistent backreacting or higher-loop extension is a natural next step. In earlier work \cite{Ali:2025wld,Ali:2024rnw} we argued that unbroken SU(3) confinement at $T\to 0$ can account for the observed value of the cosmological constant and for the origin of the fundamental constants $(\hbar,G,c)$ as effective couplings fixed by the SU(3) vacuum structure; this makes SU(3) the natural candidate for the internal clock of de~Sitter, whose radius and temperature are themselves set by the same cosmological constant. This idea is implemented with three explicit SU(3) realizations (qutrit, Cartan weight-lattice, and $U(1)^2$ rotor), for which the observer-inclusive density of states factorizes into a universal gravity factor, a universal worldline residue, and a clock-dependent SU(3) weight.

Figures (1)

• Figure 1: Left: static patch tiled by $\mathrm{SU}(3)$ vacuum atoms, with one representative cell highlighted. Right: an observer on the equatorial worldline carries an $\mathrm{SU}(3)$ clock while a nearby $\mathrm{SU}(3)$ vacuum atom contributes to the worldline parameter $\nu$; the clock energy $E_{\rm clock}$ enters the constraint $H_{\text{patch}}-H_{\text{clock}}-\nu=0$.