Canonical Quantization and Constraint Algebra of Weyl-Invariant Gravity in a Conformal Gauge: Emergence of a Geometric Cosmological Term

Abstract

We present the canonical quantization of Weyl-invariant gravity in a specific conformal gauge. Fixing the scale factor as $a(η) = 1/(H_0η)$ yields an effective geometric cosmological term $\mathcal{M}(η) = α/η^2$ on the left-hand side of the Einstein equations, guaranteeing an equation of state $w = -1$ by construction. The Bianchi identity requires a compensator scalar field; we show that this compensator is the conformal mode of the metric, a gauge artifact that does not propagate ghost instabilities in the full quantum theory. We develop the complete ADM Hamiltonian formulation and prove that the constraint algebra closes in standard first-class form. Canonical quantization yields a well-defined Wheeler-DeWitt equation where the $\mathcal{M}(η)$ term acts as a potential barrier dynamically suppressing the quantum creation of small-scale universes. The physical reduced Hamiltonian is demonstrated to be bounded from below, time evolution is strictly unitary in the physical Hilbert space, and microcausality is preserved. In the pure vacuum regime, the model admits an exact analytical solution with $Ω_{\text{DE}} = 1$, resolving the cosmic coincidence problem without fine-tuned parameters. In the matter-dominated era, the comoving horizon scales as $R_H \propto η$, so our result $\mathcal{M} \propto η^{-2} \propto R_H^{-2}$ reproduces the central Compton Mass Dark Energy (CMaDE) hypothesis from first principles and establishes $n = -2$ as the theoretical benchmark for the Conformal Holographic Dark Energy (CHDE) parametrization $\mathcal{M} \propto η^n$. From a pure conformal symmetry principle, the model delivers an effective geometric cosmological term that drives the accelerated expansion with $w = -1$, providing the missing action principle for CMaDE.