Causal Field Theory IV: Quantum Gravity and the Causal Wheeler-DeWitt Equation

We present the first complete quantization of gravity within the framework of Causal Field Theory in the minisuperspace approximation, where spacetime is endowed with a dynamical causal structure described by a scalar field φ. Starting from the principle of causal optimality, the modified Einstein equations and the causal potential V(φ) = β(e2φ-1-2φ) were derived in previous work (Papers I, II, and III). Using the ADM formalism, we obtain the causal Wheeler-DeWitt equation (Eq.~14 in the main text), where a is the scale factor and Ψ is the wave function of the universe. The field φ is shown to be monotonic, allowing it to serve as an intrinsic quantum clock. A Hermitian time operator t̂ is constructed with an accuracy better than $0.03%$ over the physical domain, transforming the Wheeler-DeWitt equation into a Schr\"odinger equation i∂ₜψ = Ĥphysψ. The physical Hamiltonian yields a discrete energy spectrum with 24 bound states and ground state energy E₀ = -3051.43 (in Planck units). Building on previous work (Papers I, II, and III), the same field φ unifies dark matter (φₘₐₓ=0.429 from SPARC data), dark energy (φvac=0.3476), the Hubble tension (Δφ ≈ 0.0786, yielding Hlocal/HCMB = 1.0829), and gravitational wave propagation (satisfying the GW170817 bound with |vg-c|/c ~ 2×10⁻²⁴ for LIGO-band frequencies). This work constitutes the first complete quantization of gravity within the causal field framework in minisuperspace, resolving the long-standing problem of time in quantum gravity and eliminating classical singularities. The discrete energy spectrum of the universe emerges as a new quantum prediction, potentially detectable through imprints on the cosmic microwave background.