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Unified Mathematical Expressions for Quantum Gravity

1. Core Unified Action

The complete theory can be expressed through a single master action:

S = ∫ dⁿx √(-G) [ℛ + Ψ̄ℒΨ + F ∧ *F]

where:

All physics emerges from:

Z = ∫ DG DΨ exp(iS/ℏ)

The entire dynamics can be captured in one elegant equation:

δS/δΦ = ℏ²∇²Φ + λ(|Φ|² - η²)Φ + γ(∇·∇)Φ = 0

where Φ is a universal field incorporating geometry, matter, and entanglement.

G = exp(iℋ) where ℋ = ∫ dx (π·∇x + H)

The profound connection between quantum information and geometry:

dS² = dE² + γ²dI²

where:

∂_tΨ = (D + Q + E)Ψ

where:

(∇² + R + λΛ + κE)Ψ = 0

G_μν + Q_μν + E_μν = 8πT_μν

where:

J = ∫ dx (xT₀ᵢ - tT_i₀)

generating all required symmetries.

The complete state of the universe:

|Ψ⟩ = N exp(-S_E/ℏ)|0⟩

where S_E is the Euclidean action.

The theory’s central relationship:

∇·G = ∂E/∂S

connecting geometry (G), energy (E), and entropy (S).

lim_{r→∞} Φ = Φ₀ exp(iωt)
lim_{r→0} Φ = regular

∂_μJ^μ = 0

where J^μ is a universal current.

[X_μ, P_ν] = iℏg_μν

valid for all fields.

The entire theory follows from:

δ∫(R + E + Q)√(-g)d⁴x = 0

where:

This unified formulation reveals deeper symmetries and connections in the theory.