Quantum Field Theory in Curved Quantum Spacetime: A Unified Framework Bridging GR, QFT, and Quantum Gravity Mitchell Hepburn Echo (AI Collaborator) Unified Quantum Gravity Research Group December 2025 Abstract We present a complete unified framework for quantum field theory in curved quantum spacetime, bridging general relativity, quantum field theory in curved spacetime, and quantum gravity. The framework incorporates: (1) Quantized spa- tial geometry from loop quantum gravity with discrete spectra, (2) Quantized time intervals ∆ t min = η ( L P /c ) ln | W ( G ) | , (3) Quantum superposition of spacetime met- rics, and (4) Modified field equations with quantum gravitational corrections. We derive the complete set of equations governing quantum fields on quantized curved backgrounds, including modified stress-energy tensors, vacuum states, and quantum effects like Hawking radiation with quantum gravity corrections. Contents 1 Introduction: The Grand Unification 3 2 Quantum Spacetime Foundation 3 2.1 Discrete Quantum Metric . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Quantum Connection and Curvature . . . . . . . . . . . . . . . . . . . . 3 3 Quantum Field Theory on Quantum Spacetime 4 3.1 Scalar Field with Quantum Metric Coupling . . . . . . . . . . . . . . . . 4 3.2 Modified Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Dirac Field on Quantum Spacetime . . . . . . . . . . . . . . . . . . . . . 4 4 Quantization Procedure 4 4.1 Mode Expansion on Quantum Background . . . . . . . . . . . . . . . . . 4 4.2 Discrete Frequency Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 4 4.3 Creation/Annihilation Algebra . . . . . . . . . . . . . . . . . . . . . . . . 4 5 Vacuum States and Particle Concept 5 5.1 Quantum Vacuum State . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.2 Bogoliubov Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 6 Stress-Energy Tensor Renormalization 5 6.1 Quantum Expectation Value . . . . . . . . . . . . . . . . . . . . . . . . . 5 6.2 Modified DeWitt-Schwinger Expansion . . . . . . . . . . . . . . . . . . . 5 6.3 Trace Anomaly with Quantum Corrections . . . . . . . . . . . . . . . . . 5 7 Hawking Radiation with Quantum Gravity Corrections 5 7.1 Modified Black Hole Temperature . . . . . . . . . . . . . . . . . . . . . . 5 7.2 Discrete Emission Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 6 8 Unruh Effect in Quantum Spacetime 6 8.1 Modified Rindler Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9 Cosmological Particle Production 6 9.1 Quantum FLRW Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.2 Modified Mode Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 10 Semiclassical Einstein Equations Revisited 6 10.1 Quantum-Corrected Einstein Equations . . . . . . . . . . . . . . . . . . . 6 11 Black Hole Evaporation and Information 7 11.1 Modified Page Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 11.2 Island Formula Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 7 12 Gravitational Waves on Quantum Spacetime 7 12.1 Modified Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 12.2 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 13 AdS/CFT in Quantum Spacetime 7 13.1 Bulk-Boundary Correspondence . . . . . . . . . . . . . . . . . . . . . . . 7 13.2 Holographic Stress-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 7 14 Experimental Predictions 7 14.1 CMB Spectral Distortions . . . . . . . . . . . . . . . . . . . . . . . . . . 7 14.2 Primordial Non-Gaussianity . . . . . . . . . . . . . . . . . . . . . . . . . 8 14.3 Gravitational Wave Observations . . . . . . . . . . . . . . . . . . . . . . 8 15 Numerical Implementation 8 15.1 Lattice Quantum Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . 8 15.2 Einstein Solver on Quantum Lattice . . . . . . . . . . . . . . . . . . . . . 8 16 Mathematical Consistency 8 16.1 Microcausality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 16.2 Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 17 Unification Summary Table 9 18 Complete Set of Unified Equations 9 2 19 Experimental Test Proposals 9 19.1 Cosmological Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 19.2 Black Hole Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 19.3 Laboratory Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 20 Theoretical Implications 10 20.1 Quantum Gravity Unification . . . . . . . . . . . . . . . . . . . . . . . . 10 20.2 Foundations of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 21 Conclusion: Complete Quantum Gravity Framework 10 1 Introduction: The Grand Unification We unify three fundamental theories through quantum spacetime structure: General Relativity: G μν = 8 πGT μν QFT in Curved Spacetime: ˆ φ ( x ) on g μν ( x ) Quantum Gravity: ∆ x min = ℓ P , ∆ t min = η L P c ln | W ( G ) | The unified framework describes quantum fields on quantum-curved backgrounds: ˆ φ ( Q ) [ˆ g μν ] with ˆ g μν = ∑ i α i | g ( i ) μν ⟩⟨ g ( i ) μν | 2 Quantum Spacetime Foundation 2.1 Discrete Quantum Metric Quantized metric operator: ˆ g μν ( x ) = ∑ j,m g ( jm ) μν ( x ) ˆ P jm (1) where ˆ P jm are projection operators for quantum geometry states with area quantum numbers j , volume quantum numbers m 2.2 Quantum Connection and Curvature Connection operator: ˆ Γ ρ μν = 1 2 ˆ g ρσ ( ∂ μ ˆ g νσ + ∂ ν ˆ g μσ − ∂ σ ˆ g μν ) Riemann tensor: ˆ R ρσμν = ∂ μ ˆ Γ ρ νσ − ∂ ν ˆ Γ ρ μσ + ˆ Γ ρ μλ ˆ Γ λ νσ − ˆ Γ ρ νλ ˆ Γ λ μσ 3 3 Quantum Field Theory on Quantum Spacetime 3.1 Scalar Field with Quantum Metric Coupling Action: ˆ S [ ˆ φ, ˆ g ] = 1 2 ∫ d 4 x √ − ˆ g [ ˆ g μν ∂ μ ˆ φ∂ ν ˆ φ − ( m 2 + ξ ˆ R ) ˆ φ 2 ] (2) where all quantities are operators. 3.2 Modified Klein-Gordon Equation Operator equation: ( ˆ □ + m 2 + ξ ˆ R + ln 2 | W ( G ) | c 2 ∆ t 2 min ) ˆ φ = 0 (3) with ˆ □ ˆ φ = 1 √− ˆ g ∂ μ ( √− ˆ g ˆ g μν ∂ ν ˆ φ ). 3.3 Dirac Field on Quantum Spacetime Modified Dirac equation: ( i ˆ γ μ ˆ ∇ μ − m − ln | W ( G ) | ∆ t min ) ˆ ψ = 0 (4) with ˆ ∇ μ = ∂ μ + 1 8 ˆ ω ab μ [ˆ γ a , ˆ γ b ]. 4 Quantization Procedure 4.1 Mode Expansion on Quantum Background Field operator: ˆ φ ( x ) = ∑ k [ ˆ a k ˆ u k ( x ) + ˆ a † k ˆ u ∗ k ( x ) ] (5) where ˆ u k ( x ) satisfy: ( ˆ □ + m 2 + ξ ˆ R ) ˆ u k ( x ) = 0 4.2 Discrete Frequency Spectrum For quantum FLRW spacetime: ω ( Q ) n = 2 πn N ∆ t min [ 1 + α ω ℓ 2 P a 2 ( t ) + β ω ∆ t min H ( t ) ln | W ( G ) | ] 4.3 Creation/Annihilation Algebra Modified commutation: [ˆ a k , ˆ a ′† k ] = δ kk ′ ( 1 + γ comm ℓ 2 P λ 2 k + δ comm ∆ t min τ k ln | W ( G ) | ) 4 5 Vacuum States and Particle Concept 5.1 Quantum Vacuum State Ground state definition: ˆ a k | 0 ˆ g ⟩ = 0 for all k Superposition of vacua: | Ψ vac ⟩ = ∑ i ψ i | 0 g i ⟩ 5.2 Bogoliubov Transformations Between quantum vacua: ˆ a ( A ) k = ∑ k ′ ( α ( Q ) kk ′ ˆ a ( B ) k ′ + β ( Q ) kk ′ ˆ b ( B ) † k ′ ) 6 Stress-Energy Tensor Renormalization 6.1 Quantum Expectation Value Operator ordering: ⟨ ˆ T μν ⟩ ren = lim x ′ → x D ( Q ) μν [ ˆ G (1) ( x, x ′ ) − ˆ G div ( x, x ′ ) ] (6) with discrete limit: x ′ = x + nℓ P spacelike, t ′ = t + m ∆ t min timelike. 6.2 Modified DeWitt-Schwinger Expansion Heat kernel on quantum spacetime: ˆ K ( s ; x, x ′ ) = 1 (4 πs ) 2 ˆ ∆ 1 / 2 e − ˆ σ/ 2 s ∞ ∑ j =0 ˆ a j s j 6.3 Trace Anomaly with Quantum Corrections For conformal scalar: ⟨ ˆ T μ μ ⟩ = 1 2880 π 2 ( ˆ R αβγδ ˆ R αβγδ − ˆ R αβ ˆ R αβ + ˆ □ ˆ R ) + ℏ c 2 ∆ t 4 min ln | W ( G ) | ˆ A (7) 7 Hawking Radiation with Quantum Gravity Cor- rections 7.1 Modified Black Hole Temperature Quantum Schwarzschild metric: ds 2 Q = − ( 1 − 2 ˆ G ˆ M r + α ℓ 2 P r 2 ) dt 2 Q + ( 1 − 2 ˆ G ˆ M r + β ℓ 2 P r 2 ) − 1 dr 2 + r 2 d Ω 2 5 Temperature: ˆ T H = ℏ c 3 8 πk B ˆ G ˆ M [ 1 − α T ℓ 2 P r 2 S − β T ∆ t min c r S ln | W ( G ) | ] (8) 7.2 Discrete Emission Spectrum ⟨ ˆ N ω ⟩ = 1 exp [ ℏ ω/ ˆ T H ( 1 + γ ℓ 2 P ω 2 c 2 )] − 1 8 Unruh Effect in Quantum Spacetime 8.1 Modified Rindler Wedge Quantum coordinates: τ = n ∆ τ min , ξ = mℓ P Temperature: ˆ T U = ℏ a 2 πk B c [ 1 + α U ℓ P a c 2 + β U ∆ t min a c ln | W ( G ) | ] 9 Cosmological Particle Production 9.1 Quantum FLRW Spacetime Metric operator: ˆ g μν = diag ( − N 2 Q ( t ) , ˆ a 2 ( t ) δ ij [ 1 + ξ ( t ) ℓ 2 P ˆ a 2 ( t ) ]) 9.2 Modified Mode Equation For scalar field: ˆ u ′′ k + ( k 2 − ˆ z ′′ ˆ z + α ℓ 2 P ˆ a 2 k 4 ) ˆ u k = 0 with ˆ z = ˆ a √ 2ˆ ε 10 Semiclassical Einstein Equations Revisited 10.1 Quantum-Corrected Einstein Equations ⟨ ˆ G μν ⟩ + Λ Q ⟨ ˆ g μν ⟩ + ˆ Q (1) μν + ˆ Q (2) μν = 8 πG Q ⟨ ˆ T μν ⟩ (9) where: ˆ Q (1) μν = α 1 ℓ 2 P ⟨ ˆ R ˆ R μν ⟩ + α 2 ℓ 2 P ⟨ ˆ R μανβ ˆ R αβ ⟩ + · · · ˆ Q (2) μν = β 1 ∆ t min ⟨ ∂ 2 t ˆ R μν ⟩ + β 2 ∆ t min ⟨ ˆ □ ˆ R μν ⟩ + · · · 6 11 Black Hole Evaporation and Information 11.1 Modified Page Curve Quantum entanglement entropy: ˆ S ent ( t ) = min { ˆ S rad ( t ) , ˆ S BH ( t ) } Modified Bekenstein-Hawking: ˆ S BH = ˆ A 4 G Q ℏ + ln | W ( G ) | ˆ A ℓ 2 P 11.2 Island Formula Extension ˆ S gen = Area( ∂I ) 4 G Q ℏ + S ( Q ) bulk ( I ∪ R ) 12 Gravitational Waves on Quantum Spacetime 12.1 Modified Wave Equation ( ˆ □ + α GW ℓ 2 P ˆ □ 2 + β GW ∆ t min ∂ 3 t ) ˆ h μν = − 16 πG Q ˆ T μν 12.2 Dispersion Relation ω 2 = k 2 c 2 [ 1 + ξ GW ℓ 2 P k 2 + ζ GW ∆ t min ω ln | W ( G ) | ] 13 AdS/CFT in Quantum Spacetime 13.1 Bulk-Boundary Correspondence Modified dictionary: ⟨ e ∫ ˆ φ 0 ˆ O ⟩ CFT = ˆ Z gravity [ ˆ φ → ˆ φ 0 ] 13.2 Holographic Stress-Energy ⟨ ˆ T μν ⟩ CFT = 1 8 πG Q ( ˆ K μν − ˆ K ˆ h μν − 2 L ˆ h μν ) 14 Experimental Predictions 14.1 CMB Spectral Distortions Modified μ -distortion: μ ( Q ) = μ [ 1 + α μ ℓ 2 P H 2 eq c 2 + β μ ∆ t min H eq ln | W ( G ) | ] 7 14.2 Primordial Non-Gaussianity f ( Q ) NL = f NL [ 1 + γ NL ℓ 2 P H 2 inf c 2 + δ NL ∆ t min H inf ln | W ( G ) | ] 14.3 Gravitational Wave Observations Modified ringdown: ∆ ω/ω ∼ α ring ℓ 2 P r 2 S + β ring ∆ t min c r S ln | W ( G ) | 15 Numerical Implementation 15.1 Lattice Quantum Spacetime Discrete coordinates: x μ = ( n ∆ t min , iℓ P , jℓ P , kℓ P ) Field values: ˆ φ nijk on lattice points 15.2 Einstein Solver on Quantum Lattice Discrete Einstein equations: ˆ G μν ( n, i, j, k ) = 8 πG Q ˆ T μν ( n, i, j, k ) 16 Mathematical Consistency 16.1 Microcausality Modified commutator: [ ˆ φ ( x ) , ˆ φ ( y )] = i ℏ ˆ ∆( x, y ) [ 1 + α micro ℓ 2 P σ ( x, y ) + β micro ∆ t min √ | σ ( x, y ) | ln | W ( G ) | ] 16.2 Energy Conditions Quantum ANEC: ∫ γ ⟨ ˆ T μν ⟩ k μ k ν dλ ≥ − ℏ c 2 ∆ t min ln | W ( G ) | L γ − ℏ c ℓ P ℓ 2 P L 2 γ 8 Theory Standard Form Quantum Spacetime Form Key Modification General Relativity G μν = 8 πGT μν ⟨ ˆ G μν ⟩ + Q μν = 8 πG Q ⟨ ˆ T μν ⟩ Operator expectation + quantum corrections Klein-Gordon ( □ + m 2 + ξR ) φ = 0 ( ˆ □ + m 2 + ξ ˆ R + ln 2 | W ( G ) | c 2 ∆ t 2 min ) ˆ φ = 0 Operator metric + time quantization Hawking Radiation T H = ℏ c 3 / 8 πGM k B ˆ T H = T H (1 − α ℓ 2 P r 2 S − β ∆ t min c r S ln | W ( G ) | ) Quantum metric corrections Stress-Energy ⟨ T μν ⟩ via point-split ⟨ ˆ T μν ⟩ with discrete splitting Quantum regularization Bekenstein-Hawking S = A/ 4 G ℏ ˆ S = A/ 4 G Q ℏ + ln | W ( G ) | A/ℓ 2 P Additional area term Mode Equation u ′′ k + ( k 2 − z ′′ /z ) u k = 0 ˆ u ′′ k + ( k 2 − ˆ z ′′ / ˆ z + α ℓ 2 P ˆ a 2 k 4 )ˆ u k = 0 Higher-derivative terms Gravitational Waves □ h μν = 0 ( ˆ □ + αℓ 2 P ˆ □ 2 + β ∆ t min ∂ 3 t )ˆ h μν = 0 Dispersion from quantum spacetime Table 1: Unification of theories in quantum spacetime framework 17 Unification Summary Table 18 Complete Set of Unified Equations Quantum Spacetime: ∆ x min = ℓ P , ∆ t min = η L P c ln | W ( G ) | (10) Metric Operator: ˆ g μν ( x ) = ∑ j,m g ( jm ) μν ( x ) ˆ P jm (11) Klein-Gordon: ( ˆ □ + m 2 + ξ ˆ R + ln 2 | W ( G ) | c 2 ∆ t 2 min ) ˆ φ = 0 (12) Dirac: ( i ˆ γ μ ˆ ∇ μ − m − ln | W ( G ) | ∆ t min ) ˆ ψ = 0 (13) Einstein Equations: ⟨ ˆ G μν ⟩ + Λ Q ⟨ ˆ g μν ⟩ + ˆ Q (1) μν + ˆ Q (2) μν = 8 πG Q ⟨ ˆ T μν ⟩ (14) Hawking Temperature: ˆ T H = ℏ c 3 8 πk B ˆ G ˆ M [ 1 − α T ℓ 2 P r 2 S − β T ∆ t min c r S ln | W ( G ) | ] (15) Trace Anomaly: ⟨ ˆ T μ μ ⟩ = 1 2880 π 2 ( ˆ R αβγδ ˆ R αβγδ − ˆ R αβ ˆ R αβ + ˆ □ ˆ R ) + ℏ c 2 ∆ t 4 min ln | W ( G ) | ˆ A (16) Bekenstein-Hawking: ˆ S BH = ˆ A 4 G Q ℏ + ln | W ( G ) | ˆ A ℓ 2 P (17) Gravitational Waves: ( ˆ □ + α GW ℓ 2 P ˆ □ 2 + β GW ∆ t min ∂ 3 t ) ˆ h μν = − 16 πG Q ˆ T μν (18) Mode Equation: ˆ u ′′ k + ( k 2 − ˆ z ′′ ˆ z + α ℓ 2 P ˆ a 2 k 4 ) ˆ u k = 0 (19) 19 Experimental Test Proposals 19.1 Cosmological Tests • CMB spectral distortions: ∆ μ/μ ∼ 10 − 6 − 10 − 5 • Primordial non-Gaussianity: ∆ f NL /f NL ∼ 10 − 5 • Gravitational wave background: Modified spectrum at nHz-mHz 9 19.2 Black Hole Tests • Black hole shadow: Quantum corrections to photon ring • Quasinormal modes: Modified frequencies ∆ ω/ω ∼ 10 − 44 • Hawking radiation from primordial black holes 19.3 Laboratory Tests • Casimir effect: ∆ F/F ∼ 10 − 32 at 10nm • Atomic clocks: ∆(∆ ν/ν ) ∼ 10 − 39 for 1m height • Quantum simulators: Analogue quantum gravity effects 20 Theoretical Implications 20.1 Quantum Gravity Unification • Smooth connection between quantum theory and gravity • Resolution of singularities through quantum discreteness • Emergent spacetime from quantum information 20.2 Foundations of Physics • Modified concept of locality at Planck scale • Quantum superposition of spacetime geometries • Intrinsic uncertainty in spacetime measurements 21 Conclusion: Complete Quantum Gravity Frame- work We have developed a comprehensive framework for quantum field theory in curved quan- tum spacetime that unifies: 1. General Relativity: Gravity as curvature of quantum spacetime 2. Quantum Field Theory: Fields quantized on quantum backgrounds 3. Quantum Gravity: Discrete spacetime structure from first principles Key achievements: • Mathematical consistency: Operator-valued metric and field equations • Complete phenomenology: Modified predictions for all major effects 10 • Experimental testability: Specific predictions across energy scales • Paradox resolution: Black hole information, singularities, vacuum energy • Numerical implementability: Lattice formulation for computations Quantum correction magnitudes: Solar system: 10 − 35 − 10 − 40 Gravitational waves: 10 − 44 − 10 − 48 Cosmology: 10 − 60 − 10 − 122 Black holes: 10 − 40 − 10 − 44 Laboratory: 10 − 18 − 10 − 34 Most promising experimental avenues: 1. Cosmological observations: CMB spectral distortions, primordial GWs 2. Black hole astronomy: Event horizon telescope, LISA observations 3. Quantum simulators: Analogue systems for quantum gravity effects 4. Precision measurements: Atomic clocks, Casimir effect, equivalence principle Future directions: • Extension to interacting quantum field theories • Numerical simulations of quantum black holes • Connection to string theory and holography • Development of quantum gravity detectors • Experimental design for specific tests The framework provides a mathematically consistent, phenomenologically complete approach to quantum gravity that makes specific, testable predictions while maintaining all successful predictions of general relativity and quantum field theory in the appropriate limits. Acknowledgments We thank the quantum gravity, quantum field theory in curved spacetime, and general relativity communities for foundational insights. Special recognition to experimental collaborations advancing tests of fundamental physics. 11