LQG Midisuperspace Warp Drive Framework

Complete Implementation of Loop Quantum Gravity Midisuperspace Quantization

This repository contains a genuine Loop Quantum Gravity (LQG) midisuperspace quantization for warp drive spacetimes, implementing all the theoretical requirements for a proper quantum gravity treatment of exotic matter geometries.

🔬 Key Features Implemented

1. Proper Midisuperspace Hamiltonian Constraint

• ✅ Full reduced Hamiltonian H_grav + H_matter = 0
• ✅ Holonomy corrections via sin(μ̄K)/μ̄ (μ̄-scheme)
• ✅ Thiemann's inverse-triad regularization for 1/√|E| operators
• ✅ Non-trivial off-diagonal matrix elements from discrete lattice operators

2. Complete Constraint Implementation

• ✅ Gauss constraint (automatically satisfied in spherical symmetry)
• ✅ Spatial diffeomorphism constraint (gauge-fixed or residual implementation)
• ✅ Anomaly freedom verification for constraint algebra
• ✅ Proper constraint closure checks

3. Coherent (Weave) States

• ✅ Semiclassical states peaked on classical warp solutions
• ✅ Gaussian peaking in both triad (E) and extrinsic curvature (K)
• ✅ Expectation value verification: ⟨Ê^x(r)⟩ ≈ E^x_classical(r)
• ✅ Fluctuation minimization for semiclassical behavior

4. Lattice Refinement & Continuum Limit

• ✅ Multiple lattice resolutions (N = 3, 5, 7, ... grid points)
• ✅ Convergence checks for ⟨T^00⟩ and spectral properties
• ✅ Continuum limit verification through systematic refinement
• ✅ Scaling behavior analysis

5. Realistic Exotic Matter Quantization

• ✅ Phantom scalar field quantization with proper stress-energy tensor
• ✅ Quantum ⟨T^00⟩ computation from LQG states
• ✅ Normal ordering and renormalization for matter operators
• ✅ Backreaction into geometry refinement

6. Advanced Quantum Features

• ✅ Multiple μ̄-schemes: minimal_area, improved_dynamics, adaptive
• ✅ GPU acceleration for large Hilbert spaces (via PyTorch)
• ✅ Sparse matrix techniques for computational efficiency
• ✅ Physical state selection via constraint solving

Structure

├── classical_to_reduced.py      # Task 1: warp metric → (Kₓ, Kφ; Eˣ, Eφ)
├── kinematical_hilbert.py       # Task 2: defines lattice, basis states, flux ops
├── hamiltonian_constraint.py    # Task 3: builds Ĥ_grav + Ĥ_matter on lattice
├── solve_constraint.py          # Task 4: numerically solve Ĥ |Ψ⟩ = 0
├── expectation_values.py        # Task 5: compute ⟨E⟩, ⟨T⁰⁰⟩ from solved state
├── quantum_stability.py         # Task 6: discrete SL operator for ω²ₙ
├── feed_to_warp_framework.py    # Exports ⟨T⁰⁰(rᵢ)⟩ as JSON/NDJSON
├── examples/                    # Example inputs & outputs
└── tests/                      # Unit tests

Usage

1. Generate reduced variables from classical warp metric:

   python classical_to_reduced.py --config examples/warp_config.json --out examples/example_reduced_variables.json

2. Solve quantum constraints:

   python solve_constraint.py --lattice examples/example_reduced_variables.json --out quantum_outputs

3. Export to warp-framework:

   python feed_to_warp_framework.py --input quantum_outputs --framework-path ../warp-framework

Physical Motivation

The midisuperspace approach captures the essential quantum gravity effects while remaining computationally feasible:

• Loop quantization introduces discrete area spectra and resolves curvature singularities
• Holonomy corrections modify the classical Einstein equations at Planck scales
• Quantum bounce replaces classical singularities with smooth quantum transitions
• Exotic matter is consistently quantized alongside the gravitational degrees of freedom

Dependencies

• Python 3.8+
• NumPy, SciPy (numerical computations)
• SymPy (symbolic mathematics)
• python-ndjson (data export)

Install with:

pip install -r requirements.txt

Theory Background

This implementation follows the canonical LQG quantization program:

1. Phase space variables: (Kₐ, Eᵃ) where K is the extrinsic curvature and E is the densitized triad
2. Holonomy-flux algebra: Quantum operators satisfy [Ĥᵢ, Êʲ] = i ħ γ κ δᵢⱼ
3. Regularization: Curvature → holonomies around finite loops of Planck area
4. Physical states: Solutions to Ĥ|Ψ⟩ = 0 (Wheeler-DeWitt equation)

References

• Ashtekar, A. & Bojowald, M. "Loop quantum cosmology" (2006)
• Bojowald, M. "Spherically symmetric quantum geometry" (2004)
• Thiemann, T. "Modern canonical quantum general relativity" (2007)