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

✅ 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

Generate reduced variables from classical warp metric:

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

Solve quantum constraints:

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

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:

Phase space variables: (Kₐ, Eᵃ) where K is the extrinsic curvature and E is the densitized triad
Holonomy-flux algebra: Quantum operators satisfy [Ĥᵢ, Êʲ] = i ħ γ κ δᵢⱼ
Regularization: Curvature → holonomies around finite loops of Planck area
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)

arcticoder/warp-lqg-midisuperspace