Psi-MCP: Advanced Quantum Systems MCP Server

""" Quantum Field Theory Module This module provides quantum field theory functionality including field quantization, path integrals, and QFT calculations. """ import logging from typing import Dict, Any, List, Optional, Union, Tuple import asyncio import numpy as np logger = logging.getLogger(__name__) async def quantize_scalar_field( field_type: str = "free", spacetime_dimensions: int = 4, lattice_size: int = 16, mass: float = 1.0, coupling: float = 0.1 ) -> Dict[str, Any]: """ Quantize scalar field theory. Args: field_type: Type of field (free, phi4, sine_gordon) spacetime_dimensions: Number of spacetime dimensions lattice_size: Lattice size for discretization mass: Field mass parameter coupling: Coupling constant Returns: Field quantization results """ logger.info(f"Quantizing {field_type} scalar field in {spacetime_dimensions}D") try: # Create momentum space lattice if spacetime_dimensions == 2: # 1+1 dimensions k_values = np.fft.fftfreq(lattice_size, d=1.0) * 2 * np.pi dispersion = np.sqrt(k_values**2 + mass**2) elif spacetime_dimensions == 4: # 3+1 dimensions (simplified to 1D for computational efficiency) k_values = np.fft.fftfreq(lattice_size, d=1.0) * 2 * np.pi dispersion = np.sqrt(k_values**2 + mass**2) else: raise ValueError(f"Unsupported spacetime dimensions: {spacetime_dimensions}") # Calculate zero-point energy zero_point_energy = 0.5 * np.sum(dispersion) # Generate field operators (creation/annihilation operators) creation_ops = {} annihilation_ops = {} for i, k in enumerate(k_values): if abs(k) > 1e-10: # Avoid k=0 issues creation_ops[k] = f"a_dag_{i}" annihilation_ops[k] = f"a_{i}" # Calculate field commutation relations commutation_relations = _calculate_field_commutators(field_type, lattice_size) # Calculate Hamiltonian spectrum n_states = min(100, 2**min(lattice_size//2, 10)) # Limit for computational efficiency energy_levels = [] for n in range(n_states): # Energy levels for harmonic oscillator modes if field_type == "free": energy = zero_point_energy + np.sum(dispersion) * n elif field_type == "phi4": # Include quartic interaction (perturbative) interaction_energy = coupling * n**2 * lattice_size / 100 energy = zero_point_energy + np.sum(dispersion) * n + interaction_energy else: energy = zero_point_energy + np.sum(dispersion) * n energy_levels.append(energy) # Calculate correlation functions correlations = _calculate_field_correlations(field_type, lattice_size, mass, coupling) return { 'success': True, 'field_type': field_type, 'spacetime_dimensions': spacetime_dimensions, 'lattice_size': lattice_size, 'mass': mass, 'coupling': coupling, 'zero_point_energy': float(zero_point_energy), 'momentum_modes': len(k_values), 'energy_levels': energy_levels[:20], # Return first 20 levels 'ground_state_energy': float(energy_levels[0]), 'first_excited_energy': float(energy_levels[1]) if len(energy_levels) > 1 else None, 'field_correlations': correlations, 'commutation_relations': len(commutation_relations) } except Exception as e: logger.error(f"Error quantizing scalar field: {e}") return {'success': False, 'error': str(e)} def _calculate_field_commutators(field_type: str, lattice_size: int) -> Dict[str, str]: """Calculate field commutation relations.""" commutators = {} # Canonical commutation relations [φ(x), π(y)] = iδ(x-y) for i in range(min(lattice_size, 10)): # Limit for demo for j in range(min(lattice_size, 10)): if i == j: commutators[f"[phi_{i}, pi_{j}]"] = "i * delta" else: commutators[f"[phi_{i}, pi_{j}]"] = "0" return commutators def _calculate_field_correlations( field_type: str, lattice_size: int, mass: float, coupling: float ) -> Dict[str, List[float]]: """Calculate field correlation functions.""" distances = np.arange(1, min(lattice_size//2, 20)) if field_type == "free": # Free field correlator: exponential decay correlations = np.exp(-mass * distances) / np.sqrt(distances) elif field_type == "phi4": # φ⁴ theory: modified by interactions correlations = np.exp(-mass * distances) / np.sqrt(distances) * (1 + coupling * distances) elif field_type == "sine_gordon": # Sine-Gordon: soliton contributions correlations = np.exp(-mass * distances) * np.cos(coupling * distances) else: # Default correlations = np.exp(-mass * distances) / np.sqrt(distances) return { 'distances': distances.tolist(), 'two_point_function': correlations.tolist(), 'correlation_length': float(1.0 / mass), 'field_type': field_type } async def calculate_path_integral( action_type: str = "scalar", path_samples: int = 1000, time_steps: int = 100, spatial_points: int = 20 ) -> Dict[str, Any]: """ Calculate path integral using Monte Carlo methods. Args: action_type: Type of action (scalar, gauge, fermion) path_samples: Number of path configurations time_steps: Temporal discretization spatial_points: Spatial discretization Returns: Path integral calculation results """ logger.info(f"Calculating path integral for {action_type} action") try: # Generate field configurations configurations = [] actions = [] for sample in range(path_samples): if action_type == "scalar": config = _generate_scalar_configuration(time_steps, spatial_points) action = _calculate_scalar_action(config) elif action_type == "gauge": config = _generate_gauge_configuration(time_steps, spatial_points) action = _calculate_gauge_action(config) else: config = _generate_scalar_configuration(time_steps, spatial_points) action = _calculate_scalar_action(config) configurations.append(config) actions.append(action) # Calculate observables mean_action = np.mean(actions) action_variance = np.var(actions) # Effective action (averaged over configurations) effective_action = -np.log(np.mean(np.exp(-np.array(actions)))) # Calculate correlation functions from path integral correlations = _calculate_path_integral_correlations(configurations) # Partition function (approximate) partition_function = np.mean(np.exp(-np.array(actions))) return { 'success': True, 'action_type': action_type, 'path_samples': path_samples, 'time_steps': time_steps, 'spatial_points': spatial_points, 'mean_action': float(mean_action), 'action_variance': float(action_variance), 'effective_action': float(effective_action), 'partition_function': float(partition_function), 'correlations': correlations, 'configurations_generated': len(configurations) } except Exception as e: logger.error(f"Error in path integral calculation: {e}") return {'success': False, 'error': str(e)} def _generate_scalar_configuration(time_steps: int, spatial_points: int) -> np.ndarray: """Generate random scalar field configuration.""" # Gaussian random field config = np.random.normal(0, 1, (time_steps, spatial_points)) # Apply some smoothing to make it more physical for _ in range(3): config[1:-1, 1:-1] = (config[1:-1, 1:-1] + 0.1 * (config[:-2, 1:-1] + config[2:, 1:-1] + config[1:-1, :-2] + config[1:-1, 2:])) / 1.4 return config def _generate_gauge_configuration(time_steps: int, spatial_points: int) -> np.ndarray: """Generate gauge field configuration.""" # U(1) gauge field: random angles config = np.random.uniform(0, 2*np.pi, (time_steps, spatial_points, 2)) # 2 gauge field components return config def _calculate_scalar_action(config: np.ndarray) -> float: """Calculate action for scalar field configuration.""" time_steps, spatial_points = config.shape action = 0.0 # Kinetic term: (∂φ/∂t)² for t in range(time_steps - 1): time_derivative = (config[t+1] - config[t])**2 action += 0.5 * np.sum(time_derivative) # Gradient term: (∇φ)² for x in range(spatial_points - 1): spatial_derivative = (config[:, x+1] - config[:, x])**2 action += 0.5 * np.sum(spatial_derivative) # Mass term: m²φ² mass_squared = 1.0 action += 0.5 * mass_squared * np.sum(config**2) # Interaction term: λφ⁴ coupling = 0.1 action += coupling * np.sum(config**4) return action def _calculate_gauge_action(config: np.ndarray) -> float: """Calculate action for gauge field configuration.""" time_steps, spatial_points, components = config.shape action = 0.0 # Wilson action: simplified beta = 1.0 # Gauge coupling for t in range(time_steps - 1): for x in range(spatial_points - 1): # Plaquette: U_μ(x) U_ν(x+μ) U_μ†(x+ν) U_ν†(x) plaquette = (config[t, x, 0] + config[t+1, x, 1] - config[t+1, x+1, 0] - config[t, x+1, 1]) # Wilson action: Re[Tr(plaquette)] action += beta * (1 - np.cos(plaquette)) return action def _calculate_path_integral_correlations(configurations: List[np.ndarray]) -> Dict[str, List[float]]: """Calculate correlation functions from path integral configurations.""" n_configs = len(configurations) if n_configs == 0: return {} # Take first configuration for size reference time_steps, spatial_points = configurations[0].shape[:2] # Calculate two-point correlation function separations = range(1, min(spatial_points//2, 10)) correlations = [] for sep in separations: corr_sum = 0.0 count = 0 for config in configurations[:min(n_configs, 100)]: # Limit for efficiency for x in range(spatial_points - sep): for t in range(time_steps): corr_sum += config[t, x] * config[t, x + sep] count += 1 if count > 0: correlations.append(corr_sum / count) else: correlations.append(0.0) return { 'separations': list(separations), 'two_point_correlations': correlations } async def analyze_quantum_anomalies( theory_type: str = "chiral", dimensions: int = 4, gauge_group: str = "U1" ) -> Dict[str, Any]: """ Analyze quantum anomalies in field theories. Args: theory_type: Type of quantum field theory dimensions: Spacetime dimensions gauge_group: Gauge group Returns: Anomaly analysis results """ logger.info(f"Analyzing quantum anomalies in {theory_type} theory") try: anomalies = {} if theory_type == "chiral": # Chiral anomaly analysis if dimensions == 4 and gauge_group == "U1": # ABJ anomaly coefficient anomaly_coefficient = 1.0 / (24 * np.pi**2) anomalies['chiral_anomaly'] = { 'coefficient': anomaly_coefficient, 'type': 'Adler-Bell-Jackiw', 'cancellation': False } elif dimensions == 4 and gauge_group == "SU3": # QCD anomaly anomaly_coefficient = 3.0 / (32 * np.pi**2) # 3 colors anomalies['chiral_anomaly'] = { 'coefficient': anomaly_coefficient, 'type': 'QCD', 'cancellation': True # Cancelled by quarks } elif theory_type == "gravitational": # Gravitational anomalies if dimensions == 4: anomalies['gravitational_anomaly'] = { 'coefficient': 1.0 / (384 * np.pi**2), 'type': 'Mixed gauge-gravitational', 'cancellation': False } elif theory_type == "conformal": # Conformal anomalies if dimensions == 4: # Weyl anomaly central_charge_a = 1.0 / 90 # Simplified central_charge_c = 1.0 / 120 anomalies['conformal_anomaly'] = { 'central_charge_a': central_charge_a, 'central_charge_c': central_charge_c, 'type': 'Weyl anomaly' } # Calculate anomaly-related observables if anomalies: # Anomalous Ward identities ward_identity_violation = sum(abs(anom.get('coefficient', 0)) for anom in anomalies.values() if isinstance(anom, dict)) # Index theorem connections atiyah_singer_index = _calculate_topological_index(theory_type, dimensions) return { 'success': True, 'theory_type': theory_type, 'dimensions': dimensions, 'gauge_group': gauge_group, 'anomalies': anomalies, 'ward_identity_violation': float(ward_identity_violation), 'topological_index': atiyah_singer_index, 'anomaly_cancellation': all(anom.get('cancellation', False) for anom in anomalies.values() if isinstance(anom, dict)) } else: return { 'success': True, 'theory_type': theory_type, 'anomalies': {}, 'anomaly_free': True } except Exception as e: logger.error(f"Error analyzing anomalies: {e}") return {'success': False, 'error': str(e)} def _calculate_topological_index(theory_type: str, dimensions: int) -> int: """Calculate topological index for Atiyah-Singer theorem.""" if theory_type == "chiral" and dimensions == 4: # Simplified index calculation # In reality, this depends on the specific bundle and operator return 1 elif theory_type == "gravitational" and dimensions == 4: # Hirzebruch signature return 0 # Simplified else: return 0 async def renormalization_group_flow( theory_type: str = "phi4", coupling_initial: float = 0.1, energy_scales: int = 50, beta_function_order: int = 1 ) -> Dict[str, Any]: """ Calculate renormalization group flow. Args: theory_type: Type of field theory coupling_initial: Initial coupling constant energy_scales: Number of energy scales to compute beta_function_order: Order of beta function (1-loop, 2-loop, etc.) Returns: RG flow analysis results """ logger.info(f"Calculating RG flow for {theory_type} theory") try: # Energy scale range (logarithmic) mu_min, mu_max = 0.1, 10.0 mu_values = np.logspace(np.log10(mu_min), np.log10(mu_max), energy_scales) # Calculate beta function and run coupling couplings = [coupling_initial] beta_values = [] g = coupling_initial for i in range(1, energy_scales): # Beta function for different theories if theory_type == "phi4": # φ⁴ theory in 4D beta = _beta_function_phi4(g, beta_function_order) elif theory_type == "qed": # QED beta function beta = _beta_function_qed(g, beta_function_order) elif theory_type == "qcd": # QCD beta function beta = _beta_function_qcd(g, beta_function_order) else: # Generic beta function beta = g**3 / (16 * np.pi**2) # 1-loop generic beta_values.append(beta) # RG equation: dg/d(ln μ) = β(g) dln_mu = np.log(mu_values[i] / mu_values[i-1]) g_new = g + beta * dln_mu # Ensure coupling remains positive and bounded g_new = max(0, min(g_new, 5.0)) couplings.append(g_new) g = g_new # Find fixed points fixed_points = _find_fixed_points(beta_values, couplings) # Calculate critical exponents critical_exponents = _calculate_critical_exponents(theory_type, fixed_points) return { 'success': True, 'theory_type': theory_type, 'coupling_initial': coupling_initial, 'energy_scales': mu_values.tolist(), 'couplings': couplings, 'beta_function': beta_values, 'fixed_points': fixed_points, 'critical_exponents': critical_exponents, 'asymptotic_freedom': beta_values[-1] < 0 if beta_values else False, 'landau_pole': any(abs(g) > 4.0 for g in couplings) } except Exception as e: logger.error(f"Error in RG flow calculation: {e}") return {'success': False, 'error': str(e)} def _beta_function_phi4(g: float, order: int) -> float: """Beta function for φ⁴ theory.""" if order == 1: # 1-loop return 3 * g**2 / (16 * np.pi**2) elif order == 2: # 2-loop (simplified) beta_1 = 3 * g**2 / (16 * np.pi**2) beta_2 = -17 * g**3 / (32 * np.pi**2)**2 return beta_1 + beta_2 else: return 3 * g**2 / (16 * np.pi**2) def _beta_function_qed(g: float, order: int) -> float: """Beta function for QED.""" if order == 1: # 1-loop return g**3 / (12 * np.pi**2) else: return g**3 / (12 * np.pi**2) def _beta_function_qcd(g: float, order: int) -> float: """Beta function for QCD.""" n_f = 3 # Number of flavors if order == 1: # 1-loop b0 = (11 * 3 - 2 * n_f) / 3 # 3 colors return -b0 * g**3 / (16 * np.pi**2) else: b0 = (11 * 3 - 2 * n_f) / 3 return -b0 * g**3 / (16 * np.pi**2) def _find_fixed_points(beta_values: List[float], couplings: List[float]) -> List[Dict[str, float]]: """Find fixed points where β(g) = 0.""" fixed_points = [] # Look for sign changes in beta function for i in range(1, len(beta_values)): if len(beta_values) > i and beta_values[i-1] * beta_values[i] < 0: # Sign change indicates fixed point g_fixed = (couplings[i-1] + couplings[i]) / 2 beta_fixed = (beta_values[i-1] + beta_values[i]) / 2 fixed_points.append({ 'coupling': g_fixed, 'beta': beta_fixed, 'stable': beta_values[i] < beta_values[i-1] # Stability criterion }) # Always include g=0 as trivial fixed point fixed_points.insert(0, { 'coupling': 0.0, 'beta': 0.0, 'stable': True }) return fixed_points def _calculate_critical_exponents(theory_type: str, fixed_points: List[Dict]) -> Dict[str, float]: """Calculate critical exponents near fixed points.""" exponents = {} if theory_type == "phi4": # φ⁴ theory critical exponents (3D Ising universality) exponents = { 'nu': 0.630, # Correlation length 'gamma': 1.237, # Susceptibility 'beta': 0.327, # Order parameter 'alpha': 0.110, # Specific heat 'eta': 0.036 # Anomalous dimension } elif theory_type == "qcd": # QCD critical exponents (rough estimates) exponents = { 'gamma_m': 0.0, # Anomalous mass dimension 'gamma_g': -11/3 # Gauge coupling anomalous dimension } else: # Generic mean-field exponents exponents = { 'nu': 0.5, 'gamma': 1.0, 'beta': 0.5, 'alpha': 0.0 } return exponents