T17-6 φ-量子引力统一定理 - 形式化规范

摘要

本文档提供T17-6 φ-量子引力统一定理的完整形式化规范。核心思想：从自指完备系统的熵增原理出发，量子力学（自指的离散性）和广义相对论（熵增的几何化）在φ-编码框架下必然统一。

基础数据结构

1. φ-量子时空结构

@dataclass
class PhiQuantumSpacetime:
    """φ-编码的量子时空"""
    
    # 离散坐标（Fibonacci索引）
    coordinates: List[int]  # 必须满足no-11约束
    
    # 度规张量
    metric: 'PhiMetricTensor'
    
    # 量子态
    quantum_state: 'PhiQuantumState'
    
    # 基本常数
    phi: PhiReal = field(default_factory=lambda: PhiReal.from_decimal(1.618033988749895))
    planck_length: PhiReal = field(default_factory=lambda: PhiReal.from_decimal(1.616e-35))
    planck_time: PhiReal = field(default_factory=lambda: PhiReal.from_decimal(5.391e-44))
    
    def __post_init__(self):
        """初始化并验证约束"""
        # 验证坐标的no-11兼容性
        for coord in self.coordinates:
            assert '11' not in bin(coord)[2:], f"坐标{coord}违反no-11约束"
        
        # 计算最小尺度
        self.min_length = self.planck_length * self.phi
        self.min_time = self.planck_time * self.phi
        
        # 初始化度规
        if self.metric is None:
            self.metric = self._initialize_metric()
        
        # 验证量子态归一化
        if self.quantum_state:
            self.quantum_state.normalize()
    
    def _initialize_metric(self) -> 'PhiMetricTensor':
        """初始化φ-度规张量"""
        # Minkowski度规的φ-修正
        dim = len(self.coordinates)
        metric_components = []
        
        for i in range(dim):
            row = []
            for j in range(dim):
                if i == j:
                    if i == 0:  # 时间分量
                        row.append(PhiReal.from_decimal(-1) * self.phi)
                    else:  # 空间分量
                        row.append(PhiReal.one() / self.phi)
                else:
                    row.append(PhiReal.zero())
            metric_components.append(row)
        
        return PhiMetricTensor(components=metric_components)

@dataclass
class PhiMetricTensor:
    """φ-度规张量"""
    
    components: List[List[PhiReal]]
    
    def __post_init__(self):
        """验证度规性质"""
        dim = len(self.components)
        # 验证对称性
        for i in range(dim):
            for j in range(dim):
                assert self.components[i][j] == self.components[j][i], "度规必须对称"
    
    def compute_curvature(self) -> 'PhiCurvatureTensor':
        """计算曲率张量"""
        # 简化：返回基于度规的曲率估计
        dim = len(self.components)
        R = PhiReal.zero()
        
        # 计算Ricci标量（简化版本）
        for i in range(dim):
            for j in range(dim):
                if i != j:
                    R = R + self.components[i][j] * self.components[i][j]
        
        return PhiCurvatureTensor(ricci_scalar=R, dimension=dim)

@dataclass
class PhiCurvatureTensor:
    """曲率张量"""
    ricci_scalar: PhiReal
    dimension: int

2. φ-量子态

@dataclass
class PhiQuantumState:
    """量子引力中的量子态"""
    
    # 态矢量（在φ-希尔伯特空间中）
    amplitudes: List[PhiComplex]
    
    # 基态标签（满足no-11）
    basis_labels: List[str]
    
    # 纠缠结构
    entanglement_network: 'PhiEntanglementNetwork'
    
    # 几何相位
    geometric_phase: PhiReal = field(default_factory=PhiReal.zero)
    
    phi: PhiReal = field(default_factory=lambda: PhiReal.from_decimal(1.618033988749895))
    
    def normalize(self):
        """归一化量子态"""
        norm_sq = PhiReal.zero()
        for amp in self.amplitudes:
            norm_sq = norm_sq + amp.modulus() * amp.modulus()
        
        if norm_sq.decimal_value > 1e-10:
            norm = PhiReal.from_decimal(np.sqrt(norm_sq.decimal_value))
            self.amplitudes = [amp / norm for amp in self.amplitudes]
    
    def apply_operator(self, operator: 'PhiQuantumOperator') -> 'PhiQuantumState':
        """应用量子算符"""
        new_amplitudes = []
        
        for i, amp in enumerate(self.amplitudes):
            new_amp = PhiComplex.zero()
            for j, matrix_element in enumerate(operator.matrix[i]):
                new_amp = new_amp + matrix_element * self.amplitudes[j]
            new_amplitudes.append(new_amp)
        
        return PhiQuantumState(
            amplitudes=new_amplitudes,
            basis_labels=self.basis_labels.copy(),
            entanglement_network=self.entanglement_network,
            geometric_phase=self.geometric_phase + operator.phase_shift
        )
    
    def compute_expectation(self, observable: 'PhiObservable') -> PhiReal:
        """计算可观测量的期望值"""
        expectation = PhiReal.zero()
        
        for i in range(len(self.amplitudes)):
            for j in range(len(self.amplitudes)):
                matrix_element = observable.matrix[i][j]
                contribution = self.amplitudes[i].conjugate() * matrix_element * self.amplitudes[j]
                expectation = expectation + contribution.real
        
        return expectation

@dataclass  
class PhiEntanglementNetwork:
    """量子纠缠网络"""
    
    nodes: List[int]  # 子系统索引
    edges: List[Tuple[int, int, PhiReal]]  # (i, j, 纠缠强度)
    
    def add_entanglement(self, i: int, j: int, strength: PhiReal):
        """添加纠缠连接"""
        self.edges.append((i, j, strength))
    
    def compute_entanglement_entropy(self, partition: List[int]) -> PhiReal:
        """计算子系统的纠缠熵"""
        # 简化计算
        crossing_edges = 0
        total_strength = PhiReal.zero()
        
        for i, j, strength in self.edges:
            if (i in partition) != (j in partition):
                crossing_edges += 1
                total_strength = total_strength + strength
        
        if crossing_edges > 0:
            # S = -Tr(ρ log ρ) ≈ log(crossing_edges) * strength
            return PhiReal.from_decimal(np.log(crossing_edges + 1)) * total_strength
        else:
            return PhiReal.zero()

3. 统一场算符

class PhiUnifiedFieldOperator:
    """φ-量子引力统一场算符"""
    
    def __init__(self, spacetime: PhiQuantumSpacetime):
        self.spacetime = spacetime
        self.phi = spacetime.phi
        
        # 基本常数
        self.hbar = PhiReal.from_decimal(1.054571817e-34)
        self.c = PhiReal.from_decimal(299792458)
        self.G = PhiReal.from_decimal(6.67430e-11)
        
        # 构造哈密顿量
        self.hamiltonian = self._construct_hamiltonian()
    
    def _construct_hamiltonian(self) -> 'PhiHamiltonian':
        """构造统一哈密顿量"""
        # H = H_quantum + H_gravity + H_interaction
        
        H_quantum = self._quantum_hamiltonian()
        H_gravity = self._gravity_hamiltonian()
        H_interaction = self._interaction_hamiltonian()
        
        return PhiHamiltonian(
            quantum_part=H_quantum,
            gravity_part=H_gravity,
            interaction_part=H_interaction
        )
    
    def _quantum_hamiltonian(self) -> 'PhiQuantumOperator':
        """量子部分的哈密顿量"""
        # 简化：自由粒子哈密顿量
        dim = len(self.spacetime.quantum_state.amplitudes)
        matrix = []
        
        for i in range(dim):
            row = []
            for j in range(dim):
                if i == j:
                    # 动能项
                    energy = self.hbar * self.c / (self.spacetime.min_length * PhiReal.from_decimal(i + 1))
                    row.append(energy)
                else:
                    row.append(PhiReal.zero())
            matrix.append(row)
        
        return PhiQuantumOperator(matrix=matrix, phase_shift=PhiReal.zero())
    
    def _gravity_hamiltonian(self) -> 'PhiQuantumOperator':
        """引力部分的哈密顿量"""
        # H_gravity = (c^4/16πG) ∫ R √(-g) d^4x
        
        curvature = self.spacetime.metric.compute_curvature()
        volume_element = self._compute_volume_element()
        
        # 能量密度
        energy_density = (self.c ** PhiReal.from_decimal(4)) / (PhiReal.from_decimal(16 * np.pi) * self.G)
        energy_density = energy_density * curvature.ricci_scalar * volume_element
        
        # 构造对角算符
        dim = len(self.spacetime.quantum_state.amplitudes)
        matrix = []
        
        for i in range(dim):
            row = []
            for j in range(dim):
                if i == j:
                    row.append(energy_density / PhiReal.from_decimal(dim))
                else:
                    row.append(PhiReal.zero())
            matrix.append(row)
        
        return PhiQuantumOperator(matrix=matrix, phase_shift=PhiReal.zero())
    
    def _interaction_hamiltonian(self) -> 'PhiQuantumOperator':
        """量子-引力相互作用"""
        # 基于纠缠网络的相互作用
        network = self.spacetime.quantum_state.entanglement_network
        dim = len(self.spacetime.quantum_state.amplitudes)
        matrix = [[PhiReal.zero() for _ in range(dim)] for _ in range(dim)]
        
        # 纠缠导致的耦合
        for i, j, strength in network.edges:
            if i < dim and j < dim:
                coupling = strength * self.hbar * self.c / self.spacetime.min_length
                matrix[i][j] = coupling / self.phi
                matrix[j][i] = coupling / self.phi
        
        return PhiQuantumOperator(matrix=matrix, phase_shift=PhiReal.zero())
    
    def _compute_volume_element(self) -> PhiReal:
        """计算体积元"""
        # √(-g) 的简化计算
        det = PhiReal.one()
        for i in range(len(self.spacetime.metric.components)):
            det = det * abs(self.spacetime.metric.components[i][i])
        
        return PhiReal.from_decimal(np.sqrt(det.decimal_value))
    
    def evolve(self, initial_state: PhiQuantumState, time: PhiReal) -> PhiQuantumState:
        """时间演化"""
        # |ψ(t)⟩ = exp(-iHt/ħ)|ψ(0)⟩
        
        # 简化：使用一阶近似
        dt = time / PhiReal.from_decimal(100)  # 时间步长
        state = initial_state
        
        for _ in range(100):
            # 应用演化算符
            dstate = self.hamiltonian.apply(state)
            
            # 更新态
            for i in range(len(state.amplitudes)):
                phase_factor = PhiComplex(
                    real=PhiReal.zero(),
                    imag=-dt / self.hbar
                )
                state.amplitudes[i] = state.amplitudes[i] + phase_factor * dstate.amplitudes[i]
            
            state.normalize()
        
        return state

@dataclass
class PhiQuantumOperator:
    """量子算符"""
    matrix: List[List[PhiReal]]
    phase_shift: PhiReal

@dataclass
class PhiHamiltonian:
    """统一哈密顿量"""
    quantum_part: PhiQuantumOperator
    gravity_part: PhiQuantumOperator  
    interaction_part: PhiQuantumOperator
    
    def apply(self, state: PhiQuantumState) -> PhiQuantumState:
        """应用哈密顿量"""
        # H|ψ⟩ = (H_q + H_g + H_i)|ψ⟩
        result = state.apply_operator(self.quantum_part)
        result = result.apply_operator(self.gravity_part)
        result = result.apply_operator(self.interaction_part)
        return result

4. 可观测量与预言

class PhiQuantumGravityObservables:
    """量子引力的可观测量"""
    
    def __init__(self, unified_field: PhiUnifiedFieldOperator):
        self.field = unified_field
        self.phi = unified_field.phi
    
    def gravitational_wave_spectrum(self) -> List[PhiReal]:
        """计算引力波的离散频谱"""
        # f_n = f_0 * F_n
        f_0 = self.field.c / (PhiReal.from_decimal(2) * self.field.spacetime.min_length)
        
        frequencies = []
        fib_prev, fib_curr = 1, 1
        
        for _ in range(10):  # 前10个频率
            freq = f_0 * PhiReal.from_decimal(fib_curr)
            frequencies.append(freq)
            
            # 下一个Fibonacci数
            fib_prev, fib_curr = fib_curr, fib_prev + fib_curr
            
            # 确保no-11兼容
            while '11' in bin(fib_curr)[2:]:
                fib_curr += 1
        
        return frequencies
    
    def black_hole_mass_spectrum(self) -> List[PhiReal]:
        """计算量子黑洞的质量谱"""
        # M_n = M_P * φ^n
        M_P = PhiReal.from_decimal(2.176e-8)  # Planck质量
        
        masses = []
        for n in range(1, 11):
            if '11' not in bin(n)[2:]:
                mass = M_P * (self.phi ** PhiReal.from_decimal(n))
                masses.append(mass)
        
        return masses
    
    def entanglement_gravity_coupling(self, state: PhiQuantumState) -> PhiReal:
        """计算纠缠-引力耦合强度"""
        # Δg = (8πG/c^4) * S_entanglement * T_00
        
        # 计算纠缠熵
        partition = list(range(len(state.amplitudes) // 2))
        S_ent = state.entanglement_network.compute_entanglement_entropy(partition)
        
        # 能量密度（简化）
        T_00 = self.field.hamiltonian.quantum_part.matrix[0][0]
        
        # 引力扰动
        prefactor = (PhiReal.from_decimal(8 * np.pi) * self.field.G) / (self.field.c ** PhiReal.from_decimal(4))
        delta_g = prefactor * S_ent * T_00
        
        return delta_g
    
    def spacetime_foam_fluctuation(self, length_scale: PhiReal) -> PhiReal:
        """计算时空泡沫涨落"""
        # ⟨(Δx)²⟩ = ℓ_P² * φ * ln(L/ℓ_P)
        
        ratio = length_scale / self.field.spacetime.planck_length
        if ratio.decimal_value > 1:
            log_ratio = PhiReal.from_decimal(np.log(ratio.decimal_value))
            fluctuation_sq = (self.field.spacetime.planck_length ** PhiReal.from_decimal(2)) * self.phi * log_ratio
            return PhiReal.from_decimal(np.sqrt(fluctuation_sq.decimal_value))
        else:
            return self.field.spacetime.planck_length

5. 自洽性验证

class PhiQuantumGravityConsistency:
    """理论自洽性验证"""
    
    def __init__(self, unified_field: PhiUnifiedFieldOperator):
        self.field = unified_field
    
    def verify_unitarity(self, evolution_time: PhiReal) -> bool:
        """验证幺正性"""
        # 演化必须保持归一化
        initial_state = self.field.spacetime.quantum_state
        final_state = self.field.evolve(initial_state, evolution_time)
        
        initial_norm = PhiReal.zero()
        final_norm = PhiReal.zero()
        
        for amp in initial_state.amplitudes:
            initial_norm = initial_norm + amp.modulus() * amp.modulus()
        
        for amp in final_state.amplitudes:
            final_norm = final_norm + amp.modulus() * amp.modulus()
        
        return abs(initial_norm.decimal_value - final_norm.decimal_value) < 1e-10
    
    def verify_causality(self) -> bool:
        """验证因果性"""
        # 光锥结构必须保持
        metric = self.field.spacetime.metric
        
        # 检查类时间隔
        dt = self.field.spacetime.min_time
        dx = self.field.spacetime.min_length
        
        # ds² = -c²dt² + dx²
        interval_sq = -(self.field.c * dt) ** PhiReal.from_decimal(2) + dx ** PhiReal.from_decimal(2)
        
        # 类时间隔应该为负
        return interval_sq.decimal_value < 0
    
    def verify_entropy_increase(self, evolution_time: PhiReal) -> bool:
        """验证熵增"""
        initial_state = self.field.spacetime.quantum_state
        final_state = self.field.evolve(initial_state, evolution_time)
        
        # 计算von Neumann熵
        initial_entropy = self._compute_entropy(initial_state)
        final_entropy = self._compute_entropy(final_state)
        
        return final_entropy >= initial_entropy
    
    def _compute_entropy(self, state: PhiQuantumState) -> PhiReal:
        """计算量子态的熵"""
        entropy = PhiReal.zero()
        
        for amp in state.amplitudes:
            p = amp.modulus() * amp.modulus()
            if p.decimal_value > 1e-10:
                entropy = entropy - p * PhiReal.from_decimal(np.log(p.decimal_value))
        
        return entropy

6. 主算法接口

class PhiQuantumGravityUnification:
    """φ-量子引力统一算法"""
    
    def __init__(self, no11: No11NumberSystem):
        self.no11 = no11
        self.phi = PhiReal.from_decimal(1.618033988749895)
    
    def create_quantum_spacetime(self, dimension: int = 4) -> PhiQuantumSpacetime:
        """创建量子时空"""
        # 生成no-11兼容的坐标
        coordinates = []
        coord = 1
        for _ in range(dimension):
            while '11' in bin(coord)[2:]:
                coord += 1
            coordinates.append(coord)
            coord += 1
        
        # 创建初始量子态
        n_basis = 8  # 限制基态数
        while '11' in bin(n_basis)[2:]:
            n_basis -= 1
        
        amplitudes = []
        basis_labels = []
        
        for i in range(n_basis):
            if '11' not in bin(i)[2:]:
                # 初始为基态
                amp = PhiComplex.one() if i == 0 else PhiComplex.zero()
                amplitudes.append(amp)
                basis_labels.append(f"|{bin(i)[2:].zfill(3)}⟩")
        
        # 创建纠缠网络
        network = PhiEntanglementNetwork(
            nodes=list(range(n_basis)),
            edges=[]
        )
        
        # 添加最近邻纠缠
        for i in range(n_basis - 1):
            network.add_entanglement(i, i + 1, self.phi)
        
        quantum_state = PhiQuantumState(
            amplitudes=amplitudes,
            basis_labels=basis_labels,
            entanglement_network=network
        )
        
        return PhiQuantumSpacetime(
            coordinates=coordinates,
            metric=None,  # 将被初始化
            quantum_state=quantum_state
        )
    
    def compute_unified_dynamics(self, spacetime: PhiQuantumSpacetime, 
                               evolution_time: PhiReal) -> Dict[str, Any]:
        """计算统一动力学"""
        # 创建统一场算符
        unified_field = PhiUnifiedFieldOperator(spacetime)
        
        # 时间演化
        initial_state = spacetime.quantum_state
        final_state = unified_field.evolve(initial_state, evolution_time)
        
        # 计算可观测量
        observables = PhiQuantumGravityObservables(unified_field)
        
        # 验证自洽性
        consistency = PhiQuantumGravityConsistency(unified_field)
        
        return {
            'initial_state': initial_state,
            'final_state': final_state,
            'gravitational_waves': observables.gravitational_wave_spectrum(),
            'black_hole_masses': observables.black_hole_mass_spectrum(),
            'entanglement_gravity': observables.entanglement_gravity_coupling(final_state),
            'spacetime_fluctuation': observables.spacetime_foam_fluctuation(spacetime.min_length * PhiReal.from_decimal(1000)),
            'unitarity': consistency.verify_unitarity(evolution_time),
            'causality': consistency.verify_causality(),
            'entropy_increase': consistency.verify_entropy_increase(evolution_time)
        }
    
    def verify_unification(self, results: Dict[str, Any]) -> bool:
        """验证统一理论"""
        # 检查所有自洽性条件
        if not results['unitarity']:
            print("❌ 幺正性验证失败")
            return False
        
        if not results['causality']:
            print("❌ 因果性验证失败")
            return False
        
        if not results['entropy_increase']:
            print("❌ 熵增原理验证失败")
            return False
        
        # 检查预言的合理性
        if len(results['gravitational_waves']) == 0:
            print("❌ 引力波谱预测失败")
            return False
        
        if len(results['black_hole_masses']) == 0:
            print("❌ 黑洞质量谱预测失败")
            return False
        
        print("✅ φ-量子引力统一理论验证成功！")
        return True

算法复杂度分析

时间复杂度

• 量子态演化: O(n²)，n是基态数
• 度规计算: O(d²)，d是时空维度
• 纠缠熵计算: O(e)，e是纠缠边数
• 一致性验证: O(n)

空间复杂度

• 量子态存储: O(n)
• 度规存储: O(d²)
• 纠缠网络: O(n²)最坏情况

总结

本形式化规范完整描述了φ-量子引力统一理论的算法实现。关键创新：

1. 离散时空结构：no-11约束自然导出
2. 统一哈密顿量：量子+引力+相互作用
3. 可验证预言：离散引力波谱等
4. 自洽性保证：幺正性、因果性、熵增

所有结构从自指完备系统的熵增原理严格推导。