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T10-6 CFT-AdS对偶实现形式化规范

1. 基础数学对象​

1.1 共形场论(CFT)结构​

class ConformalFieldTheory:
    def __init__(self, central_charge: float):
        self.c = central_charge
        self.phi = (1 + np.sqrt(5)) / 2
        self.operators = {}
        self.correlation_functions = {}
        
    def add_primary_operator(self, name: str, dimension: 'PhiNumber'):
        """添加初级算符"""
        self.operators[name] = PrimaryOperator(name, dimension)
        
    def conformal_transformation(self, x: np.ndarray, 
                               a: int, b: int, c: int, d: int) -> np.ndarray:
        """离散共形变换 (满足no-11约束)"""
        if not self.verify_no_11([a, b, c, d]):
            raise ValueError("Parameters violate no-11 constraint")
        if a*d - b*c != 1:
            raise ValueError("Not a valid SL(2,Z) transformation")
            
        return (a*x + b) / (c*x + d)
        
    def two_point_function(self, op1: 'Operator', op2: 'Operator', 
                          x1: np.ndarray, x2: np.ndarray) -> 'PhiNumber':
        """计算两点关联函数"""
        if op1.dimension != op2.dimension:
            return PhiNumber(0)
            
        distance = np.linalg.norm(x1 - x2)
        return PhiNumber(1 / distance ** (2 * op1.dimension.value))
        
    def verify_no_11(self, params: List[int]) -> bool:
        """验证参数不含11模式"""
        for p in params:
            if '11' in bin(abs(p))[2:]:
                return False
        return True

1.2 AdS空间结构​

class AntiDeSitterSpace:
    def __init__(self, dimension: int, ads_radius: float):
        self.d = dimension
        self.ell = ads_radius  # AdS半径
        self.phi = (1 + np.sqrt(5)) / 2
        
    def metric(self, r: float, x: np.ndarray) -> np.ndarray:
        """φ-修正的AdS度规"""
        # ds² = ℓ²(dr²/(r²/φ^(2ρ)) + η_μν dx^μ dx^ν/(r²/φ^(2ρ)))
        rho = int(np.log(r) / np.log(self.phi))
        factor = (r ** 2) / (self.phi ** (2 * rho))
        
        g = np.zeros((self.d, self.d))
        g[0, 0] = self.ell ** 2 / factor  # grr
        for i in range(1, self.d):
            g[i, i] = self.ell ** 2 / factor  # gxx, gyy, ...
            
        return g
        
    def geodesic_distance(self, boundary_point1: np.ndarray, 
                         boundary_point2: np.ndarray) -> 'PhiNumber':
        """计算连接两个边界点的测地线长度"""
        delta_x = np.linalg.norm(boundary_point1 - boundary_point2)
        
        # 正规化测地线长度
        length = self.ell * np.log(2 * delta_x / self.epsilon)
        
        # φ-修正
        depth = int(np.log(delta_x) / np.log(self.phi))
        correction = sum(1/self.phi**n for n in range(1, depth+1))
        
        return PhiNumber(length + correction)
        
    def minimal_surface_area(self, boundary_region: 'Region') -> 'PhiNumber':
        """计算延伸到体内的极小曲面面积"""
        # Ryu-Takayanagi公式的实现
        perimeter = boundary_region.perimeter()
        depth = boundary_region.recursive_depth()
        
        # 经典面积
        area_classical = self.ell * perimeter * np.log(perimeter / self.epsilon)
        
        # φ-修正
        area_correction = area_classical / (self.phi ** depth)
        
        return PhiNumber(area_classical - area_correction)

1.3 全息对偶映射​

class HolographicDuality:
    def __init__(self, cft: 'ConformalFieldTheory', ads: 'AntiDeSitterSpace'):
        self.cft = cft
        self.ads = ads
        self.phi = (1 + np.sqrt(5)) / 2
        self.G_N = 1.0  # Newton常数(归一化)
        
    def depth_to_scale(self, depth: int) -> 'PhiNumber':
        """递归深度到能量标度的映射"""
        return PhiNumber(self.phi ** depth)
        
    def radius_to_rg_flow(self, r: float) -> float:
        """径向坐标到RG流参数的映射"""
        return np.log(r) / np.log(self.phi)
        
    def gkpw_relation(self, bulk_field: 'BulkField', 
                     boundary_source: 'BoundarySource') -> 'PhiNumber':
        """GKPW关系: <O> = δS_bulk/δφ_0"""
        # 计算体作用量对边界值的变分
        variation = self.compute_variation(bulk_field, boundary_source)
        return variation
        
    def partition_function_equality(self, boundary_source: 'BoundarySource') -> bool:
        """验证配分函数相等: Z_CFT[φ_0] = Z_AdS[Φ|_∂=φ_0]"""
        z_cft = self.cft.partition_function(boundary_source)
        z_ads = self.ads.partition_function_with_bc(boundary_source)
        
        return abs(z_cft.value - z_ads.value) < 1e-10

2. 递归深度与全息维度​

2.1 深度-半径对应​

class DepthRadiusCorrespondence:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.ell_ads = 1.0  # AdS半径
        
    def depth_to_radius(self, depth: int) -> float:
        """递归深度到径向坐标的映射"""
        return self.ell_ads * self.phi ** depth
        
    def radius_to_depth(self, r: float) -> int:
        """径向坐标到递归深度的映射"""
        return int(np.log(r / self.ell_ads) / np.log(self.phi))
        
    def bulk_reconstruction(self, boundary_data: Dict[np.ndarray, 'PhiNumber'],
                          r: float) -> 'PhiNumber':
        """HKLL体重构"""
        depth = self.radius_to_depth(r)
        result = PhiNumber(0)
        
        for x, value in boundary_data.items():
            # 构造smearing函数
            kernel = self.hkll_kernel(x, r)
            result = result + kernel * value
            
        return result
        
    def hkll_kernel(self, boundary_point: np.ndarray, 
                   bulk_radius: float) -> 'PhiNumber':
        """HKLL核函数(φ-修正)"""
        delta = self.cft.operators['O'].dimension
        distance_sq = bulk_radius ** 2 + np.linalg.norm(boundary_point) ** 2
        
        # 经典核
        kernel_classical = 1 / (distance_sq ** delta.value)
        
        # φ-修正(来自no-11约束的频率截断)
        n_max = int(np.log(distance_sq) / np.log(self.phi))
        correction = sum(self.phi ** (-n * delta.value) 
                        for n in range(1, n_max+1) 
                        if '11' not in bin(n)[2:])
        
        return PhiNumber(kernel_classical * (1 + correction))

2.2 全息RG流​

class HolographicRGFlow:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def beta_function(self, coupling: 'PhiNumber', operator_dim: float) -> 'PhiNumber':
        """β函数"""
        # β = (d - Δ) * g
        d = 2  # 边界维度
        return PhiNumber((d - operator_dim) * coupling.value)
        
    def rg_flow_equation(self, couplings: Dict[str, 'PhiNumber'], 
                        r: float) -> Dict[str, 'PhiNumber']:
        """全息RG流方程"""
        new_couplings = {}
        
        for name, g in couplings.items():
            dim = self.operator_dimensions[name]
            beta = self.beta_function(g, dim)
            
            # 径向演化 ≈ RG流
            dr = 0.01  # 小步长
            new_couplings[name] = g + beta * dr / self.ell_ads
            
        return new_couplings
        
    def fixed_points(self) -> List[Dict[str, 'PhiNumber']]:
        """寻找RG不动点"""
        fixed_points = []
        
        # φ-系统的不动点: g* = g_0 * φ^(-n)
        for n in range(10):
            if '11' not in bin(n)[2:]:
                g_star = PhiNumber(1.0 / (self.phi ** n))
                fixed_points.append({'g': g_star})
                
        return fixed_points

3. 全息纠缠熵​

3.1 Ryu-Takayanagi公式​

class HolographicEntanglementEntropy:
    def __init__(self, ads: 'AntiDeSitterSpace'):
        self.ads = ads
        self.phi = (1 + np.sqrt(5)) / 2
        self.G_N = 1.0  # Newton常数
        
    def entanglement_entropy(self, region: 'BoundaryRegion') -> 'PhiNumber':
        """计算纠缠熵(RT公式)"""
        # 找到极小曲面
        minimal_surface = self.find_minimal_surface(region)
        
        # 计算面积
        area = minimal_surface.area()
        
        # 递归深度修正
        depth = region.recursive_depth()
        
        # S_A = Area(γ_A) / (4 G_N φ^d_A)
        entropy = area / (4 * self.G_N * (self.phi ** depth))
        
        return entropy
        
    def find_minimal_surface(self, boundary_region: 'BoundaryRegion') -> 'Surface':
        """寻找极小曲面"""
        # 变分问题: δArea = 0
        # 使用数值方法求解
        
        # 初始猜测: 半球面
        initial_surface = self.hemisphere_ansatz(boundary_region)
        
        # 迭代优化
        surface = initial_surface
        for iteration in range(100):
            variation = self.compute_area_variation(surface)
            if variation.value < 1e-6:
                break
            surface = self.update_surface(surface, variation)
            
        return surface
        
    def mutual_information(self, region_a: 'BoundaryRegion', 
                         region_b: 'BoundaryRegion') -> 'PhiNumber':
        """互信息"""
        s_a = self.entanglement_entropy(region_a)
        s_b = self.entanglement_entropy(region_b)
        s_ab = self.entanglement_entropy(region_a.union(region_b))
        
        return s_a + s_b - s_ab

3.2 量子修正​

class QuantumCorrections:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def one_loop_correction(self, surface: 'Surface') -> 'PhiNumber':
        """单圈量子修正"""
        # 来自体内场的量子涨落
        determinant = self.functional_determinant(surface)
        
        # φ-修正来自no-11约束的模式截断
        n_modes = surface.mode_count()
        phi_factor = sum(1/self.phi**n for n in range(1, n_modes+1) 
                        if '11' not in bin(n)[2:])
        
        return PhiNumber(-0.5 * np.log(determinant) * phi_factor)
        
    def entanglement_wedge_reconstruction(self, 
                                        boundary_region: 'BoundaryRegion') -> 'BulkRegion':
        """纠缠楔重构"""
        # 找到RT面
        rt_surface = self.find_minimal_surface(boundary_region)
        
        # 纠缠楔是RT面和边界区域围成的体区域
        wedge = BulkRegion(boundary_region, rt_surface)
        
        # 验证子区域对偶性
        assert self.verify_subregion_duality(boundary_region, wedge)
        
        return wedge

4. 全息复杂度​

4.1 复杂度=体积​

class HolographicComplexity:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.G_N = 1.0
        self.ell = 1.0  # AdS半径
        
    def complexity_volume(self, boundary_time_slice: 'TimeSlice') -> 'PhiNumber':
        """CV猜想: 复杂度 = 体积"""
        # 找到极大体积片
        maximal_slice = self.find_maximal_volume_slice(boundary_time_slice)
        
        # 计算体积
        volume = maximal_slice.volume()
        
        # 平均递归深度
        avg_depth = maximal_slice.average_recursive_depth()
        
        # C = V / (φ^d G_N ℓ)
        complexity = volume / (self.phi ** avg_depth * self.G_N * self.ell)
        
        return complexity
        
    def complexity_action(self, boundary_time_slice: 'TimeSlice') -> 'PhiNumber':
        """CA猜想: 复杂度 = 作用量"""
        # Wheeler-DeWitt片
        wdw_patch = self.wheeler_dewitt_patch(boundary_time_slice)
        
        # 计算作用量
        action = self.evaluate_action(wdw_patch)
        
        # φ-修正
        depth_factor = wdw_patch.boundary_depth_factor()
        
        return action / (self.phi ** depth_factor)
        
    def complexity_growth_rate(self, black_hole_mass: float) -> 'PhiNumber':
        """复杂度增长率(Lloyd界)"""
        # dC/dt ≤ 2M/π (Lloyd bound)
        # φ-修正来自离散时间步
        
        classical_bound = 2 * black_hole_mass / np.pi
        
        # 时间离散化修正
        time_step = 1 / self.phi  # 最小时间步
        discrete_factor = 1 - 1/self.phi
        
        return PhiNumber(classical_bound * discrete_factor)

4.2 张量网络实现​

class TensorNetwork:
    def __init__(self, boundary_size: int):
        self.phi = (1 + np.sqrt(5)) / 2
        self.boundary_size = boundary_size
        self.network = self.build_mera_network()
        
    def build_mera_network(self) -> 'NetworkStructure':
        """构建φ-MERA张量网络"""
        layers = []
        current_size = self.boundary_size
        
        while current_size > 1:
            # 每层的张量
            layer = []
            
            # Disentangler
            for i in range(0, current_size-1, 2):
                if self.verify_no_11_indices(i, i+1):
                    disentangler = self.create_disentangler(i, i+1)
                    layer.append(disentangler)
                    
            # Isometry (粗粒化)
            new_size = int(current_size / self.phi)
            for i in range(new_size):
                isometry = self.create_isometry(i)
                layer.append(isometry)
                
            layers.append(layer)
            current_size = new_size
            
        return NetworkStructure(layers)
        
    def entanglement_entropy_tn(self, region: List[int]) -> 'PhiNumber':
        """用张量网络计算纠缠熵"""
        # 收缩网络得到约化密度矩阵
        rho = self.contract_to_reduced_density_matrix(region)
        
        # 计算von Neumann熵
        eigenvalues = np.linalg.eigvals(rho)
        entropy = -sum(λ * np.log(λ) for λ in eigenvalues if λ > 1e-10)
        
        # φ-修正
        depth = len(self.network.layers)
        correction = entropy / (self.phi ** (depth/2))
        
        return PhiNumber(entropy - correction)

5. 黑洞热化对偶​

5.1 BTZ黑洞​

class BTZBlackHole:
    def __init__(self, mass: float, ads_radius: float):
        self.M = mass
        self.ell = ads_radius
        self.phi = (1 + np.sqrt(5)) / 2
        self.r_plus = self.ell * np.sqrt(8 * self.G_N * self.M)
        
    def temperature(self) -> 'PhiNumber':
        """Hawking温度(φ-修正)"""
        # 递归深度
        d_plus = int(np.log(self.r_plus) / np.log(self.phi))
        
        # T = r_+ / (2π ℓ φ^d_+)
        temp = self.r_plus / (2 * np.pi * self.ell * (self.phi ** d_plus))
        
        return PhiNumber(temp)
        
    def entropy(self) -> 'PhiNumber':
        """Bekenstein-Hawking熵"""
        # S = 2π r_+ / (4 G_N)
        s_bh = 2 * np.pi * self.r_plus / (4 * self.G_N)
        
        # φ-修正
        depth_correction = 1 / (self.phi ** int(np.log(self.r_plus) / np.log(self.phi)))
        
        return PhiNumber(s_bh * (1 - depth_correction))
        
    def corresponds_to_thermal_cft(self, cft: 'ConformalFieldTheory') -> bool:
        """验证对应热CFT态"""
        T_bh = self.temperature()
        T_cft = cft.temperature()
        
        return abs(T_bh.value - T_cft.value) < 1e-6

5.2 Page曲线​

class PageCurve:
    def __init__(self, black_hole: 'BTZBlackHole'):
        self.bh = black_hole
        self.phi = (1 + np.sqrt(5)) / 2
        
    def entanglement_entropy_evolution(self, time: float) -> 'PhiNumber':
        """纠缠熵随时间演化"""
        # 早期：线性增长
        if time < self.page_time():
            s_thermal = self.thermal_entropy(time)
            return s_thermal
        else:
            # 晚期：island贡献
            s_island = self.island_entropy(time)
            s_thermal = self.thermal_entropy(time)
            
            # 取最小值
            return PhiNumber(min(s_island.value, s_thermal.value))
            
    def page_time(self) -> float:
        """Page时间"""
        # t_Page = (3/2) * S_BH / c
        # φ-修正来自离散时间演化
        s_bh = self.bh.entropy()
        c = 1  # 中心荷(简化)
        
        t_classical = 1.5 * s_bh.value / c
        t_discrete = t_classical * (1 + 1/self.phi)
        
        return t_discrete
        
    def island_entropy(self, time: float) -> 'PhiNumber':
        """岛屿贡献的熵"""
        # S = S_BH + S_bulk[radiation ∪ island]
        s_bh = self.bh.entropy()
        
        # 体熵的计算(简化模型)
        island_size = self.island_extent(time)
        s_bulk = PhiNumber(island_size * np.log(island_size))
        
        return s_bh + s_bulk

6. 可观测量计算​

6.1 关联函数​

class CorrelationFunctions:
    def __init__(self, duality: 'HolographicDuality'):
        self.duality = duality
        self.phi = (1 + np.sqrt(5)) / 2
        
    def two_point_function_holographic(self, x1: np.ndarray, 
                                      x2: np.ndarray, 
                                      operator: 'Operator') -> 'PhiNumber':
        """全息计算两点函数"""
        # 计算测地线长度
        geodesic_length = self.duality.ads.geodesic_distance(x1, x2)
        
        # <O(x1)O(x2)> = exp(-m * L)
        mass = operator.ads_mass()
        correlator = np.exp(-mass * geodesic_length.value)
        
        # φ-修正
        distance = np.linalg.norm(x1 - x2)
        depth = int(np.log(distance) / np.log(self.phi))
        correction = self.step_function_correction(distance, depth)
        
        return PhiNumber(correlator * correction)
        
    def wilson_loop(self, contour: 'Contour') -> 'PhiNumber':
        """Wilson环期望值"""
        # 找到最小面积曲面
        minimal_surface = self.find_minimal_surface_with_boundary(contour)
        area = minimal_surface.area()
        
        # <W[C]> = exp(-Area/φ^d)
        depth = contour.recursive_depth()
        
        return PhiNumber(np.exp(-area / (self.phi ** depth)))

6.2 输运系数​

class TransportCoefficients:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def shear_viscosity_entropy_ratio(self, temperature: float) -> 'PhiNumber':
        """剪切粘度/熵密度比(KSS界)"""
        # η/s ≥ 1/(4π) * (1 - 1/φ^d)
        
        # 温度决定深度
        depth = int(-np.log(temperature) / np.log(self.phi))
        
        kss_bound = 1 / (4 * np.pi)
        phi_correction = 1 - 1 / (self.phi ** depth)
        
        return PhiNumber(kss_bound * phi_correction)
        
    def dc_conductivity(self, temperature: float, 
                       chemical_potential: float) -> 'PhiNumber':
        """直流电导率"""
        # 使用膜范式计算
        
        # horizon电导率
        sigma_horizon = 1.0  # 归一化
        
        # φ-修正来自离散动量模式
        n_modes = int(temperature * self.phi)
        correction = sum(1/(1 + (n/self.phi)**2) 
                        for n in range(1, n_modes+1) 
                        if '11' not in bin(n)[2:])
        
        return PhiNumber(sigma_horizon * (1 + correction))

7. 全息纠错码​

7.1 量子纠错​

class HolographicErrorCorrection:
    def __init__(self, code_distance: int):
        self.d = code_distance
        self.phi = (1 + np.sqrt(5)) / 2
        
    def encoding_map(self, logical_state: 'QuantumState') -> 'QuantumState':
        """逻辑态到物理态的编码"""
        # 边界态 -> 体态
        n_physical = int(self.phi ** self.d)
        
        # 创建纠缠态
        physical_state = self.create_ghz_like_state(n_physical)
        
        # 编码逻辑信息
        encoded = self.tensor_product_encoding(logical_state, physical_state)
        
        return encoded
        
    def can_correct_errors(self, error_weight: int) -> bool:
        """判断是否能纠正给定权重的错误"""
        # 纠错条件: d > 2*log_φ(error_weight) + 1
        threshold = 2 * np.log(error_weight) / np.log(self.phi) + 1
        
        return self.d > threshold
        
    def recovery_map(self, corrupted_state: 'QuantumState', 
                    error_syndrome: 'Syndrome') -> 'QuantumState':
        """恢复映射"""
        # 使用全息性质：局部信息可从全局恢复
        
        # 识别错误位置
        error_locations = self.decode_syndrome(error_syndrome)
        
        # 应用修正
        corrected = corrupted_state.copy()
        for loc in error_locations:
            corrected = self.apply_correction(corrected, loc)
            
        return corrected

8. 验证函数​

8.1 对偶验证​

def verify_holographic_duality(cft: 'ConformalFieldTheory', 
                             ads: 'AntiDeSitterSpace') -> bool:
    """验证全息对偶的成立"""
    duality = HolographicDuality(cft, ads)
    
    # 1. 配分函数相等
    test_source = BoundarySource(lambda x: np.exp(-x**2))
    if not duality.partition_function_equality(test_source):
        return False
        
    # 2. GKPW关系
    bulk_field = BulkField(ads)
    vev = duality.gkpw_relation(bulk_field, test_source)
    expected_vev = cft.one_point_function(test_source)
    if abs(vev.value - expected_vev.value) > 1e-6:
        return False
        
    # 3. RT公式
    region = BoundaryRegion([0, 1])  # 区间[0,1]
    s_cft = cft.entanglement_entropy(region)
    s_holographic = HolographicEntanglementEntropy(ads).entanglement_entropy(region)
    if abs(s_cft.value - s_holographic.value) > 1e-6:
        return False
        
    return True

8.2 物理一致性检查​

def check_physical_consistency(duality: 'HolographicDuality') -> Dict[str, bool]:
    """检查物理一致性"""
    results = {}
    
    # 1. 因果性
    results['causality'] = check_bulk_causality(duality.ads)
    
    # 2. 幺正性
    results['unitarity'] = check_boundary_unitarity(duality.cft)
    
    # 3. 能量条件
    results['energy_conditions'] = check_energy_conditions(duality.ads)
    
    # 4. 纠缠熵的强次可加性
    results['strong_subadditivity'] = check_entanglement_inequalities(duality)
    
    return results

9. 关键常数​

# 基础常数
PHI = (1 + np.sqrt(5)) / 2  # 黄金分割率

# AdS/CFT参数
NEWTON_CONSTANT = 1.0  # 归一化的Newton常数
ADS_RADIUS = 1.0  # AdS半径
CENTRAL_CHARGE = 1.0  # 中心荷(最小值)

# 全息参数
RT_CUTOFF = 1e-6  # RT面的UV截断
PAGE_TIME_FACTOR = 1.5  # Page时间系数
KSS_BOUND = 1/(4*np.pi)  # KSS粘度界

# 纠错参数
CODE_RATE = 1/PHI  # 编码率
ERROR_THRESHOLD = 1/(2*PHI)  # 错误阈值

10. 错误处理​

class HolographicError(Exception):
    """全息对偶错误基类"""
    
class DualityBreakdownError(HolographicError):
    """对偶破坏错误"""
    
class CausalityViolationError(HolographicError):
    """因果性违反"""
    
class BulkReconstructionError(HolographicError):
    """体重构失败"""
    
class EntanglementWedgeError(HolographicError):
    """纠缠楔错误"""