AI 中文总结
该研究针对微扰玻色出射映射的伴随逆问题，基于CCR代数证明构造性可积性定理，将其应用于NSF引力子散射，重构出自伴生成元并分离出射映射的贡献分支。
AI 中文摘要
我们研究微扰玻色出射映射的伴随逆问题。在多项式玻色正则对易关系（CCR）代数中，我们证明了一个构造性可积性定理：逐阶保持CCR足以递归重构一个形式自伴生成元，其指数通过伴随共轭实现出射映射。在每一阶，由先前重构生成元确定的贝克-坎贝尔-豪斯多夫（Baker--Campbell--Hausdorff）贡献从出射系数中减去，剩余的齐次CCR条件明确确定下一个生成元。我们将该构造应用于零曲面公式（NSF）中的引力子散射，专门针对具有闵可夫斯基度规、零生成元、仿射参数和锥测度的平坦解空间分支。NSF出射映射在(δa₃)阶（对应ε²阶）保持Ashtekar辐射CCR。在该阶，标量CCR分支唯一选择非平凡相互作用核的对称序（c=1）。由同一三次核生成的相关线性收缩抵消了([δa₂,δa₂†])的双收缩贡献。随后，该定理重构出自伴生成元(δT₁)和(δT₂)，提供了形式微扰幺正实现。所得正则组织还将直接(2→2)贡献分离为不可约圈、单粒子可约和因子化分支。
英文摘要
We study the inverse adjoint problem for a perturbative bosonic outgoing map. Within the polynomial bosonic canonical-commutation-relation (CCR) algebra, we prove a constructive integrability theorem: order-by-order preservation of the CCRs is sufficient to reconstruct recursively a formal self-adjoint generator whose exponential implements the outgoing map by adjoint conjugation. At each order, the Baker--Campbell--Hausdorff contribution determined by previously reconstructed generators is subtracted from the outgoing coefficient, and the remaining homogeneous CCR conditions determine the next generator explicitly. We apply the construction to graviton scattering in the null-surface formulation (NSF), specialized to the flat solution-space sector with Minkowski metric, null generators, affine parameter, and cone measure. The NSF outgoing map preserves the Ashtekar radiative CCRs through (δa_3), corresponding to order (\varepsilon^2). At this order, the scalar CCR sector uniquely selects the symmetric ordering (c=1) for the nontrivial interacting kernel. The associated linear contraction, generated by the same cubic kernel, cancels the double-contraction contribution from ([δa_2,δa_2^\dagger]). The theorem then reconstructs the self-adjoint generators (δT_1) and (δT_2), providing a formal perturbative unitary implementation. The resulting canonical organization also separates the direct (2\to2) contribution into irreducible-loop, one-particle-reducible, and factorized sectors.
