Perturbative Reconstruction of Self-Adjoint Generators from Bosonic Canonical Commutation Relations: Application to the Null-Surface Formulation

Abstract

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 (\delta a_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 ([\delta a_2,\delta a_2^\dagger]). The theorem then reconstructs the self-adjoint generators (\delta T_1) and (\delta T_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.