Rigorous computation of the modular Hamiltonian for null-deformed Rindler wedges

Establish a rigorous derivation, analogous to the Bisognano–Wichmann theorem, of the vacuum modular Hamiltonian associated with null-deformed Rindler regions L_{γ,s}, thereby justifying the first-order expression in the deformation parameter and its domain of validity.

Background

For the undeformed Rindler wedge, the modular Hamiltonian of the Minkowski vacuum is rigorously known (Bisognano–Wichmann) to generate boosts. For a null-deformed wedge, Faulkner, Leigh, Parrikar, and Wang provided a path-integral argument for the modular Hamiltonian’s first-order form in the deformation, which plays a central role in the algebraic argument for the ANEC.

However, unlike the undeformed case, a fully rigorous derivation of the modular Hamiltonian for null-deformed regions—parallel to Bisognano–Wichmann’s approach—has not yet been produced. Such a derivation would put the ANEC argument on a firm mathematical foundation and clarify potential subtleties about operator domains and UV regularization.