Modular Self-Duality, Symmetrized Relative Entropy, and Bogoliubov--Kubo--Mori Susceptibility in Quantum Field Theory

We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the Bogoliubov--Kubo--Mori quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type III Bogoliubov--Kubo--Mori susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations.