FROM DYSON–SCHWINGER EQUATIONS TO QUANTUM ENTANGLEMENT

We apply combinatorial Dyson–Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory Φ in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra H FG cut ( Φ ) and lattices of intermediate structures. Feynman diagrams in H FG ( Φ ) are applied to describe states in Φ where we build the Fisher information metric on finite dimensional linear subspaces of states in terms of homomorphism densities of Feynman graphons which are continuous functionals on the topological space S graphon Φ , M ⊆ [ 0 , ∞ ) ( [ 0 , 1 ] ) . We associate Hopf subalgebras of H FG ( Φ ) generated by quantum motions to separated regions of space-time to address some new correlations. These correlations are encoded by assigning a statistical manifold to the space of 1PI Green’s functions of Φ . These correlations are applied to build lattices of Hopf subalgebras, Lie subgroups, and Tannakian subcategories, derived from towers of combinatorial Dyson–Schwinger equations, which contribute to separated but correlated cut-distance topological regions. This lattice setting is applied to formulate a new tower of renormalization groups which encodes quantum entanglement of space-time separated particles under different energy scales.