FROM DYSON–SCHWINGER EQUATIONS TO QUANTUM ENTANGLEMENT

Abstract

We apply combinatorial Dyson–Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory ${\varPhi }$ in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra $H^{\text {cut}}_{\text {FG}}({\varPhi })$ and lattices of intermediate structures. Feynman diagrams in $H_{\text {FG}}({\varPhi })$ are applied to describe states in ${\varPhi }$ 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 $\mathcal {S}^{{\varPhi },M \subseteq [0,\infty )}_{\text {graphon}}([0,1])$. We associate Hopf subalgebras of $H_{\text {FG}}({\varPhi })$ 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 ${\varPhi }$. 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.