Intersection Numbers in Quantum Mechanics and Field Theory

Abstract: By elaborating on the recent progress made in the area of Feynman integrals, we apply the intersection theory for twisted de Rham cohomologies to simple integrals involving orthogonal polynomials, matrix elements of operators in Quantum Mechanics and Green's functions in Field Theory, showing that the algebraic identities they obey are related to the decomposition of twisted cocycles within cohomology groups, and which, therefore, can be derived by means of intersection numbers. Our investigation suggests an algebraic approach generically applicable to the study of higher-order moments of probability distributions, where the dimension of the cohomology groups corresponds to the number of independent moments; the intersection numbers for twisted cocycles can be used to derive linear and quadratic relations among them. Our study offers additional evidence of the intertwinement between physics, geometry, and statistics.