Feynman diagrams and rooted maps

Abstract

Abstract:The rooted maps theory, a branch of the theory of homology, is shown to be a powerful tool for investigating the topological properties of Feynman diagrams, related to the single particle propagator in the quantum manybody systems. The numerical correspondence between the number of this class of Feynman diagrams as a function of perturbative order and the number of rooted maps as a function of the number of edges is studied. A graphical procedure to associate Feynman diagrams and rooted maps is then stated. Finally, starting from rooted maps principles, an original definition of the genus of a Feynman diagram, which totally differs from the usual one, is given. Keywords

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

“…If we have a different number of external legs for each component, the total number of possibilities is simply N ! multiplied by the multinomial coefficient expressed in (15). This is an over-counting if there are components with equal number of external legs, see fig 3. In particular, if we have only r < l components with different number of external legs such that N = d 1 n 1 + d 2 n 2 + · · · + d r n r , where d i is the number of components with the same number of external legs (and, evidently, l = d 1 + · · · + d r ), the correct counting, in this case, is given by…”

Feynman diagrams, ribbon graphs, and topological recursion of Eynard-Orantin

“…transforms the curve (4) into the curve (19)x +ỹ + 1/ỹ = 0 , which implies the following relation on the respective local parameters: t = − z. With this identification, the recursion kernel K H coincides with the kernel (21) from [13] if we put c 2 = 2 , that is if we specialize to the curve (22).…”

“…It has been known for some time that using these numbers, one can count the number of rooted maps (that is maps with a distinguished half-edge) of genus g with e edges [25]. It has been recently realized, see [18] and [22], that the rooted maps are in one-to-one correspondence with the Feynman diagrams of the two-point function of a charged scalar field interacting with a neutral scalar field through a cubic term φ † Aφ. These diagrams also correspond to the electron propagator in QED if Furry's theorem is not applied or in many-body physics, including tadpoles.…”

Equivalence between the Arquès-Walsh sequence formula and the number of connected Feynman diagrams for every perturbation order in the fermionic many-body problem

“…In the many-body non-relativistic case, topological connections between Feynman diagrams and rooted maps (objects in homology theory) have been established. In particular, it can be assumed that the topology of the m-order different connected Feynman diagrams and the topology of rooted maps with m edges are the same [6]. This hypothesis implies that, for each order m, the number of those objects (connected Feynman diagrams and rooted maps) is the same, leading to the sequence 2, 10, 74, 706, · · · (1)…”

“…It is remarkable that there exists a topological connection between such objects and Feynman diagrams. Further considerations about topological similarities between those different objects can be found in [6] and references therein.…”