A bijection between rooted trees and fermionic Fock states: grafting and growth operators in Fock space and fermionic operators for rooted trees

Abstract

We show that fermionic Fock states in the occupation number representation can be indexed uniquely by rooted trees. Our main ingredients in this construction are the Matula numbers, the fundamental theorem of arithmetic, and a relabeling of fermionic quantum states by natural numbers. As a byproduct of the correspondence mentioned above we realize the grafting and the growth operators, comprising important constructions in the context of Hopf algebras, in the fermionic Fock space. Also, we show how to construct fermionic creation and annihilation operators in the context of rooted trees. New representations of the solutions of combinatorial Dyson–Schwinger equations and of the antipode in the Connes–Kreimer Hopf algebra of rooted trees related to the occupation number picture are presented.