By making use of Matula numbers, which give a 1-1 correspondence between rooted trees and natural numbers, and a G?del type relabelling of quantum states, we construct a bijection between rooted trees and vectors in the Fock space. As a by product of the aforementioned correspondence (rooted trees ? Fock space) we show that the fundamental theorem of arithmetic is related to the grafting operator, a basic construction in many Hopf algebras. Also, we introduce the Heisenberg-Weyl algebra built in the vector space of rooted trees rather than the usual Fock space. This work is a cross-fertilization of concepts from combinatorics (Matula numbers), number theory (G?del numbering) and quantum mechanics (Fock space).
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