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Matula numbers, Gödel numbering and Fock space

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Abstract

By making use of Matula numbers, which give a 1-1 correspondence between rooted trees and natural numbers, and a Gödel 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 \(\leftrightarrow \) 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 (Gödel numbering) and quantum mechanics (Fock space).

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Acknowledgments

This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) Grant 307617/2012-2. I would like to thank Profs. Ennio Gozzi and Michael O’Carroll for support and Università Degli Studi di Trieste and ICTP for hospitality during November 2011 when part of this work was done. I also thank Prof. Loïc Foissy for kindly providing me his packages for drawing trees in latex and Prof. John Butcher for encouragement.

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  1. DEPRO, Escola de Minas, UFOP, Campus Morro do Cruzeiro, Ouro Preto, MG, 35400-000, Brazil

    Antônio Francisco Neto

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  1. Antônio Francisco Neto
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Correspondence to Antônio Francisco Neto.

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Neto, A.F. Matula numbers, Gödel numbering and Fock space. J Math Chem 51, 1802–1814 (2013). https://doi.org/10.1007/s10910-013-0178-z

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