An interface between physics and number theory

Gérard Henry Edmond Duchamp (1) , Vincel Hoang Ngoc Minh (1) , Allan I. Solomon (2, 3) , Silvia Goodenough (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 Department of Physics and Astronomy [Milton Keynes]
3 LPTMC - Laboratoire de Physique Théorique de la Matière Condensée

We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a {\em mathematical} route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate {\it inter alia} a basis for concluding that the Euler gamma constant $\gamma$ may be rational.

Mots clés: Hopf algebra; Drinfel'd Associator; polyzeta; Euler's gamma

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Combinatoire [math.CO]; Calcul formel [cs.SC]

Gérard Henry Edmond Duchamp, Vincel Hoang Ngoc Minh, Allan I. Solomon, Silvia Goodenough. An interface between physics and number theory. 2010. ⟨hal-00531100v2⟩