Algebras of diagrams and functions coming from physics
by Gérard H.E. Duchamp presentation


Abstract.
We consider two aspects of the product formula for formal power series applied to combinatorial field theories. Firstly, we remark that the case when the functions involved in the product formula have a constant term is of special interest as often these functions give rise to substitutional groups. The groups arising from the normal
ordering problem of boson strings are naturally associated with explicit vector fields, or their conjugates in the case when there is only one annihilation operator.
Secondly, we discuss the Feynman-type graph representation resulting from the product formula. We show that there is a correspondence between the packed integer matrices of the theory of noncommutative symmetric functions and these Feynman-type graphs. We show that, for some values of the parameters, one recovers the algebra of Noncommutative Matrix Quasi-symmetric functions.
On the way, this deformation provides, as a specialized homomorphic image, the algebra of polylogarithms.


Nous considérons deux aspects de la formule du produit appliquée aux séries formelles de la théorie combinatoire des champs. D'abord, on remarque que le cas où les fonctions ont un terme constant est d'un intérêt spécial parce qu'elle donnent lieu à des groupes de substitution. Les groupes issus du problème 


References

[1] C. M. Bender, D. C. Brody, and B. K. Meister, Quantum field theory of partitions, J. Math. Phys. Vol 40 (1999)

[2] Gerard Duchamp, Karol A. Penson, Allan I. Solomon, Andrej Horzela, Pawel BlasiakOne-parameter groups and combinatorial physics, 
arXiv:quant-ph/0401126v1 [quant-ph]

[3] An interface between physics and number theory
Gérard Henry Edmond Duchamp (LIPN), Vincel Hoang Ngoc Minh (LIPN), Allan I. Solomon (LPTMC), Silvia Goodenough (LIPN), 
arXiv:1011.0523v2 [math-ph]