10 research outputs found
Rational Hadamard products via Quantum Diagonal Operators
We use the remark that, through Bargmann-Fock representation, diagonal
operators of the Heisenberg-Weyl algebra are scalars for the Hadamard product
to give some properties (like the stability of periodic fonctions) of the
Hadamard product by a rational fraction. In particular, we provide through this
way explicit formulas for the multiplication table of the Hadamard product in
the algebra of rational functions in \C[[z]]
Higher loop corrections to a Schwinger--Dyson equation
We consider the effects of higherloop corrections to a Schwinger--Dyson
equations for propagators. This is made possible by the efficiency of the
methods we developed in preceding works, still using the supersymmetric
Wess--Zumino model as a laboratory. We obtain the dominant contributions of the
three and four loop primitive divergences at high order in perturbation theory,
without the need for their full evaluations. Our main conclusion is that the
asymptotic behavior of the perturbative series of the renormalization function
remains unchanged, and we conjecture that this will remain the case for all
finite order corrections.Comment: 12 pages, 2 imbedded TiKZ pictures. A few clarifications matching the
published versio
Some Open Problems in Combinatorial Physics
We point out four problems which have arisen during the recent research in
the domain of Combinatorial Physics
A Three Parameter Hopf Deformation of the Algebra of Feynman-like Diagrams
We construct a three-parameter deformation of the Hopf algebra \LDIAG. This
is the algebra that appears in an expansion in terms of Feynman-like diagrams
of the {\em product formula} in a simplified version of Quantum Field Theory.
This new algebra is a true Hopf deformation which reduces to \LDIAG for some
parameter values and to the algebra of Matrix Quasi-Symmetric Functions
(\MQS) for others, and thus relates \LDIAG to other Hopf algebras of
contemporary physics. Moreover, there is an onto linear mapping preserving
products from our algebra to the algebra of Euler-Zagier sums
Combinatorial Deformations of Algebras: Twisting and Perturbations
The framework used to prove the multiplicative law deformation of the algebra
of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact,
twice). We give here a clear interpretation of its two parameters. The crossing
parameter is a deformation of the tensor structure whereas the superposition
parameters is a perturbation of the shuffle coproduct of Hoffman type which, in
turn, can be interpreted as the diagonal restriction of a superproduct. Here,
we systematically detail these constructions
Higher loop corrections to a Schwinger-Dyson equation
We consider the effects of higher loop corrections to a Schwinger–Dyson equation for propagators. This is made possible by the efficiency of the methods we developed in preceding works, still using the supersymmetric Wess–Zumino model as a laboratory. We obtain the dominant contributions of the three and four-loop primitive divergences at high order in perturbation theory, without the need for their full evaluations. Our main conclusion is that the asymptotic behavior of the perturbative series of the renormalization function remains unchanged, and we conjecture that this will remain the case for all finite order corrections.Facultad de Ciencias Exacta
Renormalization in combinatorially non-local field theories: the Hopf algebra of 2-graphs
It is well known that the mathematical structure underlying renormalization
in perturbative quantum field theory is based on a Hopf algebra of Feynman
diagrams. A precondition for this is locality of the field theory.
Consequently, one might suspect that non-local field theories such as matrix or
tensor field theories cannot benefit from a similar algebraic understanding.
Here I show that, on the contrary, the renormalization and perturbative
diagramatics of a broad class of such field theories is based in the same way
on a Hopf algebra. These theories are characterized by interaction vertices
with graphs as external structure leading to Feynman diagrams which can be
summed up under the concept of "2-graphs". From the renormalization
perspective, such graph-like interactions are as much local as point-like
interactions. They differ in combinatorial details as I exemplify with the
central identity for the perturbative series of combinatorial correlation
functions. This sets the stage for a systematic study of perturbative
renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger
equations, for a number of combinatorially non-local field theories with
possible applications to quantum gravity, statistical models and more.Comment: 22 pages, v2 minor adaptions for consistency with arXiv:2103.0113
Hopf algebras of diagrams
23 pages, LaTEXInternational audienceWe investigate several Hopf algebras of diagrams related to Quantum Field Theory of Partitions and whose product comes from the Hopf algebras WSym or WQSym respectively built on integer set partitions and set compositions. Bases of these algebras are indexed either by bipartite graphs (labelled or unlabbeled) or by packed matrices (with integer or set coefficients). Realizations on biword are exhibited, and it is shown how these algebras fit into a commutative diagram. Hopf deformations and dendriform structures are also considered for some algebras in the picture