Spitzer's identity and the algebraic Birkhoff decomposition in pQFT

In this paper we continue to explore the notion of Rota–Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the solutions of the recursively defined formulae for the Birkhoff factorization of regularized Hopf algebra characters, i.e. Feynman rules, naturally give a non-commutative generalization of the well-known Spitzer's identity. The underlying abstract algebraic structure is analysed in terms of complete filtered Rota–Baxter algebras.

[1]  Li Guo,et al.  Mixable shuffles, quasi-shuffles and Hopf algebras , 2006 .

[2]  Alain Connes,et al.  Renormalization in quantum field theory and the Riemann-Hilbert problem , 1999 .

[3]  Li Guo.  ON THE TYPE AND TYPE PRODUCT OF DENDRIFORM ALGEBRAS , 2003 .

[4]  J. Kingman.  Spitzer's Identity and its use in Probability Theory , 1962 .

[5]  Alain Connes,et al.  Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem , 2000 .

[6]  M. A. Semenov-Tyan-Shanskii.  What is a classical r-matrix? , 1983 .

[7]  Alain Connes,et al.  Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group , 2001 .

[8]  K. Ebrahimi-Fard.  Loday-Type Algebras and the Rota–Baxter Relation , 2002 .

[9]  J. G. Wendel.  Brief Proof of a Theorem of Baxter. , 1962 .

[10]  Dirk Kreimer.  Chen’s iterated integral represents the operator product expansion , 1999 .

[11]  Integrable Renormalization II: The General Case , 2004, hep-th/0403118.

[12]  F. Spitzer.  A Combinatorial Lemma and its Application to Probability Theory , 1956 .

[13]  Gian-Carlo Rota,et al.  Baxter algebras and combinatorial identities. II , 1969 .

[14]  A. G. Reyman,et al.  Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems , 1994 .

[15]  Li Guo,et al.  Baxter Algebras and Shuffle Products , 2000 .

[16]  Le Bois-Marie,et al.  Integrable Renormalization I : the Ladder Case , 2004 .

[17]  Le Bois-Marie,et al.  Integrable Renormalization II : the general case , 2004 .

[18]  D. Manchon.  Hopf algebras, from basics to applications to renormalization , 2004, math/0408405.

[19]  G. Baxter,et al.  AN ANALYTIC PROBLEM WHOSE SOLUTION FOLLOWS FROM A SIMPLE ALGEBRAIC IDENTITY , 1960 .

[20]  Pierre Cartier,et al.  On the structure of free baxter algebras , 1972 .

[21]  Joseph C. Várilly,et al.  Elements of Noncommutative Geometry , 2000 .

[22]  M. Semenov-Tian-Shansky.  Integrable Systems and Factorization Problems , 2002, nlin/0209057.

[23]  Li Guo.  BAXTER ALGEBRAS AND DIFFERENTIAL ALGEBRAS , 2002 .

[24]  Li Guo,et al.  On Free Baxter Algebras: Completions and the Internal Construction , 2000 .

[25]  Denis Bernard,et al.  Introduction to classical integrable systems , 2003 .

[26]  F. Atkinson,et al.  Some aspects of Baxter's functional equation , 1963 .

[27]  Dirk Kreimer,et al.  On the Hopf algebra structure of perturbative quantum field theories , 1997 .

[28]  Li Jin-q,et al.  Hopf algebras , 2019, Graduate Studies in Mathematics.

[29]  Alain Connes,et al.  Hopf Algebras, Renormalization and Noncommutative Geometry , 1998 .

[30]  Marcelo Aguiar,et al.  Pre-Poisson Algebras , 2000 .

[31]  Gian-Carlo Rota,et al.  Ten Mathematics Problems I will never solve , 1998 .