Spitzer's identity and the algebraic Birkhoff decomposition in pQFT

By Kurusch Ebrahimi-Fard, Li Guo, Dirk Kreimer. 2004

157 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/29f637379d6fad8837968e3a0314fb8c144b118a

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Citations in the corpus

Operated semigroups, Motzkin paths and rooted trees (2007, unknown): “Given the recent progresses on Rota-Baxter algebra in both theoretical and application aspects [1, 2, 10, 16, 17, 19, 20, 26, 27, 29, 30, 31], it is desirable to obtain convenient constructions of free Rota-Baxter algebras.”
FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES (2005, methodology): “Later, a third construction was obtained by the second named author and Keigher [36, 37] as a generalization of shuffle product algebras.”
On integro-differential algebras (2012, background): “Rota-Baxter algebras have found extensive applications in mathematics and physics, including quantum field theory and the classical Yang-Baxter equation [4, 12, 13, 16, 17, 23].”
Renormalization of multiple zeta values (2006, methodology): “The renormalization procedure of QFT was put in the framework of Hopf algebra and Rota-Baxter algebra by the recent works of Connes and Kreimer [12, 13], continued in [14, 18, 19], and thus made possible for applications beyond QFT.”
Mixable shuffles, quasi-shuffles and Hopf algebras (2005, background): “In recent years, Rota-Baxter algebras have found applications in quantum field theory [8, 9, 15, 16, 17], dendriform algebras [1, 10, 13, 31], number theory [22], Hopf algebras [2] and combinatorics [21].”
Rota–Baxter algebras and dendriform algebras (2005, background): “The most prominent of these is the work [14, 15] of Connes and Kreimer in their Hopf algebraic approach to renormalization theory in perturbative quantum field theory, continued in a series of papers [18, 22, 25, 26, 27, 28, 29, 30, 31].”
On Differential Rota-Baxter Algebras (2007, methodology): “Likewise, the integral operator P and summation operator have been abstracted to give the notion of Rota-Baxter operators (previously called Baxter operators) and Rota-Baxter algebras [3, 25, 26].”