The aim of this exposition is to provide a detailed description of the use of combinatorial algebra in quantum field theory in the planar setting. Particular emphasis is placed on the relations between different types of planar Green’s functions. The primary object is a Hopf algebra that is naturally defined on variables representing non-commuting sources, and whose coproduct splits into two halfcoproducts. The latter give rise to the notion of an unshuffle bialgebra. This setting allows a description of the relation between full and connected planar Green’s functions to be given by solving a simple linear fixed point equation. We also include a brief outline of the consequences of our approach in the framework of ordinary quantum field theory.
I. Singer, The master field for two-dimensional Yang– Mills theory, in: Proceedings 1994 Paris Conference on Mathematical Physics
6
D. Voiculescu, K. Dykema, and A. Nica, Free random variables, CRM Monograph Series 1, AMS, Providence, RI, 1992
7
D. Voiculescu, Free Probability Theory: Random Matrices and von Neumann Algebras, Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994, Birkhäusser Verlag, Basel, Switzerland, 1995
8
D. Voiculescu (Ed.), Free Probability Theory, Fields Institute Communications 12, 1997
P. Biane, Free probability and combinatorics, Proceedings of the International Congress of Mathematicians, Vol. II, Beijing: Higher Education Press, 2002, pp765–774
12
A. Nica and R. Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series 335, Cambridge University Press, 2006
13
J. Novak and P. Sniady, What is ... a free cumulant? Not. Am. Math. Soc. 58(2), 300 (2011)
14
J. Novak, Three lectures on free probability (with Michael LaCroix), “Random Matrix Theory, Interacting Particle Systems and Integrable Systems, MSRI Publications 65, 309 (2014)
15
R. Speicher, Free probability theory and non-crossing partitions, Sém., Lothar. Combin. 39, 38 (1997)
16
R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Memoir of the AMS 627, 1998
17
K. Ebrahimi-Fard and F. Patras, Cumulants, free cumulants and half-shuffles, Proc. R. Soc. A 471(2176), 20140843 (2015)
https://doi.org/10.1098/rspa.2014.0843
18
K. Ebrahimi-Fard and F. Patras, The splitting process in free probability, arXiv: 1502.02748
19
R. J. Rivers, Path Integral Methods in Quantum Field Theory, Cambridge Monographs on Mathematical Physics, 1988
20
C. Itzykson and J. B. Zuber, Quantum Field Theory, McGraw-Hill, 1980
21
M. E. Peskin and D. V. Schroeder, An Introduction To Quantum Field Theory, Westview Press, First Edition, 1995
K. Ebrahimi-Fard, A. Lundervold, and D. Manchon, Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras, Int. J. Algebra Comput. 24(05), 671 (2014)
https://doi.org/10.1142/S0218196714500283
26
J. Collins, Renormalization, Cambridge monographs in mathematical physics, Cambridge, 1984
27
O. I. Zavialov, Renormalized Quantum Field Theory, Kluwer Acad. Publ., 1990
28
S. Blanes, F. Casas, J. A. Oteo, and J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470(5- 6), 151 (2009)
29
W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7(4), 649 (1954)
https://doi.org/10.1002/cpa.3160070404
30
P. Cartier, A primer of Hopf algebras, in: Frontiers in Number Theory, Physics, and Geometry II, Berlin Heidelberg: Springer, 2007, pp 537–615
https://doi.org/10.1007/978-3-540-30308-4_12
31
C. Reutenauer, Free Lie Algebras, Oxford University Press, 1993
32
M. E. Sweedler, Hopf Algebras, New-York: Benjamin, 1969
33
A. Connes and D. Kreimer, Hopf Algebras, Renormalization and Noncommutative Geometry, Commun. Math. Phys. 199(1), 203 (1998)
https://doi.org/10.1007/s002200050499
34
A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann–Hilbert problem I: The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210(1), 249 (2000)
https://doi.org/10.1007/s002200050779
35
A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann–Hilbert problem II: The β-function, diffeomorphisms and the renormalization group, Commun. Math. Phys. 216(1), 215 (2001)
https://doi.org/10.1007/PL00005547
36
K. Ebrahimi-Fard, J. M. Gracia-Bondía, and F. Patras, A Lie theoretic approach to renormalization, Commun. Math. Phys. 276(2), 519 (2007)
https://doi.org/10.1007/s00220-007-0346-8
P. Cartier, Vinberg algebras, Lie groups and combinatorics, Clay Mathematical Proceedings 11, 107 (2011)
43
D. Manchon, A short survey on pre-Lie algebras, E. Schrödinger Institut Lectures in Math. Phys., “Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory”, Eur. Math. Soc., A. Carey (<Eds/>.), 2011
44
E. F. Kurusch, J. M. Gracia-Bondía, and F. Patras, Rota–Baxter algebras and new combinatorial identities, Lett. Math. Phys. 81(1), 61 (2007)
https://doi.org/10.1007/s11005-007-0168-9
45
K. Ebrahimi-Fard, D. Manchon, and F. Patras, A noncommutative Bohnenblust–Spitzer identity for Rota– Baxter algebras solves Bogolioubov’s recursion, J. Noncommut. Geom. 3(2), 181 (2009)
https://doi.org/10.4171/JNCG/35
46
K. Ebrahimi-Fard and F. Patras, The pre-Lie structure of the time-ordered exponential, Lett. Math. Phys. 104(10), 1281 (2014)
https://doi.org/10.1007/s11005-014-0703-4
K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fertype formula in dendriform algebras, Found. Comput. Math. 9(3), 295 (2009)
https://doi.org/10.1007/s10208-008-9023-3
49
F. Chapoton and F. Patras, Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula, Int. J. Algebra Comput. 23(04), 853 (2013)
https://doi.org/10.1142/S0218196713400134