The combinatorics of Green’s functions in planar field theories

Abstract

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.

Keywords

planar field theory / Green’s functions / free probability / Hopf algebra / shuffle algebra / partitions

References

[1] P. Cvitanović, Planar perturbation expansion, Phys. Lett. B 99(1), 49 (1981)

[2] P. Cvitanović, P. G. Lauwers, and P. N. Scharbach, The planar sector of field theories, Nucl. Phys. B 203(3), 385 (1982)

[3] G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72(3), 461 (1974)

[4] T. P. Speed, Cumulants and partition lattices, Austral. J. Statist. 25(2), 378 (1983)

[5] 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

[9] M. Douglas, Stochastic master fields, Phys. Lett. B 344(1–4), 117 (1995)

[10] R. Gopakumar and D. J. Gross, Mastering the Master Field, Nucl. Phys. B 451(1–2), 379 (1995)

[11] 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)

[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

[22] G. Sterman, An Introduction to Quantum Field Theory, Cambridge: Cambridge University Press, 1993

[23] J. Schwinger, On the Green’s functions of quantized fields I+ II, Proc. Natl. Acad. Sci. USA 37 (7), 452–455, 455–459 (1951)

[24] J. S. Beissinger, The enumeration of irreducible combinatorial objects, J. Comb. Theory Ser. A 38(2), 143 (1985)

[25] K. Ebrahimi-Fard, A. Lundervold, and D. Manchon, Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras, Int. J. Algebra Comput. 24(05), 671 (2014)

[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)

[30] P. Cartier, A primer of Hopf algebras, in: Frontiers in Number Theory, Physics, and Geometry II, Berlin Heidelberg: Springer, 2007, pp 537–615

[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)

[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)

[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)

[36] K. Ebrahimi-Fard, J. M. Gracia-Bondía, and F. Patras, A Lie theoretic approach to renormalization, Commun. Math. Phys. 276(2), 519 (2007)

[37] J. M. Gracia-Bondía, J. C. Várilly, and H. Figueroa, Elements of Noncommutative Geometry, Boston: Birkhäuser, 2001

[38] D. Manchon, Hopf algebras and renormalisation, Handbook of Algebra 5, edited by M. Hazewinkel, 2008, pp 365–427

[39] L. Foissy and F. Patras, Natural endomorphisms of shuffle algebras, Int. J. Algebra Comput. 23(04), 989 (2013)

[40] L. Foissy, Bidendriform bialgebras, trees, and free quasisymmetric functions, J. Pure Appl. Algebra 209(2), 439 (2007)

[41] W. E. Caswell and A. D. Kennedy, A simple approach to renormalisation theory, Phys. Rev. D 25(2), 392 (1982)

[42] 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)

[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)

[46] K. Ebrahimi-Fard and F. Patras, The pre-Lie structure of the time-ordered exponential, Lett. Math. Phys. 104(10), 1281 (2014)

[47] K. Ebrahimi-Fard and D. Manchon, Dendriform equations, J. Algebra 322(11), 4053 (2009)

[48] K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fertype formula in dendriform algebras, Found. Comput. Math. 9(3), 295 (2009)

[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)

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