Algèbre de Hopf des diagrammes de Feynman, renormalisation et factorisation de Wiener-Hopf

Boutet de Monvel, Louis

Séminaire Bourbaki : volume 2001/2002, exposés 894-908, Astérisque, no. 290 (2003), Exposé no. 900, 17 p.

[CK1] A. Connes & D. Kreimer - « Renormalization in quantum field theory and the Riemann-Hilbert problem I : The Hopf algebra structure of graphs and the main theorem », Comm. Math. Phys. 210 (2000), no. 1, p. 249-273. | MR | Zbl

[CK2] _, « Renormalization in quantum field theory and the Riemann-Hilbert problem. II : the β-function, diffeomorphisms and the renormalization group », Comm. Math. Phys. 216 (2001), no. 1, p. 215-241. | Zbl

[BK1] D. J. Broadhurst & D. Kreimer - « Knots and numbers in Ø4 theory to 7 loops and beyond », Internat. J. Modern Phys. C 6 (1995), no. 4, p. 519-524. | MR | Zbl

[BK2] _, « Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops », Phys. Lett. B 393 (1997), no. 3-4, p. 403-412. | MR | Zbl

[BK3] D. J. Broadhurst, J. A. Gracey & D. Kreimer - « Beyond the triangle and uniqueness relations : non-zeta counterterms at large N from positive knots », Z. Phys. C 75 (1997), no. 3, p. 559-574. | MR

[BK4] D. J. Broadhurst & D. Kreimer - « Feynman diagrams as a weight system : four-loop test of a four-term relation », Phys. Lett. B 426 (1998), no. 3-4, p. 339-346. | MR | Zbl

[BK5] _, « Renormalization automated by Hopf algebra », J. Symbolic Comput. 27 (1999), no. 6, p. 581-600, and hep-th/9810087. | MR | Zbl

[BK6] _, « Combinatoric explosion of renormalization tamed by Hopf algebra : 30-loop Padé-Borel resummation », Phys. Lett. B 475 (2000), no. 1-2, p. 63- 70, and hep-th/9912093. | MR | Zbl

[BK7] _, « Towards cohomology of renormalization : bigrading the combinatorial Hopf algebra of rooted trees », Comm. Math. Phys. 215 (2000), no. 1, p. 217-236. | MR | Zbl

[CK3] A. Connes & D. Kreimer - « Hopf algebras, renormalization and noncommutative geometry », Comm. Math. Phys. 199 (1998), no. 1, p. 203-242. | MR | Zbl

[CK4] _, « Hopf algebras, renormalization and noncommutative geometry », in Quantum field theory : perspective and prospective (Les Houches 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 530, Kluwer Acad. Publ., Dordrecht, 1999, p. 59-108. | MR | Zbl

[CK5] _, « Renormalization in quantum field theory and the Riemann-Hilbert problem », J. High Energy Phys. (1999), no. 9, p. Paper 24, 8 pp., (electronic) and hep-th/9909126. | MR | Zbl

[CK6] _, « Lessons from quantum field theory : Hopf algebras and spacetime geometries, Moshé Flato (1937-1998) », Lett. Math. Phys. 48 (1999), no. 1, p. 85-96, and hep-th/9904044. | MR | Zbl

[CK7] _, « From local perturbation theory to Hopfand Lie-algebras of Feynman graphs », in Mathematical physics in mathematics and physics (Siena, 2000), Fields Inst. Commun., vol. 30, Amer. Math. Soc., 2001, p. 105-114. | MR | Zbl

[DK] R. Delbourgo & D. Kreimer - « Using the Hopf algebra structure of QFT in calculations », Phys. Rev. D (3) 60 (1999), no. 10, and hep-th/9903249. | MR

[K1] D. Kreimer - « Renormalization and knot theory », J. Knot Theory and Ramifications 6 (1997), no. 4, p. 479-581. | MR | Zbl

[K2] _, « On the Hopf algebra structure of perturbative quantum field theories », Adv. Theor. Math. Phys. 2 (1998), no. 2, p. 303-334. | MR | Zbl

[K3] _, « On overlapping divergences », Comm. Math. Phys. 204 (1999), no. 3, p. 669-689, and hep-th/9810022. | MR

[K4] _, « Chen's iterated integral represents the operator product expansion », Adv. Theor. Math. Phys. 3 (1999), no. 3, and hep-th/9901099. | Zbl

[K5] _, Knots and Feynman diagrams, Cambridge Lecture Notes in Physics, vol. 13, Cambridge University Press, Cambridge, 2000. | MR | Zbl

[K6] _, « Shuffling quantum field theory », Lett. Math. Phys. 51 (2000), no. 3, p. 179-191. | MR | Zbl

[Coll] J. Collins - Renormalization,, Cambridge monographs in math. phys., Cambridge University Press, Cambridge, 1984. | MR | Zbl

[Dres] M. Dresden - « Renormalization in historical perspective - The first stage », in Renormalization, Springer-Verlag, New York, Berlin, Heidelberg, 1994. | MR

[Drou] J.-M. Drouffe & C. Itzykson - Théorie statistique des champs, Savoirs actuels, InterEditions/Editions du C.N.R.S., 1989, 2 volumes.

[EG] H. Epstein & V. Glaser - « The role of locality in perturbation theory », Ann. Inst. H. Poincaré A 19 (1973), p. 211-295. | Numdam | MR

[FMRS] J. Feldman, J. Magnen, V. Rivasseau & R. Seneor - « Massive Gross-Neveu model : a rigorous perturbative construction », Phys. Rev. Lett. 54 (1985). | MR

[GK] K. Gawedski & A. Kupianen - « Exact renormalization of the Gross-Neveu model of quantum fields », Phys. Rev. Lett 54 (1985).

[GJ] J. Glimm & A. Jaffe - Quantum Physics, Springer-Verlag, New York, Berlin, Heidelberg, 1987. | MR | Zbl

[LeBe] M. Le Bellac - Des phénomènes critiques aux champs de jauge, Savoirs actuels, InterEditions/Editions du C.N.R.S., 1988. | MR

[ZJ] J. Zinn-Justin - Quantum Field Theory and Critical Phenomena, International series of monographies on physics, vol. 92, Oxford science publications, 1996. | MR | Zbl

[Beau] A. Beauville - « Monodromie des systèmes différentiels linéaires à pôles simples sur la sphère de Riemann », in Sém. Bourbaki, Astérisque, vol. 216, Soc. Math. France, Paris, 1993, exp. n° 765 (mars 1993), p. 103-119. | Numdam | MR | Zbl

[Boli] A. Bolibruch - « Fuchsian systems with reducible monodromy and the Riemann-Hilbert problem », Lecture Notes in Math., vol. 1250, Springer, 1992, p. 139-155. | MR | Zbl

[BKI] N. Bourbaki - Éléments de mathématique. Algèbre. Chapitres 1 à 3, Masson, Paris, 1982. | MR

[ENS] L. Boutet De Monvel, A. Douady & J.-L. Verdier (éds.) - Mathématique et Physique, Séminaire de l'E.N.S. 1979-82, Progress in Math., vol. 35, Birkhäuser, 1983. | MR | Zbl

[Drin] V. G. Drinfel'D - « Almost cocommutative Hopf algebras », Algebra i Analiz 1 (1989), no. 2, p. 30-46, and translation in Leningrad Math. J. 1 (1990), no. 2, p. 321-342. | MR | Zbl

[WH] I. Gohberg & M. A. Kaashoek (éds.) - Constructive methods of Wiener-Hopf factorization, Operator Theory : Advances and Applications, vol. 21, Birkhäuser Verlag, Basel, 1986. | MR | Zbl

[LP] I. Lappo-Danilevskii - Mémoire sur la théorie des systèmes d'équations différentielles linéaires, Chelsea, New York, 1953. | Zbl

[Pat1] F. Patras & C. Reutenauer - « Higher Lie idempotents », J. Algebra 222 (1999), no. 1, p. 51-64. | MR | Zbl

[Pat2] F. Patras - « La décomposition en poids des algèbres de Hopf », Ann. Inst. Fourier 43 (1993), no. 4, p. 1067-1087. | Numdam | MR | Zbl

[Spe] F.-O. Speck - General Wiener-Hopf factorization methods, vol. 119, Pitman (Advanced Publishing Program), Boston, MA, 1985, With a foreword by E. Meister. | MR | Zbl

[BCE] J. Brodzki, A. Connes & D. Ellwood - « Polarized modules and Fredholm modules », Mat. Fiz. Anal. Geom. 2 (1995), no. 1, p. 15-24. | MR | Zbl

[C1] A. Connes - Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994, 661 pp. | MR | Zbl

[C2] _, « Non-commutative geometry and physics », in Gravitation et quantifications (Les Houches, 1992), North-Holland, Amsterdam, 1995, p. 805- 950. | MR | Zbl

[C3] _, « Géométrie non commutative et physique quantique », in Mathématiques quantiques, SMF Journ. Annu., Soc. Math. France, Paris, 1992, 20 pp. | MR | Zbl

[C4] _, « The action functional in noncommutative geometry », Comm. Math. Phys. 117 (1988), no. 4, p. 673-683. | MR | Zbl

[CM] A. Connes & H. Moscovici - « Hopf algebras, cyclic cohomology and the transverse index theorem », Comm. Math. Phys. 198 (1998), no. 1, p. 199- 246. | MR | Zbl

[CS] A. Connes & E. Størmer - « A connection between the classical and the quantum mechanical entropies », in Operator algebras and group representations (Neptun, 1980), vol. I, Monographs Stud. Math., vol. 17, Pitman, Boston, Mass.-London, 1984, p. 113-123. | MR | Zbl