Journal of Mathematical Chemistry

, Volume 50, Issue 3, pp 552–576 | Cite as

Nonlocal, noncommutative diagrammatics and the linked cluster theorems

Original Paper

Abstract

Recent developments in quantum chemistry, perturbative quantum field theory, statistical physics or stochastic differential equations require the introduction of new families of Feynman-type diagrams. These new families arise in various ways. In some generalizations of the classical diagrams, the notion of Feynman propagator is extended to generalized propagators connecting more than two vertices of the graphs. In some others (introduced in the present article), the diagrams, associated to noncommuting product of operators inherit from the noncommutativity of the products extra graphical properties. The purpose of the present article is to introduce a general way of dealing with such diagrams. We prove in particular a “universal” linked cluster theorem and introduce, in the process, a Feynman-type “diagrammatics” that allows to handle simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams arising from the study of interacting systems (such as the ones where the ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or electric field, by impurities...) or Wightman fields (that is, expectation values of products of interacting fields). Our diagrammatics seems to be the first attempt to encode in a unified algebraic framework such a wide variety of situations. In the process, we promote two ideas. First, Feynman-type diagrammatics belong mathematically to the theory of linear forms on combinatorial Hopf algebras. Second, linked cluster-type theorems rely ultimately on Möbius inversion on the partition lattice. The two theories should therefore be introduced and presented accordingly. Among others, our theorems encompass the usual versions of the theorem (although very different in nature, from Goldstone diagrams in solid state physics to Feynman diagrams in QFT or probabilistic Wick theorems).

Keywords

Linked cluster theorem Feynman diagram Combinatorial Hopf algebra Cumulant Truncated moment function 

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References

  1. 1.
    Brouder Ch., Patras F.: Hyperoctahedral Chen calculus for effective Hamiltonians. J. Algebra 322, 4105–4120 (2009)CrossRefGoogle Scholar
  2. 2.
    Brouder Ch., Mestre A., Patras F.: Tree expansions in time-dependent perturbation theory. J. Math. Phys. 51, 072104 (2010)CrossRefGoogle Scholar
  3. 3.
    Ch. Brouder, G. Duchamp, F. Patras, G.Z. Toth. The Rayleigh-Schrödinger perturbation series of quasi-degenerate systems, 2010. arXiv:1011.1751v1 [quant-ph].Google Scholar
  4. 4.
    Brouder Ch., Patras F.: Decomposition into one-particle irreducible Green functions in many-body physics. Proceedings of the conference on combinatorics and physics, Bonn. Contemp. Math. 539, 1–25 (2011)Google Scholar
  5. 5.
    Djah S.H., Gottschalk H., Ouerdiane H.: Feynman graph representation of the perturbation series for general functional measures. J. Funct. Anal. 227, 153–187 (2005)CrossRefGoogle Scholar
  6. 6.
    Itzykson C., Zuber J.-B.: Quantum Field Theory. McGraw-Hill, New York (1980)Google Scholar
  7. 7.
    Stora R.: Renormalized perturbation theory: A missing chapter. Int. J. Geometr. Methods Modern Phys. 5(8), 1345–1360 (2008)CrossRefGoogle Scholar
  8. 8.
    Joni S.A., Rota G.-C.: Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61, 93–139 (1979)Google Scholar
  9. 9.
    Patras F., Reutenauer C.: On Dynkin and Klyachko idempotents in graded bialgebras. Adv. Appl. Math. 28, 560–579 (2002)CrossRefGoogle Scholar
  10. 10.
    Patras F.: La décomposition en poids des algèbres de Hopf. Ann. Inst. Fourier 43(4), 1067–1087 (1993)CrossRefGoogle Scholar
  11. 11.
    Patras F., Reutenauer C.: On descent algebras and twisted bialgebras. Moscow Math. J. 4(1), 199–216 (2004)Google Scholar
  12. 12.
    Patras F., Schocker M.: Twisted descent algebras and the Solomon-Tits algebra. Adv. Math. 199, 151–184 (2006)CrossRefGoogle Scholar
  13. 13.
    M. Aguiar, S. Mahajan, Monoidal Functors, Species and Hopf Algebras. CRM Monograph Series, Vol. 29, Montréal (2010)Google Scholar
  14. 14.
    Cassam-Chenai P., Patras F.: The Hopf algebra of identical, fermionic particle systems. Fundamental concepts and properties. J. Math. Phys. 44, 4484–4906 (2003)CrossRefGoogle Scholar
  15. 15.
    Brouder Ch., Fauser B., Frabetti A., Oeckl R.: Quantum field theory and Hopf algebra cohomology. J. Phys. A Math. Gen. 37, 5895–5927 (2004)CrossRefGoogle Scholar
  16. 16.
    Patras F., Schocker M.: Trees, set compositions and the twisted descent algebra. J. Algebr. Comb. 28, 3–23 (2008)CrossRefGoogle Scholar
  17. 17.
    Gurau R., Magnen J., Rivasseau V.: Tree quantum field theory. Ann. Henri Poincaré 10, 867–891 (2009)CrossRefGoogle Scholar
  18. 18.
    Leroux P., Bergeron F., Labelle G.: Combinatorial Species and Tree-like Structures, volume 67 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1998)Google Scholar
  19. 19.
    Ecalle J.: Singularités non abordables par la géométrie. Ann. Inst. Fourier 42, 73–143 (1992)CrossRefGoogle Scholar
  20. 20.
    Menous F.: On the stability of some groups of formal diffeomorphisms by the birkhoff decomposition. Adv. Math. 216, 1–28 (2007)CrossRefGoogle Scholar
  21. 21.
    Mattuck R.D.: A Guide to Feynman Diagrams in the Many-Body Problem 2nd edn. McGraw-Hill, New York (1976)Google Scholar
  22. 22.
    Connes A., Kreimer D.: 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, 249–273 (2000)CrossRefGoogle Scholar
  23. 23.
    Kleinert H., Schulte-Frohlinde V.: Critical Properties of \({\phi^4}\) Theories. World Scientific, Singapore (2001)CrossRefGoogle Scholar
  24. 24.
    Comtet L.: Advanced Combinatorics. Reidel, Dordrecht (1974)CrossRefGoogle Scholar
  25. 25.
    Brouder Ch.: Quantum field theory meets Hopf algebra. Math. Nachr. 282, 1664–1690 (2009)CrossRefGoogle Scholar
  26. 26.
    Gross E.K.U., Runge E., Heinonen O.: Many-Particle Theory. Adam Hilger, Bristol (1991)Google Scholar
  27. 27.
    Kira M., Koch S.W.: Cluster-expansion representation in quantum optics. Phys. Rev. A 78, 022102 (2008)CrossRefGoogle Scholar
  28. 28.
    Kutzelnigg W., Mukherjee D.: Normal order and extended Wick theorem for a multiconfiguration reference wave function. J. Chem. Phys. 107, 432–449 (1997)CrossRefGoogle Scholar
  29. 29.
    Kutzelnigg W., Mukherjee D.: Cumulant expansion of the reduced density matrices. J. Chem. Phys. 110, 2800–2809 (1999)CrossRefGoogle Scholar
  30. 30.
    Kutzelnigg W., Mukherjee D.: Direct determination of the cumulants of the reduced density matrices. Chem. Phys. Lett. 317, 567–574 (2000)CrossRefGoogle Scholar
  31. 31.
    Kong L., Nooijen M., Mukherjee D.: An algebraic proof of generalized Wick theorem. J. Chem. Phys. 132, 234107 (2010)CrossRefGoogle Scholar
  32. 32.
    S.H. Djah, H. Gottschalk, H. Ouerdiane, Feynman graphs for non-Gaussian measures. in Analyse et Probabilité, vol 16 of Séminaires et Congrès, ed. by P. Biane, J. Faraut, H. Ouerbiane (Soc. Math. Paris, France, 2008), pp. 35–54Google Scholar
  33. 33.
    Gottschalk H., Ouerdiane H., Smii B.: Convolution calculus on white noise spaces and Feynman diagrams representation of generalized renormalization flows. In: Cruzeiro, A.B., Ouerbiane, H., Obata, N. (eds) Mathematical Analysis of Random Phenomena, pp. 101–110. World Scientific, Singapore (2007)CrossRefGoogle Scholar
  34. 34.
    Gottschalk H., Smii B., Thaler H.: The Feynman graph representation of convolution semigroups and its applications to Lévy statistics. Bernoulli 14, 322–351 (2008)CrossRefGoogle Scholar
  35. 35.
    Lehner F., Belinschi S., Bozejko M., Speicher R.: The normal distribution is \({\boxplus}\) -infinitely divisible. Adv. Math. 226, 3677–3698 (2011)CrossRefGoogle Scholar
  36. 36.
    Ostendorf A.: Feynman rules for Wightman functions. Commun. Math. Phys. 40, 273–290 (1984)Google Scholar
  37. 37.
    Brunetti R., Fredenhagen K., Köhler M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633–652 (1996)CrossRefGoogle Scholar
  38. 38.
    Haag R.: Quantum field theories with composite particles and asymptotic conditions. Phys. Rev. 112, 669–673 (1958)CrossRefGoogle Scholar
  39. 39.
    Sanders K.: Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime. Commun. Math. Phys. 295, 485–501 (2010)CrossRefGoogle Scholar
  40. 40.
    Epstein H., Glaser V.: The role of locality in perturbation theory. Ann. Inst. Henri Poincaré 19, 211–295 (1973)Google Scholar
  41. 41.
    Araki H.: On asymptotic behavior of vacuum expectation values at large space-like separation. Ann. Phys. 11, 260–274 (1960)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut de Minéralogie et de Physique des Milieux Condensés, CNRS UMR 7590Université Pierre et Marie Curie-Paris 6Paris Cedex 05France
  2. 2.Laboratoire J.-A. Dieudonné, CNRS UMR 6621Université de NiceNice Cedex 02France

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