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Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory

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Abstract

Graph grammars extend the theory of formal languages in order to model distributed parallelism in theoretical computer science. We show here that to certain classes of context-free and context-sensitive graph grammars one can associate a Lie algebra, whose structure is reminiscent of the insertion Lie algebras of quantum field theory. We also show that the Feynman graphs of quantum field theories are graph languages generated by a theory dependent graph grammar.

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References

  1. Abeille, A., Rambow, O.: Tree adjoining grammars, center for the study of language and information (2001)

  2. Agrachev A., Gamkrelidze R.: Chronological algebras and nonstationary vector fields. J. Sov. Math. 17(1), 1650–1675 (1981)

    Article  MATH  Google Scholar 

  3. Bachmann M., Kleinert H., Pelster A.: Recursive graphical construction for Feynman diagrams of quantum electrodynamics. Phys.Rev. D 61, 085017 (2000)

    Article  MathSciNet  Google Scholar 

  4. Burde D.: Left-symmetric algebras, or pre-Lie algebras in geometry and physics. Cent. Eur. J. Math. 4(3), 323–357 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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(1), 249–273 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Connes A., Kreimer D.: Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs. Ann. Henri Poincaré 3(3), 411–433 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. Colloquium Publications, Vol. 55. American Mathematical Society, Providence, RI (2008)

  8. Delaney, C., Marcolli, M.: Dyson–Schwinger equations in the theory of computation. arXiv:1302.5040, to appear in “Periods and Motives”, Clay Math Institute and AMS

  9. Ebrahimi-Fard K., Gracia-Bondia J.M., Patras F.: A Lie theoretic approach to renormalization. Commun. Math. Phys 276(2), 519–549 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ehrig H., Ehrig K., Prange U., Taentzer G.: Fundamentals of Algebraic Graph Transformation. Springer, New York (2010)

    Book  MATH  Google Scholar 

  11. Ehrig, H., Kreowski, H.J., Rozenberg, G.: Graph-grammars and their application to computer science. Lecture Notes in Computer Science, Vol. 532, Springer, New York (1990)

  12. Foissy L.: Lie algebras associated to systems of Dyson-Schwinger equations. Adv. Math. 226(6), 4702–4730 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Itzykson C., Zuber J.B.: Quantum Field Theory. Dover Publications, New York (2012)

    MATH  Google Scholar 

  14. Kleinert H., Pelster A., Kastening B., Bachmann M.: Recursive graphical construction of Feynman diagrams and their multiplicities in ϕ 4 and in ϕ 2 A theory. Phys. Rev. E 62, 1537–1559 (2000)

    Article  Google Scholar 

  15. Kreimer D.: On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2(2), 303–334 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Manchon, D.: A short survey on pre-Lie algebras, In: Noncommutative geometry and physics: renormalisation, motives, index theory, pp. 89–102, ESI Lect. Math. Phys., Eur. Math. Soc. (2011)

  17. Manchon D., Saidi A.: Lois pré-Lie en interaction. Commun. Algebra 39(10), 3662–3680 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Manin, Y.I.: Infinities in quantum field theory and in classical computing: renormalization program. In: Programs, proofs, processes, pp. 307–316. Lecture Notes in Computer Science, Vol. 6158, Springer, New York (2010)

  19. Nagl M.: Graph-Grammatiken: Theorie, Implementirung, Anwendung. Vieweg, Braunschweig (1979)

    Book  MATH  Google Scholar 

  20. Oudom J.M., Guin D.: On the Lie enveloping algebra of a pre-Lie algebra. J. K Theory 2(1), 147–167 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rozenberg G.: Handbook of Graph Grammars and Computing by Graph Transformation. Volume 1: Foundations. World Scientific, Singapore (1997)

    Book  MATH  Google Scholar 

  22. Rozenberg, G.: An introduction to the NLC way of rewriting graphs, In: Graph-grammars and their application to computer science, pp. 55–66. Lecture Notes in Computer Science, Vol. 532, Springer, New York (1990)

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Authors and Affiliations

  1. Mathematics Department, Caltech, 1200 E. California Blvd, Pasadena, CA, 91125, USA

    Matilde Marcolli & Alexander Port

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  1. Matilde Marcolli
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  2. Alexander Port
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Correspondence to Matilde Marcolli.

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Marcolli, M., Port, A. Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory. Math.Comput.Sci. 9, 391–408 (2015). https://doi.org/10.1007/s11786-015-0236-y

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  • DOI: https://doi.org/10.1007/s11786-015-0236-y

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