Skip to main content
SpringerLink
Log in

The renormalization flow, spaces of two-dimensional field theories, and Connes' geometry

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We formulate a local renormalization flow using Connes' non-commutative geometry. This formulation allows a geometric description of the renormalization flow, and an intrinsic characterization of the operator product expansion. We define spaces for string theory, in terms of a ring of correlation functions and a renormalization flow on this ring, which are the analogues of the category of Riemannian manifolds with metric for general relativity. The beta function is related to a differential form of relative entropy between two renormalization flow trajectories.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys.5, 848–861 (1964)

    Google Scholar 

  2. Wilson, K. G., Kogut, J. B.: The renormalization group and theε expansion. Phys. Rep.12C, 75–199 (1974)

    Google Scholar 

  3. Wilson, K. G.: The renormalization group: Critical phenomena and the Kondo problem. Rev. Mod. Phys.47, 773–840 (1975)

    Google Scholar 

  4. Kadanoff, L. P.: Scaling, Universality and operator algebras. Phase transitions and critical phenomena, Domb, C., Green, M. S., (eds.), v. 5a, 1–34, New York: Academic Press 1976

    Google Scholar 

  5. Polyakov, A. M.: Microscopic description of critical phenomena. Sov. Phys.JETP 28, 533–539, (1969); Properties of long and short range correlations in the critical region.30, 151–157, (1970); Non-Hamiltonian approach to conformal field theory.39, 10–18 (1974)

    Google Scholar 

  6. Friedan, D. H.: Nonlinear models in 2+ε dimensions. Ann. Phys.163, 318–419 (1985)

    Google Scholar 

  7. Polyakov, A. M.: Directions in string theory. Lecture at the International Congress of Mathematicians, Berkeley (July 1986)

  8. Polchinski, J.: Renormalization and effective Lagrangians. Nucl. Phys.B231, 269–295 (1984)

    Google Scholar 

  9. Warr, B.: Renormalization of gauge theories using effective Lagrangians. Caltech preprints CALT-68-1334, 1393 (1986)

  10. Phase transitions and critical phenomena. Domb, C., Green, M. S. (eds.), v. 6, New York: Academic Press 1976

    Google Scholar 

  11. Amit, D. J.: Field theory, the renormalization group, and critical phenomena, 2nd, ed., Singapore: World Scientific 1984

    Google Scholar 

  12. Patashinskii, A. Z., Pokrovskii, V. L.: Fluctuation theory of phase transitions. Shepherd, Tr. P. J. (ed.) Oxford: Academic Press 1979

    Google Scholar 

  13. Lovelace, C.: Stability of string vacua (I). A new picture of the renormalization group. Nucl. Phys.B273, 413–467 (1986)

    Google Scholar 

  14. Banks, T., Martinec, E. J.: Renormalization group and string field theory. Nucl. Phys.B294, 733–746 (1987)

    Google Scholar 

  15. Belavin, A. A., Polyakov A. M., Zamolodchikov, A. B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys.B241, 333–380 (1984); Infinite Conformal Symmetry of Critical Fluctuations in Two Dimensions. J. Stat. Phys.34, 763–774 (1984)

    Google Scholar 

  16. Gross, D. J.: Applications of the renormalization group to high-energy physics. Les Houches Summer Institute 1975. Balian, R., Zinn-Justin, J., (eds.), pp. 141–250, Amsterdam: North-Holland 1976

    Google Scholar 

  17. Schoen, R.: Morse lectures. The Institute for Advanced Study, Princeton (November 1987)

  18. Connes, A.: Sur la theorie non commutative de l'integration. Algèbres de Opérateurs. Harpe, P. de la (ed.), Lecture. Notes in Mathematics vol. 725, pp. 19–143, Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  19. In preparation; to appear in Mod. Phys. Lett. A

  20. Emch, G. G.: TheC*-algebraic approach to phase transitions. Phase transitions and critical phenomena, Domb, C., Green, M. S., (eds.), v. 1, 137–175, New York: Academic Press 1972

    Google Scholar 

  21. Thirring, W.: A course in mathematical physics, vol. 4, Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  22. Periwal, V.: Princeton University dissertation (1988)

  23. Connes, A.:C* algebres et géométrie differentielle. C. R. Acad. Sci. Paris Ser A290 599–604 (1980); Non-commutative Differential Geometry. Pub. Math. I.H.E.S.62, 41–144 (1986)

    Google Scholar 

  24. Connes, A.: A survey of foliations and operator algebras. Proc. Symp. in Pure Math., A.M.S,38, (Part 1) 521–628 (1982)

    Google Scholar 

  25. Connes, A.: Lectures at the NATO-ASI Cargèse Summer School, July 1987

  26. Karoubi, M.: K-theory. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  27. Bott, R.: On a topological obstruction to integrability. Proc. Symp. in Pure Math., A.M.S.,16, 127–132 (1970); On topological obstructions to integrability, Proc. International Congress of Mathematicians (Nice, 1970). v. 1, 27–36. Paris: Gauthier-Villars 1971

    Google Scholar 

  28. Polchinski, J.: Vertex operators in the polyakov path integral. Nucl. Phys.B289, 465–483 (1987)

    Google Scholar 

  29. Wightman, A. S., Gårding, A.: Fields as operator-valued distributions in relativistic quantum theory. Arkiv. Fysik28, 129–184 (1964)

    Google Scholar 

  30. Callan, C. G., Gan, Z.: Vertex operators in background fields. Nucl. Phys.B272, 647–660 (1986)

    Google Scholar 

  31. Ripper, J.: In: Dr. Strangelove, dir. S. Kubrick, 1962

  32. Wilson, K. G.: Non-Lagrangian models of current algebras. Phys. Rev.179, 1499–1512 (1969); Renormalization group and strong interactions.D3, 1818–1846 (1971)

    Google Scholar 

  33. Zimmerman, W.: Local operator products and renormalization in quantum field theory, Brandeis Summer Inst. in Theor. Phys., 1970. vol 1, pp. 395–589. Deser, S., Grisaru, M., Pendleton, H., (eds.). Cambridge: MIT Press 1970

    Google Scholar 

  34. Callan Jr C. G.: Introduction to renormalization theory. Les Houches Summer Institute 1975, Balian, R., Zinn-Justin J., (eds.), pp. 41–78. Amsterdam: North Holland 1976

    Google Scholar 

  35. Brézin, E.: Applications of the renormalization group to critical phenomena, Les Houches Summer Institute 1975, Balian, R., Zinn-Justin, J., (eds.), pp. 329–386. Amsterdam: North-Holland 1976

    Google Scholar 

  36. Gross, D. J.: The Heterotic String. Proc. Santa Barbara Workshop on Unified String Theories, pp. 357–399. Green, M. B., Gross, D. J., (eds.), Singapore: World Scientific, 1986, and references therein

    Google Scholar 

  37. Fischler, W., Susskind, L.: Dilaton tadpoles, string condensates and scale invariance. Phys. Lett171B, 383–389 (1986),173B, 262–266 (1986)

    Google Scholar 

  38. Connes, A., Narnhofer, H., Thirring, W.: Dynamical entropy ofC* algebras and von Neumann algebras. Commun. Math. Phys.112, 691–719 (1987)

    Google Scholar 

  39. Zamolodchikov, A. B.: Irreversibility of the flux of the renormalization group in a 2D field theory. JET Lett.43, 730–732 (1986)

    Google Scholar 

  40. Klebanov, I., Susskind, L.: Renormalization group and string amplitudes. Phys. Lett.200B, 446 (1988)

    Google Scholar 

  41. Stueckelberg, E. C. G., Petermann, E.: La normalization des constantes dans la theorie des quanta. Helv. Phys. Acta26, 499–520 (1953)

    Google Scholar 

  42. Gell-Mann, M., Low, F. E.: Quantum electrodynamics at small distances. Phys. Rev.95, 1300–1312 (1954)

    Google Scholar 

Download references

Author information

Author notes

  1. Vipul Periwal

    Present address: Institute for Theoretical Physics, University of California, 93106, Santa Barbara, CA, USA

Authors and Affiliations

  1. Joseph Henry Laboratories, Princeton University, 08544, Princeton, New Jersey, USA

    Vipul Periwal

Authors
  1. Vipul Periwal
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by L. Alvarez-Gaumé

Rights and permissions

Reprints and permissions

About this article

Cite this article

Periwal, V. The renormalization flow, spaces of two-dimensional field theories, and Connes' geometry. Commun.Math. Phys. 120, 71–95 (1988). https://doi.org/10.1007/BF01223206

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01223206

Keywords

Search

Navigation