Communications in Mathematical Physics

, Volume 242, Issue 1–2, pp 31–65 | Cite as

From Tracial Anomalies to Anomalies in Quantum Field Theory

  • Alexander CardonaEmail author
  • Catherine Ducourtioux
  • Sylvie Paycha


ζ-regularized traces, resp. super-traces, are defined on a classical pseudo-differential operator A by: \({{{{tr}}^Q(A):= {{f.p.}} \, {{tr}}(A Q^{{-z}})_{{\vert_{{z=0}}}}, \quad{{resp.}} \quad {{str}}^Q(A):= {{f.p.}} \, {{str}}(A Q^{{-z}})_{{\vert_{{z=0}}}},}}\) where f.p. refers to the finite part and Q is an (invertible and admissible) elliptic reference operator with positive order. They are commonly used in quantum field theory in spite of the fact that, unlike ordinary traces on matrices, they are neither cyclic nor do they commute with exterior differentiation, thus giving rise to tracial anomalies. The purpose of this article is to show, on two examples, how tracial anomalies can lead to anomalous phenomena in quantum field theory.


Field Theory Quantum Field Theory Positive Order Finite Part Reference Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, S.: Axial vector vertex in spinor electrodynamics. Phys. Rev. 177, 2426–2438 (1969)CrossRefGoogle Scholar
  2. 2.
    Adams, D., Sen, S.: Phase and scaling properties of determinants arising in topological field theories. Phys. Lett. 353, 495–500 (1995)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Alvarez-Gaume, L.: Supersymmetry and the Atiyah-Singer index theorem. Commun. Math. Phys. 90, 161–173 (1983)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Alvarez-Gaume, L., Della Pietra, S., Moore, G.: Anomalies and odd dimensions. Ann. Phys. 163, 288–317 (1985)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Arnlind, J., Mickelsson, J.: Trace extensions, determinant bundles, and gauge group cocycles. hep-th/0205126, 2002Google Scholar
  6. 6.
    Atiyah, M.: The Geometry and Physics of Knots. Cambridge: Cambridge University Press, 1990Google Scholar
  7. 7.
    Atiyah, M., Patodi, V., Singer, I.M.: Spectral asymmetry and Riemannian Geometry I. Math. Proc. Camb. Phil. Soc. 77, 43–69 (1975)zbMATHGoogle Scholar
  8. 8.
    Atiyah, M., Patodi, V., Singer, I.M.: Spectral asymmetry and Riemannian Geometry II. Math. Proc. Camb. Phil. Soc. 78, 405–432 (1975)zbMATHGoogle Scholar
  9. 9.
    Atiyah, M., Patodi, V., Singer, I.M.: Spectral asymmetry and Riemannian Geometry III. Math. Proc. Camb. Phil. Soc. 79, 71–99 (1976)zbMATHGoogle Scholar
  10. 10.
    Atiyah, M., Singer, I.M.: Dirac operators coupled to vector potentials. Proc. Nath. Acad. Sci. USA 81, 2597–2600 (1984)zbMATHGoogle Scholar
  11. 11.
    Baadhio, R.: Quantum Topology and Global Anomalies. Adv. Ser. Math. Phys. 23, Singapore: World Scientific, 1996Google Scholar
  12. 12.
    Bell, J.S., Jackiw, R.: A PCAC Puzzle: π0→ γ γ in the σ model. Il Nuovo Cimento LX A, 47–61 (1969)Google Scholar
  13. 13.
    Bardeen, W.A.: Anomalous Ward identities in spinor field theories. Phys. Rev. 184, 1848–1859 (1969)CrossRefGoogle Scholar
  14. 14.
    Bertlmann, R.: Anomalies in Quantum Field Theory. Oxford: Oxford University Press, 1996Google Scholar
  15. 15.
    Bismut, J.-M., Freed, D.: The analysis of elliptic families I. Commun. Math. Phys. 106, 159–176 (1986)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Berlin-Heidelberg-New York: Springer-Verlag, 1992Google Scholar
  17. 17.
    Booss-Bavnek, B., Lesch, M., Phillips, J.: Spectral flow of paths of self-adjoint Fredholm operators. Nucl. Phys. (Proc. Suppl.) 104, 177–180 (2002); Unbounded Fredholm Operators and Spectral Flow. Preprint TEKST Nr 407, Roskilde University (2001)CrossRefGoogle Scholar
  18. 18.
    Cardona, A.: Geometry of Families of Elliptic Complexes, Duality and Anomalies. Ph.D. thesis, Mathematics Department, Université Blaise Pascal, 2002Google Scholar
  19. 19.
    Cardona, A., Ducourtioux, C., Magnot, J.-P., Paycha, S.: Weighted traces on algebras of pseudo-differential opertors and geometry on loop groups. Infinite Dim. Anal. Quant. Prob. Rel Top. 5(4), 503–540 (2002)CrossRefGoogle Scholar
  20. 20.
    Chern, S.-S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 99, 48–69 (1974)zbMATHGoogle Scholar
  21. 21.
    Cognola, G., Zerbini, S.: Consistent, covariant and multiplicative anomalies. hep-th-98110398, 1998Google Scholar
  22. 22.
    Dowker, J.S.: On the relevance of the multiplicative anomaly. hep-th/9803200, 1998Google Scholar
  23. 23.
    Ducourtioux, C.: Weighted Traces on Pseudo-differential Operators and Associated Determinants. Ph.D. thesis, Mathematics Department, Université Blaise Pascal, 2001Google Scholar
  24. 24.
    Eckstrand, C.: A simple algebraic derivation of the covariant anomaly and Schwinger term. J. Math. Phys. 41(11), 7294–7303 (2000)CrossRefGoogle Scholar
  25. 25.
    Eckstrand, C., Mickelsson, J.: Gravitational anomalies, gerbes and hamiltonian quantization. Commun. Math. Phys. 212, 613–624 (2000)CrossRefGoogle Scholar
  26. 26.
    Elizalde, E., Cognola, G., Zerbini, S.: Applications in physics of the multiplicative anomaly formula involving some basic differential operators. Nucl. Phys. B 532(1-2), 407–428 (1998)Google Scholar
  27. 27.
    Elizalde, E., Filippi, A., Vanzo, L., Zerbini, S.: Is the multiplicative anomaly relevant? hep-th/9804072, 1998Google Scholar
  28. 28.
    Freed, D., Uhlenbeck, K.: Instantons and Four-manifolds. Berlin-Heidelberg-New York: Springer-Verlag, 1984Google Scholar
  29. 29.
    Friedrich, R.: Dirac Operatoren in der Riemannschen Geometrie. Advanced Lectures in Mathematics, Vieweg, 1997Google Scholar
  30. 30.
    Fujikawa, K.: Path integral measure for gauge invariant fermion theories. Phys. Rev. Lett. 42, 1195 (1979)CrossRefGoogle Scholar
  31. 31.
    Gross, D.J., Jackiw, R.: Effect of anomalies on quasirenormalizable theories. Phys. Rev. D6, 477–493 (1972)Google Scholar
  32. 32.
    Grubb, G.: Functional calculus of pseudodifferential boundary problems. Progress in Mathematics 65, Basel-Boston: Birkhäuser, 1996Google Scholar
  33. 33.
    Kontsevich, M., Vishik, S.: Determinants of elliptic pseudo-differential operators. Max Planck Institut preprint, 1994Google Scholar
  34. 34.
    Lesch, M.: On the non commutative residue for pseudo-differential operators with log-polyhomogeneous symbols. Annals of Global Analysis and Geometry 17, 151–187 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Langmann, E., Mickelsson, J.: Elementary derivation of the chiral anomaly. Lett. Math. Phys. 36, 45–54 (1996)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lawson, H., Michelsohn, M.-L.: Spin Geometry. Princeton: Princeton University Press, 1989Google Scholar
  37. 37.
    Melrose, R.: The Atiyah-Patodi-Singer Index Theorem. Research Notes in Mathematics, Vol. 4, Wellesley, MA: A K Peters, Ltd., 1993Google Scholar
  38. 38.
    Melrose, R., Nistor, V.: Homology of pseudo-differential operators I. Manifolds with boundary. funct-an/9606005, June 1999Google Scholar
  39. 39.
    Mickelsson, J.: Second Quantization, anomalies and group extensions. In: Lecture notes given at the “Colloque sur les Méthodes Géométriques en physique, C.I.R.M, Luminy, June 1997; Wodzicki residue and anomalies on current algebras. In: ‘‘Integrable Models and Strings’’ A. Alekseev and al., (eds.), Lecture Notes in Physics 436, Berlin-Heidelberg-New York: Springer, 1994Google Scholar
  40. 40.
    Mickelsson, J., Rajeev, S.: Current algebras in d+1 dimensions and determinant bundles over infinite-dimensional Grassmainnians. Commun. Math. Phys. 116, 365–400 (1985)zbMATHGoogle Scholar
  41. 41.
    Nakahara, M.: Geometry, Topology and Physics. Bristol: Adam Hilger, 1990Google Scholar
  42. 42.
    Okikiolu, K.: The Campbell-Hausdorff theorem for elliptic operators and a related trace formula. Duke. Math. J. 79, 687–722 (1995); The multiplicative anomaly for determinants of elliptic operators. Duke. Math. J. 79, 723–750 (1995)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Paycha, S.: Renormalized traces as a looking glass into infinite dimensional geometry. Inf. Dim. Anal. Quant. Prob. Rel. Top. 4(2), 221–266 (2001)CrossRefGoogle Scholar
  44. 44.
    Paycha, S., Rosenberg, S.: Curvature on determinant bundles and first Chern forms. J. Geom. Phys. 45, 393–429 (2003)CrossRefzbMATHGoogle Scholar
  45. 45.
    Paycha, S., Rosenberg, S.: Traces and characteristic classes on Loop spaces. To appear in ‘‘Infinite dimensional groups and manifolds’’. Proceedings of the 70th Meeting of Theoretical Physicists and Mathematicians held in Strasbourg, May 23–25, 2002. Edited by Tilmann Wurzbacher. IRMA Lectures in Mathematics and Theoretical Physics. Berlin: Walter de Gruyter & Co.Google Scholar
  46. 46.
    Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl. 19, 37–41 (1985)zbMATHGoogle Scholar
  47. 47.
    Quillen, D.: Superconnections and the Chern character. Topology 24, 89–95 (1985)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Radul, A.O.: Lie algebras of differential operators, their central extensions, and W-algebras. Funct. Anal. Appl. 25, 25–39 (1991)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Ray, D.B., Singer, I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)zbMATHGoogle Scholar
  50. 50.
    Schwarz, A.: The partition function of a degenerate functional. Commun. Math. Phys. 67, 1–16 (1979)MathSciNetGoogle Scholar
  51. 51.
    Singer, I.M.: Families of Dirac operators with applications to physics. Astérisque (hors série), 323–340 (1985)Google Scholar
  52. 52.
    Treiman, S., Jackiw, R., Zumino, B., Witten, E.: Current Algebra and Anomalies. Singapore: World Scientific, 1985Google Scholar
  53. 53.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Wodzicki, M.: Non-commutative residue. In: Lecture Notes in Mathematics, 1289, Berlin-Heidelberg-New York: Springer Verlag, 1987Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Alexander Cardona
    • 1
    Email author
  • Catherine Ducourtioux
    • 1
  • Sylvie Paycha
    • 1
  1. 1.Laboratoire de Mathématiques AppliquéesUniversité Blaise Pascal (Clermont II), Complexe Universitaire des CézeauxAubière CedexFrance

Personalised recommendations