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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
Article

Abstract

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.

Keywords

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.

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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

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