A master functional for quantum field theory

  • Damiano AnselmiEmail author
Regular Article - Theoretical Physics


We study a new generating functional of one-particle irreducible diagrams in quantum field theory, called master functional, which is invariant under the most general perturbative changes of field variables. The usual functional Γ does not behave as a scalar under the transformation law inherited from its very definition as the Legendre transform of W=lnZ, although it does behave as a scalar under an unusual transformation law. The master functional, on the other hand, is the Legendre transform of an improved functional W with respect to the sources coupled to both elementary and composite fields. The inclusion of certain improvement terms in W and Z is necessary to make this new Legendre transform well defined. The master functional behaves as a scalar under the transformation law inherited from its very definition. Moreover, it admits a proper formulation, obtained extending the set of integrated fields to so-called proper fields, which allows us to work without passing through Z, W or Γ. In the proper formulation the classical action coincides with the classical limit of the master functional, and correlation functions and renormalization are calculated applying the usual diagrammatic rules to the proper fields. Finally, the most general change of field variables, including the map relating bare and renormalized fields, is a linear redefinition of the proper fields.


Radiative Correction Field Variable Classical Limit Conventional Form Proper Formulation 
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.


  1. 1.
    G. ’t Hooft, M. Veltman, One-loop divergences in the theory of gravitation. Ann. Inst. Henri Poincaré 20, 69 (1974) MathSciNetADSGoogle Scholar
  2. 2.
    M.H. Goroff, A. Sagnotti, The ultraviolet behavior of Einstein gravity. Nucl. Phys. B 266, 709 (1986) ADSCrossRefGoogle Scholar
  3. 3.
    A.E.M. van de Ven, Two loop quantum gravity. Nucl. Phys. B 378, 309 (1992) ADSCrossRefGoogle Scholar
  4. 4.
    D. Anselmi, A general field-covariant formulation of quantum field theory. Eur. Phys. J. C 73, 2338 (2013). arXiv:1205.3279 [hep-th]. doi: 10.1140/epjc/s10052-013-2338-5 ADSCrossRefGoogle Scholar
  5. 5.
    D. Anselmi, Renormalization and causality violations in classical gravity coupled with quantum matter. J. High Energy Phys. 0701, 062 (2007). arXiv:hep-th/0605205 MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in An Einstein Centenary Survey, ed. by S. Hawking, W. Israel (Cambridge University Press, Cambridge, 1979) Google Scholar
  7. 7.
    J.M. Cornwall, R. Jackiw, E. Tomboulis, Effective action for composite operators. Phys. Rev. D 10, 2428 (1974) ADSzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Authors and Affiliations

  1. 1.Dipartimento di Fisica “Enrico Fermi”Università di PisaPisaItaly

Personalised recommendations