String Chopping and Time-ordered Products of Linear String-localized Quantum Fields

  • Lucas T. Cardoso
  • Jens Mund
  • Joseph C. Várilly


For a renormalizability proof of perturbative models in the Epstein–Glaser scheme with string-localized quantum fields, one needs to know what freedom one has in the definition of time-ordered products of the interaction Lagrangian. This paper provides a first step in that direction. The basic issue is the presence of an open set of n-tuples of strings which cannot be chronologically ordered. We resolve it by showing that almost all such string configurations can be dissected into finitely many pieces which can indeed be chronologically ordered. This fixes the time-ordered products of linear field factors outside a nullset of string configurations. (The extension across the nullset, as well as the definition of time-ordered products of Wick monomials, will be discussed elsewhere).


Relativistic quantum field theory Locality Epstein-Glaser perturbative construction Time-ordered products 

Mathematics Subject Classification (2010)

81T15 81T05 



This research was generously supported by the program “Research in Pairs” of the Mathematisches Forschungsinstitut at Oberwolfach in November 2015. JM and JCV are grateful to José M. Gracia-Bondía for helpful discussions, at Oberwolfach and later. We thank the referee for pertinent comments, which helped to fine-tune the paper. JM and LC have received financial support by the Brasilian research agencies CNPq and CAPES, respectively. They are also grateful to Finep. The project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 690575. JCV acknowledges support from the Vicerrectoría de Investigación of the Universidad de Costa Rica.


  1. 1.
    Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cardoso, L.T.: Proof of renormalizability of scalar field theories using the Epstein–Glaser scheme and techniques of microlocal analysis. J. Adv. Phys. 13, 5004–5014 (2017)Google Scholar
  4. 4.
    Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann. Inst. Henri Poincaré A 19, 211–295 (1973)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gracia-Bondía, J.M., Mund, J., Várilly, J.C.: The chirality theorem. Ann. Henri Poincaré 19 (2018).
  6. 6.
    Mund, J., Oliveira, E.T.: String-localized free vector and tensor potentials for massive particles with any spin: I. Bosons. Commun. Math. Phys. 355, 1243–1282 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mund, J., Rehren, K.-H., Schroer, B.: Relations between positivity, localization and degrees of freedom: the Weinberg–Witten theorem and the van Dam–Veltman–Zakharov discontinuity. Phys. Lett. B 773, 625–631 (2017)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Mund, J., dos Santos, J.A.: Singularity structure of the two-point functions of string-localized free quantum fields, in preparationGoogle Scholar
  9. 9.
    Mund, J., Schroer, B.: How the Higgs potential got its shape. Work in progress (2017)Google Scholar
  10. 10.
    Mund, J., Schroer, B., Yngvason, J.: String-localized quantum fields from Wigner representations. Phys. Lett. B 596, 156–162 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mund, J., Schroer, B., Yngvason, J.: String-localized quantum fields and modular localization. Commun. Math. Phys. 268, 621–672 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Plaschke, M., Yngvason, J.: Massless, string localized quantum fields for any helicity. J. Math. Phys. 53, 042301 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Academic Press, New York (1980)zbMATHGoogle Scholar
  14. 14.
    Schroer, B.: Beyond gauge theory: positivity and causal localization in the presence of vector mesons. Eur. Phys. J. C 76, 378 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    Schulz, R.M.: Microlocal analysis of tempered distributions. Ph. D. thesis, Georg-August University School of Science (GAUSS), Göttingen (2014)zbMATHGoogle Scholar
  16. 16.
    Strocchi, F.: Selected Topics on the General Properties of Quantum Field Theory. Lecture Notes in Physics, vol. 51. World Scientific, Singapore (1993)CrossRefzbMATHGoogle Scholar
  17. 17.
    Thomas, L.J., Wichmann, E.H.: On the causal structure of Minkowski spacetime. J. Math. Phys. 38, 5044–5086 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Weinberg, S., Witten, E.: Limits on massless particles. Phys. Lett. B96, 59–62 (1980)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Wigner, E.P.: Relativistische Wellengleichungen. Z. Physik 124, 665–684 (1948)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal de Juiz de ForaJuiz de ForaBrazil
  2. 2.Escuela de MatemáticaUniversidad de Costa RicaSan JoséCosta Rica

Personalised recommendations