Linking numbers in local quantum field theory

  • Detlev BuchholzEmail author
  • Fabio Ciolli
  • Giuseppe Ruzzi
  • Ezio Vasselli


Linking numbers appear in local quantum field theory in the presence of tensor fields, which are closed two-forms on Minkowski space. Given any pair of such fields, it is shown that the commutator of the corresponding intrinsic (gauge-invariant) vector potentials, integrated about spacelike separated, spatial loops, are elements of the center of the algebra of all local fields. Moreover, these commutators are proportional to the linking numbers of the underlying loops. If the commutators are different from zero, the underlying two-forms are not exact (i.e. there do not exist local vector potentials for them). The theory then necessarily contains massless particles. A prominent example of this kind, due to J.E. Roberts, is given by the free electromagnetic field and its Hodge dual. Further examples with more complex mass spectrum are presented in this article.


Intrinsic vector potential Linking numbers Massless particles 

Mathematics Subject Classification

81T05 83C47 57T15 



DB gratefully acknowledges the hospitality and support extended to him by Roberto Longo and the University of Rome “Tor Vergata”, which made this collaboration possible. FC and GR are supported by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”. EV is supported in part by OPAL “Consolidate the Foundations”.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenGermany
  2. 2.Dipartimento di MatematicaUniversitá di Roma “Tor Vergata”RomaItaly

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