Skip to main content

Intersection of world-lines on curved surfaces and path-ordering of the Wilson loop

A preprint version of the article is available at arXiv.


We study contact interactions for long world-lines on a curved surface, focusing on the average number of times two world-lines intersect as a function of their end-points. The result can be used to extend the concept of path-ordering, as employed in the Wilson loop, from a closed curve into the interior of a surface spanning the curve. Taking this surface as a string world-sheet yields a generalisation of the string contact interaction previously used to represent the Abelian Wilson loop as a tensionless string. We also describe a supersymmetric generalisation.


  1. J.C. Maxwell, A treatise on electricity and magnetism, Clarendon Press, Oxford U.K. (1998).

    MATH  Google Scholar 

  2. M. Faraday, Thoughts on ray-vibrations. Letter to Richard Phillips, Esq. Phil. Mag. 28 (1846) 345, reprinted in Experimental researches in chemistry and physics, M. Faraday, R. Taylor and W. Francis, London U.K. (1859).

  3. P.A.M. Dirac, Gauge-invariant formulation of quantum electrodynamics, Canadian J. Phys. 33 (1955) 650.

  4. P. Mansfield, Faraday’s lines of force as strings: from Gauss’ law to the arrow of time, JHEP 10 (2012) 149 [arXiv:1108.5094] [INSPIRE].

  5. J.P. Edwards and P. Mansfield, Delta-function Interactions for the bosonic and spinning strings and the generation of abelian gauge theory, JHEP 01 (2015) 127 [arXiv:1410.3288] [INSPIRE].

    ADS  Article  Google Scholar 

  6. J.P. Edwards and P. Mansfield, QED as the tensionless limit of the spinning string with contact interaction, Phys. Lett. B 746 (2015) 335 [arXiv:1409.4948] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  7. M.J. Strassler, Field theory without Feynman diagrams: one loop effective actions, Nucl. Phys. B 385 (1992) 145 [hep-ph/9205205] [INSPIRE].

  8. A. Ilderton, Localisation in worldline pair production and lightfront zero-modes, JHEP 09 (2014) 166 [arXiv:1406.1513] [INSPIRE].

  9. A. Ilderton, G. Torgrimsson and J. Wårdh, Pair production from residues of complex worldline instantons, Phys. Rev. D 92 (2015) 025009 [arXiv:1503.08828] [INSPIRE].

  10. C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rept. 355 (2001) 73 [hep-th/0101036] [INSPIRE].

  11. J.P. Edwards, Contact interactions between particle worldlines, JHEP 01 (2016) 033 [arXiv:1506.08130] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. L. Brink, P. Di Vecchia and P.S. Howe, A lagrangian formulation of the classical and quantum dynamics of spinning particles, Nucl. Phys. B 118 (1977) 76 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  13. S. Samuel, Color Zitterbewegung, Nucl. Phys. B 149 (1979) 517 [INSPIRE].

    ADS  Article  Google Scholar 

  14. E. D’Hoker and D.G. Gagne, Worldline path integrals for fermions with general couplings, Nucl. Phys. B 467 (1996) 297 [hep-th/9512080] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. A.P. Balachandran, P. Salomonson, B.-S. Skagerstam and J.-O. Winnberg, Classical description of particle interacting with nonabelian gauge field, Phys. Rev. D 15 (1977) 2308 [INSPIRE].

    ADS  Google Scholar 

  16. A. Barducci, R. Casalbuoni and L. Lusanna, Classical scalar and spinning particles interacting with external Yang-Mills fields, Nucl. Phys. B 124 (1977) 93 [INSPIRE].

    ADS  Article  Google Scholar 

  17. P. Salomonson, B.-S. Skagerstam and J.-O. Winnberg, On the equations of motion of a Yang-Mills particle, Phys. Rev. D 16 (1977) 2581 [INSPIRE].

    ADS  Google Scholar 

  18. F. Bastianelli, R. Bonezzi, O. Corradini and E. Latini, Particles with non abelian charges, JHEP 10 (2013) 098 [arXiv:1309.1608] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. B. Broda, NonAbelian Stokes theorem, in Advanced electromagnetism, T.W. Barrett ed., World Scientific, Singapore (1995), hep-th/9511150 [INSPIRE].

  20. M.E. Knutt-Wehlau and R.B. Mann, Supergravity from a massive superparticle and the simplest super black hole, Nucl. Phys. B 514 (1998) 355 [hep-th/9708126] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Author information



Corresponding author

Correspondence to Paul Mansfield.

Additional information

ArXiv ePrint: 1712.04760

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Curry, C., Mansfield, P. Intersection of world-lines on curved surfaces and path-ordering of the Wilson loop. J. High Energ. Phys. 2018, 81 (2018).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Bosonic Strings
  • Long strings
  • Wilson, ’t Hooft and Polyakov loops
  • Field Theories in Lower Dimensions