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

Abstract

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.

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References

  1. [1]

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

    MATH  Google Scholar 

  2. [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. [3]

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

  4. [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. [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. [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. [7]

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

  8. [8]

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

  9. [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. [10]

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

  11. [11]

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. [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. [13]

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

    ADS  Article  Google Scholar 

  14. [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. [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. [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. [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. [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. [19]

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

  20. [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 

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Correspondence to Paul Mansfield.

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ArXiv ePrint: 1712.04760

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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). https://doi.org/10.1007/JHEP06(2018)081

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Keywords

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