Asymptotic symmetries of QED and Weinberg’s soft photon theorem

  • Miguel CampigliaEmail author
  • Alok Laddha
Open Access
Regular Article - Theoretical Physics


Various equivalences between so-called soft theorems which constrain scattering amplitudes and Ward identities related to asymptotic symmetries have recently been established in gauge theories and gravity. So far these equivalences have been restricted to the case of massless matter fields, the reason being that the asymptotic symmetries are defined at null infinity. The restriction is however unnatural from the perspective of soft theorems which are insensitive to the masses of the external particles.

In this work we remove the aforementioned restriction in the context of scalar QED. Inspired by the radiative phase space description of massless fields at null infinity, we introduce a manifold description of time-like infinity on which the asymptotic phase space for massive fields can be defined. The “angle dependent” large gauge transformations are shown to have a well defined action on this phase space, and the resulting Ward identities are found to be equivalent to Weinberg’s soft photon theorem.


Scattering Amplitudes Gauge Symmetry Global Symmetries 


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.


  1. [1]
    A. Strominger, Asymptotic Symmetries of Yang-Mills Theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinbergs soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New Symmetries of Massless QED, JHEP 10 (2014) 112 [arXiv:1407.3789] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    A. Ashtekar, Asymptotic Quantization of the Gravitational Field, Phys. Rev. Lett. 46 (1981) 573 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    A. Ashtekar, Radiative Degrees of Freedom of the Gravitational Field in Exact General Relativity, J. Math. Phys. 22 (1981) 2885 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    A. Ashtekar and M. Streubel, Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity, Proc. Roy. Soc. Lond. A 376 (1981) 585 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    A. Ashtekar, Quantization of the Radiative Modes of the Gravitational Field, in Quantum Gravity 2, C.J. Isham, R. Penrose and D.W. Sciama (eds.), Oxford University Press, Oxford U.K. (1981).Google Scholar
  10. [10]
    A. Ashtekar, Asymptotic Quantization, Bibliopolis, Naples Italy (1987).zbMATHGoogle Scholar
  11. [11]
    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516.ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    A. Mohd, A note on asymptotic symmetries and soft-photon theorem, JHEP 02 (2015) 060 [arXiv:1412.5365] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028.ADSGoogle Scholar
  14. [14]
    M. Campiglia and A. Laddha, New symmetries for the Gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. Fröhlich, G. Morchio and F. Strocchi, Infrared problem and spontaneous breaking of the Lorentz group in QED, Phys. Lett. B 89 (1979) 61 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    A.P. Balachandran, S. Vaidya, Spontaneous Lorentz Violation in Gauge Theories, Eur. Phys. J. Plus 128 (2013) 118.CrossRefGoogle Scholar
  17. [17]
    A.P. Balachandran, S. Kurkcuoglu, A.R. de Queiroz and S. Vaidya, Spontaneous Lorentz Violation: The Case of Infrared QED, Eur. Phys. J. C 75 (2015) 89 [arXiv:1406.5845] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    S. Pasterski, Asymptotic Symmetries and Electromagnetic Memory, arXiv:1505.00716 [INSPIRE].
  19. [19]
    A. Ashtekar and J.D. Romano, Spatial infinity as a boundary of space-time, Class. Quant. Grav. 9 (1992) 1069 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    D. Kapec, M. Pate and A. Strominger, New Symmetries of QED, arXiv:1506.02906 [INSPIRE].
  21. [21]
    V.P. Frolov, Null Surface Quantization and Quantum Field Theory in Asymptotically Flat Space-Time, Fortsch. Phys. 26 (1978) 455 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    C. Dappiaggi, V. Moretti and N. Pinamonti, Rigorous steps towards holography in asymptotically flat spacetimes, Rev. Math. Phys. 18 (2006) 349 [gr-qc/0506069] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    A. Ashtekar, L. Bombelli and O. Reula, The covariant phase space of asymptotically flat gravitational fields, in Analysis, Geometry and Mechanics: 200 Years After Lagrange, ed. M Francaviglia, North-Holland (1991).Google Scholar
  24. [24]
    J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    P.P. Kulish and L.D. Faddeev, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys. 4 (1970) 745 [INSPIRE].CrossRefzbMATHGoogle Scholar
  26. [26]
    J. Ware, R. Saotome and R. Akhoury, Construction of an asymptotic S matrix for perturbative quantum gravity, JHEP 10 (2013) 159 [arXiv:1308.6285] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    A. Ashtekar and K.S. Narain, Infrared problems and Penroses null infinity, Syracuse University preprint (1981).Google Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Instituto de Física, Facultad de CienciasMontevideoUruguay
  2. 2.Chennai Mathematical InstituteSiruseriIndia

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