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Journal of High Energy Physics

, 2019:126 | Cite as

Asymptotic renormalization in flat space: symplectic potential and charges of electromagnetism

  • Laurent Freidel
  • Florian Hopfmüller
  • Aldo RielloEmail author
Open Access
Regular Article - Theoretical Physics
  • 22 Downloads

Abstract

We present a systematic procedure to renormalize the symplectic potential of the electromagnetic field at null infinity in Minkowski space. We work in D ≥ 6 spacetime dimensions as a toy model of General Relativity in D ≥ 4 dimensions. Total variation counterterms as well as corner counterterms are both subtracted from the symplectic potential to make it finite. These counterterms affect respectively the action functional and the Hamiltonian symmetry generators. The counterterms are local and universal. We analyze the asymptotic equations of motion and identify the free data associated with the renormalized canonical structure along a null characteristic. This allows the construction of the asymptotic renormalized charges whose Ward identity gives the QED soft theorem, supporting the physical viability of the renormalization procedure. We touch upon how to extend our analysis to the presence of logarithmic anomalies, and upon how our procedure compares to holographic renormalization.

Keywords

Field Theories in Higher Dimensions Gauge Symmetry Anomalies in Field and String Theories Global Symmetries 

Notes

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

References

  1. [1]
    A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, Princeton University Press (2018) [arXiv:1703.05448] [INSPIRE].
  2. [2]
    A.P. Balachandran, L. Chandar and A. Momen, Edge states in gravity and black hole physics, Nucl. Phys.B 461 (1996) 581 [gr-qc/9412019] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    S. Carlip, Statistical mechanics and black hole thermodynamics, Nucl. Phys. Proc. Suppl.57 (1997) 8 [gr-qc/9702017] [INSPIRE].
  4. [4]
    T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys.88 (1974) 286 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New Symmetries of Massless QED, JHEP10 (2014) 112 [arXiv:1407.3789] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    A. Strominger, Magnetic Corrections to the Soft Photon Theorem, Phys. Rev. Lett.116 (2016) 031602 [arXiv:1509.00543] [INSPIRE].
  7. [7]
    Y. Hamada, M.-S. Seo and G. Shiu, Electromagnetic Duality and the Electric Memory Effect, JHEP02 (2018) 046 [arXiv:1711.09968] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Campiglia, L. Freidel, F. Hopfmueller and R.M. Soni, Scalar Asymptotic Charges and Dual Large Gauge Transformations, JHEP04 (2019) 003 [arXiv:1810.04213] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    V. Hosseinzadeh, A. Seraj and M.M. Sheikh-Jabbari, Soft Charges and Electric-Magnetic Duality, JHEP08 (2018) 102 [arXiv:1806.01901] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    L. Freidel and D. Pranzetti, Electromagnetic duality and central charge, Phys. Rev.D 98 (2018) 116008 [arXiv:1806.03161] [INSPIRE].
  11. [11]
    H. Godazgar, M. Godazgar and C.N. Pope, New dual gravitational charges, Phys. Rev.D 99 (2019) 024013 [arXiv:1812.01641] [INSPIRE].
  12. [12]
    G. Compére, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, JHEP11 (2018) 200 [arXiv:1810.00377] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP09 (2016) 102 [arXiv:1601.04744] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    A. Ashtekar, M. Campiglia and A. Laddha, Null infinity, the BMS group and infrared issues, Gen. Rel. Grav.50 (2018) 140 [arXiv:1808.07093] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP05 (2010) 062 [arXiv:1001.1541] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev.D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
  18. [18]
    M. Campiglia and A. Laddha, New symmetries for the Gravitational S-matrix, JHEP04 (2015) 076 [arXiv:1502.02318] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S -matrix, JHEP08 (2014) 058 [arXiv:1406.3312] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    J. Distler, R. Flauger and B. Horn, Double-soft graviton amplitudes and the extended BMS charge algebra, JHEP08 (2019) 021 [arXiv:1808.09965] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    É . É . Flanagan and D.A. Nichols, Conserved charges of the extended Bondi-Metzner-Sachs algebra, Phys. Rev.D 95 (2017) 044002 [arXiv:1510.03386] [INSPIRE].
  22. [22]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav.19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  23. [23]
    I. Papadimitriou, Holographic renormalization as a canonical transformation, JHEP11 (2010) 014 [arXiv:1007.4592] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    I. Papadimitriou, Lectures on Holographic Renormalization, Springer Proc. Phys.176 (2016) 131 [INSPIRE].CrossRefGoogle Scholar
  25. [25]
    V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys.208 (1999) 413 [hep-th/9902121] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    S. Hollands, A. Ishibashi and D. Marolf, Counter-term charges generate bulk symmetries, Phys. Rev.D 72 (2005) 104025 [hep-th/0503105] [INSPIRE].
  27. [27]
    G. Compere and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav.25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    R.B. Mann and D. Marolf, Holographic renormalization of asymptotically flat spacetimes, Class. Quant. Grav.23 (2006) 2927 [hep-th/0511096] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    M. Park and R.B. Mann, Holographic Renormalization of Asymptotically Flat Gravity, JHEP12 (2012) 098 [arXiv:1210.3843] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev.D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
  31. [31]
    D. Kapec, V. Lysov and A. Strominger, Asymptotic Symmetries of Massless QED in Even Dimensions, Adv. Theor. Math. Phys.21 (2017) 1747 [arXiv:1412.2763] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  32. [32]
    A. Herdegen, Asymptotic structure of electrodynamics revisited, Lett. Math. Phys.107 (2017) 1439 [arXiv:1604.04170] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    R. Penrose, Asymptotic properties of fields and space-times, Phys. Rev. Lett.10 (1963) 66 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    R. Penrose and W. Rindler, Spinors and Space-Time. VOL. 1: Two-Spinor Calculus and Relativistic Fields, Cambridge Monographs on Mathematical Physics, Cambridge University Press (1984).
  35. [35]
    R. Penrose and W. Rindler, Spinors and Space-Time. VOL. 2: Spinor and Twistor Method in Space-Time Geometry, Cambridge Monographs on Mathematical Physics, Cambridge University Press (1986).
  36. [36]
    S. Hollands and R.M. Wald, Conformal null infinity does not exist for radiating solutions in odd spacetime dimensions, Class. Quant. Grav.21 (2004) 5139 [gr-qc/0407014] [INSPIRE].
  37. [37]
    P.T. Chrusciel, E. Delay, J.M. Lee and D.N. Skinner, Boundary regularity of conformally compact Einstein metrics, J. Diff. Geom.69 (2005) 111 [math/0401386] [INSPIRE].
  38. [38]
    J. Winicour, Logarithmic asymptotic flatness, Found. Phys.15 (1985) 605.ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    P.T. Chrusciel, M.A.H. MacCallum and D.B. Singleton, Gravitational waves in general relativity: XIV. Bondi expansions and the “polyhomogeneity” of I , Phil. Trans. Roy. Soc. Lond.A 350 (1995) 113.Google Scholar
  40. [40]
    H. Friedrich, Smoothness at null infinity and the structure of initial data, in The Einstein Equations and the Large Scale Behavior of Gravitational Fields, P.T. Chruściel and H. Friedrich eds., Basel, pp. 121–203, Birkhäuser Basel (2004) [DOI:10.1007/978-3-0348-7953-8_4.CrossRefGoogle Scholar
  41. [41]
    J. Kijowski and W. Szczyrba, A Canonical Structure for Classical Field Theories, Commun. Math. Phys.46 (1976) 183 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in Three hundred years of gravitation (1986) [INSPIRE].
  43. [43]
    K. Gawędzki, Classical origin of quantum group symmetries in Wess-Zumino-Witten conformal field theory, Commun. Math. Phys.139 (1991) 201 [INSPIRE].
  44. [44]
    J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys.31 (1990) 725 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys.217 (2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    R.P. Geroch and J. Winicour, Linkages in general relativity, J. Math. Phys.22 (1981) 803 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    S. Hollands and A. Ishibashi, Asymptotic flatness and Bondi energy in higher dimensional gravity, J. Math. Phys.46 (2005) 022503 [gr-qc/0304054] [INSPIRE].
  48. [48]
    F. Hopfmüller and L. Freidel, Null Conservation Laws for Gravity, Phys. Rev.D 97 (2018) 124029 [arXiv:1802.06135] [INSPIRE].
  49. [49]
    V. Lysov, S. Pasterski and A. Strominger, Low’s Subleading Soft Theorem as a Symmetry of QED, Phys. Rev. Lett.113 (2014) 111601 [arXiv:1407.3814] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    M. Campiglia and A. Laddha, Subleading soft photons and large gauge transformations, JHEP11 (2016) 012 [arXiv:1605.09677] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    A. Laddha and P. Mitra, Asymptotic Symmetries and Subleading Soft Photon Theorem in Effective Field Theories, JHEP05 (2018) 132 [arXiv:1709.03850] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    A. Laddha and A. Sen, Logarithmic Terms in the Soft Expansion in Four Dimensions, JHEP10 (2018) 056 [arXiv:1804.09193] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    T. He and P. Mitra, Asymptotic Symmetries and Weinberg’s Soft Photon Theorem in Mink d+2 , arXiv:1903.02608 [INSPIRE].
  54. [54]
    H. Gomes and A. Riello, Quasilocal degrees of freedom in Yang-Mills theory, arXiv:1906.00992 [INSPIRE].
  55. [55]
    A. Riello, Soft charges from the geometry of field space, arXiv:1904.07410 [INSPIRE].
  56. [56]

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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