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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, Princeton University Press (2018) [arXiv:1703.05448] [INSPIRE].
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].
S. Carlip, Statistical mechanics and black hole thermodynamics, Nucl. Phys. Proc. Suppl.57 (1997) 8 [gr-qc/9702017] [INSPIRE].
T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys.88 (1974) 286 [INSPIRE].
T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New Symmetries of Massless QED, JHEP10 (2014) 112 [arXiv:1407.3789] [INSPIRE].
A. Strominger, Magnetic Corrections to the Soft Photon Theorem, Phys. Rev. Lett.116 (2016) 031602 [arXiv:1509.00543] [INSPIRE].
Y. Hamada, M.-S. Seo and G. Shiu, Electromagnetic Duality and the Electric Memory Effect, JHEP02 (2018) 046 [arXiv:1711.09968] [INSPIRE].
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].
V. Hosseinzadeh, A. Seraj and M.M. Sheikh-Jabbari, Soft Charges and Electric-Magnetic Duality, JHEP08 (2018) 102 [arXiv:1806.01901] [INSPIRE].
L. Freidel and D. Pranzetti, Electromagnetic duality and central charge, Phys. Rev.D 98 (2018) 116008 [arXiv:1806.03161] [INSPIRE].
H. Godazgar, M. Godazgar and C.N. Pope, New dual gravitational charges, Phys. Rev.D 99 (2019) 024013 [arXiv:1812.01641] [INSPIRE].
G. Compére, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, JHEP11 (2018) 200 [arXiv:1810.00377] [INSPIRE].
W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP09 (2016) 102 [arXiv:1601.04744] [INSPIRE].
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].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev.D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
M. Campiglia and A. Laddha, New symmetries for the Gravitational S-matrix, JHEP04 (2015) 076 [arXiv:1502.02318] [INSPIRE].
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].
J. Distler, R. Flauger and B. Horn, Double-soft graviton amplitudes and the extended BMS charge algebra, JHEP08 (2019) 021 [arXiv:1808.09965] [INSPIRE].
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].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav.19 (2002) 5849 [hep-th/0209067] [INSPIRE].
I. Papadimitriou, Holographic renormalization as a canonical transformation, JHEP11 (2010) 014 [arXiv:1007.4592] [INSPIRE].
I. Papadimitriou, Lectures on Holographic Renormalization, Springer Proc. Phys.176 (2016) 131 [INSPIRE].
V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys.208 (1999) 413 [hep-th/9902121] [INSPIRE].
S. Hollands, A. Ishibashi and D. Marolf, Counter-term charges generate bulk symmetries, Phys. Rev.D 72 (2005) 104025 [hep-th/0503105] [INSPIRE].
G. Compere and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav.25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].
R.B. Mann and D. Marolf, Holographic renormalization of asymptotically flat spacetimes, Class. Quant. Grav.23 (2006) 2927 [hep-th/0511096] [INSPIRE].
M. Park and R.B. Mann, Holographic Renormalization of Asymptotically Flat Gravity, JHEP12 (2012) 098 [arXiv:1210.3843] [INSPIRE].
T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev.D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
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].
A. Herdegen, Asymptotic structure of electrodynamics revisited, Lett. Math. Phys.107 (2017) 1439 [arXiv:1604.04170] [INSPIRE].
R. Penrose, Asymptotic properties of fields and space-times, Phys. Rev. Lett.10 (1963) 66 [INSPIRE].
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).
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).
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].
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].
J. Winicour, Logarithmic asymptotic flatness, Found. Phys.15 (1985) 605.
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.
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.
J. Kijowski and W. Szczyrba, A Canonical Structure for Classical Field Theories, Commun. Math. Phys.46 (1976) 183 [INSPIRE].
C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in Three hundred years of gravitation (1986) [INSPIRE].
K. Gawędzki, Classical origin of quantum group symmetries in Wess-Zumino-Witten conformal field theory, Commun. Math. Phys.139 (1991) 201 [INSPIRE].
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys.31 (1990) 725 [INSPIRE].
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].
R.P. Geroch and J. Winicour, Linkages in general relativity, J. Math. Phys.22 (1981) 803 [INSPIRE].
S. Hollands and A. Ishibashi, Asymptotic flatness and Bondi energy in higher dimensional gravity, J. Math. Phys.46 (2005) 022503 [gr-qc/0304054] [INSPIRE].
F. Hopfmüller and L. Freidel, Null Conservation Laws for Gravity, Phys. Rev.D 97 (2018) 124029 [arXiv:1802.06135] [INSPIRE].
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].
M. Campiglia and A. Laddha, Subleading soft photons and large gauge transformations, JHEP11 (2016) 012 [arXiv:1605.09677] [INSPIRE].
A. Laddha and P. Mitra, Asymptotic Symmetries and Subleading Soft Photon Theorem in Effective Field Theories, JHEP05 (2018) 132 [arXiv:1709.03850] [INSPIRE].
A. Laddha and A. Sen, Logarithmic Terms in the Soft Expansion in Four Dimensions, JHEP10 (2018) 056 [arXiv:1804.09193] [INSPIRE].
T. He and P. Mitra, Asymptotic Symmetries and Weinberg’s Soft Photon Theorem in Minkd+2 , arXiv:1903.02608 [INSPIRE].
H. Gomes and A. Riello, Quasilocal degrees of freedom in Yang-Mills theory, arXiv:1906.00992 [INSPIRE].
A. Riello, Soft charges from the geometry of field space, arXiv:1904.07410 [INSPIRE].
D. Greser, Polyhomogeneous functions, https://www.uni-math.gwdg.de/iwitt/SpecGeo2014/phg-fcns.pdf.
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
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1904.04384
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 https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Freidel, L., Hopfmüller, F. & Riello, A. Asymptotic renormalization in flat space: symplectic potential and charges of electromagnetism. J. High Energ. Phys. 2019, 126 (2019). https://doi.org/10.1007/JHEP10(2019)126
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)126