Abstract
We use the asymptotic data at conformal null-infinity to formulate Weinberg’s soft-photon theorem for Abelian gauge theories with massless charged particles. We show that the angle-dependent gauge transformations at are not merely a gauge redundancy, instead they are genuine symmetries of the radiative phase space. In the presence of these symmetries, Poisson bracket between gauge potentials is not well-defined. This does not pose an obstacle for the quantization of the radiative phase space, which proceeds by treating the conjugate electric field as the fundamental variable. Denoting by \( {\mathcal{G}}_{+} \) and \( {\mathcal{G}}_{-} \) as the group of gauge transformations at and respectively, Strominger has shown that a certain diagonal subgroup \( {\mathcal{G}}_{\mathrm{diag}}\subset {\mathcal{G}}_{+}\times {\mathcal{G}}_{-} \) is the symmetry of the S-matrix and Weinberg’s soft-photon theorem is the corresponding Ward identity. We give a systematic derivation of this result for Abelian gauge theories with massless charged particles. Our derivation is a slight generalization of the existing derivations since it is applicable even when the bulk spacetime is not exactly flat, but is only “almost” Minkowskian.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Sundrum, From Fixed Points to the Fifth Dimension, Phys. Rev. D 86 (2012) 085025 [arXiv:1106.4501] [INSPIRE].
J. Kaplan, Lectures on AdS/CFT from the Bottom Up, Lectures on AdS/CFT from the Bottom Up, course notes.
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 1410 (2014) 30 [arXiv:1312.2007] [INSPIRE].
A. Strominger, Asymptotic Symmetries of Yang-Mills Theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].
A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New Symmetries of Massless QED, JHEP 10 (2014) 112 [arXiv:1407.3789] [INSPIRE].
T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, arXiv:1401.7026 [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [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].
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].
E. Casali, Soft sub-leading divergences in Yang-Mills amplitudes, JHEP 08 (2014) 077 [arXiv:1404.5551] [INSPIRE].
A.J. Larkoski, Conformal Invariance of the Subleading Soft Theorem in Gauge Theory, Phys. Rev. D 90 (2014) 087701 [arXiv:1405.2346] [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 [arXiv:1102.4632] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
T. Banks, A Critique of pure string theory: Heterodox opinions of diverse dimensions, hep-th/0306074 [INSPIRE].
G. Barnich and C. Troessaert, Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity, JHEP 11 (2013) 003 [arXiv:1309.0794] [INSPIRE].
G. Barnich and P.-H. Lambert, Einstein- Yang-Mills theory: Asymptotic symmetries, Phys. Rev. D 88 (2013) 103006 [arXiv:1310.2698] [INSPIRE].
T. Adamo, E. Casali and D. Skinner, Perturbative gravity at null infinity, Class. Quant. Grav. 31 (2014) 225008 [arXiv:1405.5122] [INSPIRE].
Y. Geyer, A.E. Lipstein and L. Mason, Ambitwistor strings at null infinity and (subleading) soft limits, Class. Quant. Grav. 32 (2015) 055003 [arXiv:1406.1462] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
R. Penrose, Asymptotic properties of fields and space-times, Phys. Rev. Lett. 10 (1963) 66 [INSPIRE].
A. Ashtekar, Geometry and Physics of Null Infinity, arXiv:1409.1800 [INSPIRE].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].
S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations, Cambridge University Press, Cambridge U.K. (1995).
A.J. Larkoski, D. Neill and I.W. Stewart, Soft Theorems from Effective Field Theory, arXiv:1412.3108 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516.
A. Ashtekar, Asymptotic quantization: based on 1984 Naples Lectures, (1987).
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].
V.P. Frolov, Null Surface Quantization and Quantum Field Theory in Asymptotically Flat Space-Time, Fortsch. Phys. 26 (1978) 455 [INSPIRE].
A.P. Balachandran, L. Chandar and E. Ercolessi, Edge states in gauge theories: Theory, interpretations and predictions, Int. J. Mod. Phys. A 10 (1995) 1969 [hep-th/9411164] [INSPIRE].
A.P. Balachandran, Gauge symmetries, topology and quantization, AIP Conf. Proc. 317 (1994) 1 [hep-th/9210111] [INSPIRE].
D. Christodoulou and S. Klainerman, The Global nonlinear stability of the Minkowski space, Princeton University Press, Princeton U.S.A. (1993).
M. Herberthson and M. Ludvigsen, A relationship between future and past null infinity, Gen. Relat. Grav. 24 (1992) 1185.
F.E. Low, Scattering of light of very low frequency by systems of spin 1/2, Phys. Rev. 96 (1954) 1428 [INSPIRE].
F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, arXiv:1411.5745 [INSPIRE].
W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, arXiv:1412.1895 [INSPIRE].
K.-W. Huang, Central Charge and Entangled Gauge Fields, arXiv:1412.2730 [INSPIRE].
A.P. Balachandran, L. Chandar and A. Momen, Edge states and entanglement entropy, Int. J. Mod. Phys. A 12 (1997) 625 [hep-th/9512047] [INSPIRE].
S. Carlip, Entropy from conformal field theory at Killing horizons, Class. Quant. Grav. 16 (1999) 3327 [gr-qc/9906126] [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].
C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories., Cambridge University Press, Cambridge U.K. (1987), pp. 676-684..
A. Ashtekar, L. Bombelli and O. Reula, The covariant phase space of asymptotically flat gravitational fields, Elsevier (1991), p. 417.
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: 1412.5365
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Mohd, A. A note on asymptotic symmetries and soft-photon theorem. J. High Energ. Phys. 2015, 60 (2015). https://doi.org/10.1007/JHEP02(2015)060
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2015)060