Journal of High Energy Physics

, 2019:205 | Cite as

Null infinity and unitary representation of the Poincare group

  • Shamik BanerjeeEmail author
Open Access
Regular Article - Theoretical Physics


Following Pasterski-Shao-Strominger we construct a new basis of states in the single-particle Hilbert space of massless particles as a linear combination of standard Wigner states. Under Lorentz transformation the new basis states transform in the Unitary Principal Continuous Series representation. These states are obtained if we consider the little group of a null momentum direction rather than a null momentum. The definition of the states in terms of the Wigner states makes it easier to study the action of space-time translation in this basis. We show by taking into account the effect of space-time translation that the dynamics of massless particles described by these states takes place completely on the null-infinity of the Minkowski space. We then second quantize the theory in this basis and obtain a manifestly Poincare invariant (field) theory of free massless particles living on null-infinity. This theory has unitary time evolution. The null-infinity arises in this case purely group-theoretically without any reference to bulk space-time. Action of BMS is particularly natural in this picture.

As a by-product we generalize the conformal primary wave-functions for massless particles in a way which makes the action of space-time translation simple. Using these wave-functions we write down a modified Mellin(-Fourier) transformation of the S-matrix elements. The resulting amplitude is Poincare covariant. Under Poincare transformation it transforms like products of primaries of inhomogeneous SL(2, ℂ) (ISL(2, ℂ)) inserted at various points of null-infinity. ISL(2, ℂ) primaries are defined in the paper.


Conformal Field Theory Global Symmetries Space-Time 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]
    S. Pasterski, S.-H. Shao and A. Strominger, Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D 96 (2017) 065026 [arXiv:1701.00049] [INSPIRE].ADSMathSciNetGoogle Scholar
  2. [2]
    S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev. D 96 (2017) 065022 [arXiv:1705.01027] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    S. Pasterski, S.-H. Shao and A. Strominger, Gluon Amplitudes as 2d Conformal Correlators, Phys. Rev. D 96 (2017) 085006 [arXiv:1706.03917] [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    C. Cheung, A. de la Fuente and R. Sundrum, 4D scattering amplitudes and asymptotic symmetries from 2D CFT, JHEP 01 (2017) 112 [arXiv:1609.00732] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. de Boer and S.N. Solodukhin, A holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    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].
  8. [8]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
  9. [9]
    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].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 (2010) [arXiv:1102.4632] [INSPIRE].
  11. [11]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    T. Banks, A critique of pure string theory: Heterodox opinions of diverse dimensions, hep-th/0306074 [INSPIRE].
  14. [14]
    V. Bargmann, Irreducible Unitary Representations of the Lorentz Group, Annals Math. 48 (1947) 568.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
  16. [16]
    A. Gadde, In search of conformal theories, arXiv:1702.07362 [INSPIRE].
  17. [17]
    M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP 11 (2017) 193 [arXiv:1702.08471] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. Ashtekar and R.O. Hansen, A unified treatment of null and spatial infinity in general relativity. I — Universal structure, asymptotic symmetries and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978) 1542 [INSPIRE].
  20. [20]
    A. Ashtekar, Asymptotic Quantization of the Gravitational Field, Phys. Rev. Lett. 46 (1981) 573 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    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
  22. [22]
    A. Ashtekar, Asymptotic Quantization: Based On 1984 Naples Lectures, Monographs and Textbooks in Physical Science, 2, Bibliopolis, Naples, Italy, (1987).Google Scholar
  23. [23]
    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    A. Strominger, Asymptotic Symmetries of Yang-Mills Theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity \( \mathcal{S} \) -matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D Stress Tensor for 4D Gravity, Phys. Rev. Lett. 119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].ADSGoogle Scholar
  30. [30]
    M. Campiglia and A. Laddha, New symmetries for the Gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    M. Campiglia and A. Laddha, Asymptotic symmetries of gravity and soft theorems for massive particles, JHEP 12 (2015) 094 [arXiv:1509.01406] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  32. [32]
    D. Kapec and P. Mitra, A d-Dimensional Stress Tensor for Mink d+2 Gravity, JHEP 05 (2018) 186 [arXiv:1711.04371] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
  34. [34]
    F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
  35. [35]
    B.U.W. Schwab and A. Volovich, Subleading Soft Theorem in Arbitrary Dimensions from Scattering Equations, Phys. Rev. Lett. 113 (2014) 101601 [arXiv:1404.7749] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    Z. Bern, S. Davies and J. Nohle, On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons, Phys. Rev. D 90 (2014) 085015 [arXiv:1405.1015] [INSPIRE].ADSGoogle Scholar
  37. [37]
    J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, Constraining subleading soft gluon and graviton theorems, Phys. Rev. D 90 (2014) 065024 [arXiv:1406.6574] [INSPIRE].ADSGoogle Scholar
  38. [38]
    S.G. Avery and B.U.W. Schwab, Burg-Metzner-Sachs symmetry, string theory and soft theorems, Phys. Rev. D 93 (2016) 026003 [arXiv:1506.05789] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    A. Sen, Soft Theorems in Superstring Theory, JHEP 06 (2017) 113 [arXiv:1702.03934] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A. Sen, Subleading Soft Graviton Theorem for Loop Amplitudes, JHEP 11 (2017) 123 [arXiv:1703.00024] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    A. Laddha and A. Sen, Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity, JHEP 10 (2017) 065 [arXiv:1706.00759] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    S. Chakrabarti, S.P. Kashyap, B. Sahoo, A. Sen and M. Verma, Subleading Soft Theorem for Multiple Soft Gravitons, JHEP 12 (2017) 150 [arXiv:1707.06803] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    S. Chakrabarti, S.P. Kashyap, B. Sahoo, A. Sen and M. Verma, Testing Subleading Multiple Soft Graviton Theorem for CHY Prescription, JHEP 01 (2018) 090 [arXiv:1709.07883] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    A. Laddha and P. Mitra, Asymptotic Symmetries and Subleading Soft Photon Theorem in Effective Field Theories, JHEP 05 (2018) 132 [arXiv:1709.03850] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    A. Campoleoni, D. Francia and C. Heissenberg, On higher-spin supertranslations and superrotations, JHEP 05 (2017) 120 [arXiv:1703.01351] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    S.W. Hawking, M.J. Perry and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat Holography: Aspects of the dual field theory, JHEP 12 (2016) 147 [arXiv:1609.06203] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    A. Bagchi, M. Gary and Zodinmawia, Bondi-Metzner-Sachs bootstrap, Phys. Rev. D 96 (2017) 025007 [arXiv:1612.01730] [INSPIRE].
  49. [49]
    A. Bagchi, M. Gary and Zodinmawia, The nuts and bolts of the BMS Bootstrap, Class. Quant. Grav. 34 (2017) 174002 [arXiv:1705.05890] [INSPIRE].
  50. [50]
    G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: I. Induced representations, JHEP 06 (2014) 129 [arXiv:1403.5803] [INSPIRE].
  51. [51]
    G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: II. Coadjoint representation, JHEP 03 (2015) 033 [arXiv:1502.00010] [INSPIRE].
  52. [52]
    A. Campoleoni, H.A. Gonzalez, B. Oblak and M. Riegler, BMS Modules in Three Dimensions, Int. J. Mod. Phys. A 31 (2016) 1650068 [arXiv:1603.03812] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  53. [53]
    H.T. Lam and S.-H. Shao, Conformal Basis, Optical Theorem and the Bulk Point Singularity, Phys. Rev. D 98 (2018) 025020 [arXiv:1711.06138] [INSPIRE].ADSGoogle Scholar
  54. [54]
    C. Cardona and Y.-t. Huang, S-matrix singularities and CFT correlation functions, JHEP 08 (2017) 133 [arXiv:1702.03283] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    N. Banerjee, S. Banerjee, S. Atul Bhatkar and S. Jain, Conformal Structure of Massless Scalar Amplitudes Beyond Tree level, JHEP 04 (2018) 039 [arXiv:1711.06690] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    A. Schreiber, A. Volovich and M. Zlotnikov, Tree-level gluon amplitudes on the celestial sphere, Phys. Lett. B 781 (2018) 349 [arXiv:1711.08435] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  57. [57]
    R. Penrose, Twistor algebra, J. Math. Phys. 8 (1967) 345 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    S. Weinberg and O.W. Greenberg, The Quantum theory of fields. Vol. 1: Foundations, Phys. Today 48 (1995) 78.Google Scholar

Copyright information

© SISSA, Trieste, Italy 2019

Authors and Affiliations

  1. 1.Institute of PhysicsBhubaneshwarIndia
  2. 2.Homi Bhabha National InstituteMumbaiIndia

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