A spacetime derivation of the Lorentzian OPE inversion formula

  • David Simmons-Duffin
  • Douglas StanfordEmail author
  • Edward Witten
Open Access
Regular Article - Theoretical Physics


Caron-Huot has recently given an interesting formula that determines OPE data in a conformal field theory in terms of a weighted integral of the four-point function over a Lorentzian region of cross-ratio space. We give a new derivation of this formula based on Wick rotation in spacetime rather than cross-ratio space. The derivation is simple in two dimensions but more involved in higher dimensions. We also derive a Lorentzian inversion formula in one dimension that sheds light on previous observations about the chaos regime in the SYK model.


Conformal and W Symmetry Conformal Field Theory 


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]
    G. Mack, Group theoretical approach to conformal invariant quantum field theory, in Renormalization and invariance in quantum field theory, E.R. Caianiello ed., NATO Adv. Stud. Inst. Ser. B 5, Springer, Boston, MA, U.S.A., (1974).Google Scholar
  2. [2]
    G. Mack, Osterwalder-Schrader positivity in conformal invariant quantum field theory, Lect. Notes Phys. 37 (1975) 66 [INSPIRE].CrossRefADSGoogle Scholar
  3. [3]
    V.K. Dobrev, V.B. Petkova, S.G. Petrova and I.T. Todorov, Dynamical derivation of vacuum operator product expansion in Euclidean conformal quantum field theory, Phys. Rev. D 13 (1976) 887 [INSPIRE].ADSGoogle Scholar
  4. [4]
    V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Harmonic analysis on the N-dimensional Lorentz group and its application to conformal quantum field theory, Lect. Notes Phys. 63 (1977) 1 [INSPIRE].CrossRefzbMATHGoogle Scholar
  5. [5]
    M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP 11 (2017) 193 [arXiv:1702.08471] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  6. [6]
    M. Hogervorst, Crossing kernels for boundary and crosscap CFTs, arXiv:1703.08159 [INSPIRE].
  7. [7]
    A. Gadde, In search of conformal theories, arXiv:1702.07362 [INSPIRE].
  8. [8]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].MathSciNetADSGoogle Scholar
  9. [9]
    S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  10. [10]
    J. Murugan, D. Stanford and E. Witten, More on supersymmetric and 2d analogs of the SYK model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  11. [11]
    J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  12. [12]
    D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  13. [13]
    D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE convergence in conformal field theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].ADSGoogle Scholar
  14. [14]
    M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].ADSGoogle Scholar
  15. [15]
    T. Hartman, S. Kundu and A. Tajdini, Averaged null energy condition from causality, JHEP 07 (2017) 066 [arXiv:1610.05308] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  16. [16]
    F.A. Dolan and H. Osborn, Conformal partial waves: further mathematical results, arXiv:1108.6194 [INSPIRE].
  17. [17]
    K. Bulycheva, A note on the SYK model with complex fermions, JHEP 12 (2017) 069 [arXiv:1706.07411] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  18. [18]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  19. [19]
    L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal approximation in AdS/CFT: conformal partial waves and finite N four-point functions, Nucl. Phys. B 767 (2007) 327 [hep-th/0611123] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  20. [20]
    L. Cornalba, Eikonal methods in AdS/CFT: Regge theory and multi-reggeon exchange, arXiv:0710.5480 [INSPIRE].
  21. [21]
    M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  22. [22]
    Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].CrossRefADSGoogle Scholar
  23. [23]
    A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  24. [24]
    L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].zbMATHADSGoogle Scholar
  25. [25]
    L.F. Alday and A. Zhiboedov, Conformal bootstrap with slightly broken higher spin symmetry, JHEP 06 (2016) 091 [arXiv:1506.04659] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  26. [26]
    L.F. Alday and A. Zhiboedov, An algebraic approach to the analytic bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  27. [27]
    A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP 11 (2015) 083 [arXiv:1502.01437] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  28. [28]
    L.F. Alday, Large spin perturbation theory for conformal field theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].CrossRefADSGoogle Scholar
  29. [29]
    D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  30. [30]
    P. Dey, K. Ghosh and A. Sinha, Simplifying large spin bootstrap in Mellin space, JHEP 01 (2018) 152 [arXiv:1709.06110] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  31. [31]
    O. Nachtmann, Positivity constraints for anomalous dimensions, Nucl. Phys. B 63 (1973) 237 [INSPIRE].CrossRefADSGoogle Scholar
  32. [32]
    M.S. Costa, T. Hansen and J. Penedones, Bounds for OPE coefficients on the Regge trajectory, JHEP 10 (2017) 197 [arXiv:1707.07689] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  33. [33]
    A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, KITP seminar,, University of California, Santa Barbara, U.S.A., 12 February 2015.
  34. [34]
    D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  35. [35]
    P. Kravchuk, Casimir recursion relations for general conformal blocks, JHEP 02 (2018) 011 [arXiv:1709.05347] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  36. [36]
    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  37. [37]
    D. Karateev, P. Kravchuk and D. Simmons-Duffin, in progress.Google Scholar
  38. [38]
    F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].CrossRefADSGoogle Scholar
  39. [39]
    J. Penedones, E. Trevisani and M. Yamazaki, Recursion relations for conformal blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  40. [40]
    M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap III: higher dimensional amplitudes, arXiv:1708.06765 [INSPIRE].
  41. [41]
    L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    S. Coleman and R.E. Norton, Singularities in the physical region, Nuovo Cim. 38 (1965) 438 [INSPIRE].CrossRefGoogle Scholar
  43. [43]
    R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429 [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  44. [44]
    T. Hartman, S. Jain and S. Kundu, Causality constraints in conformal field theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • David Simmons-Duffin
    • 1
    • 2
  • Douglas Stanford
    • 2
    Email author
  • Edward Witten
    • 2
  1. 1.Walter Burke Institute for Theoretical PhysicsCaltechPasadenaU.S.A.
  2. 2.Institute for Advanced StudyPrincetonU.S.A.

Personalised recommendations