A spacetime derivation of the Lorentzian OPE inversion formula

doi:10.1007/JHEP07(2018)085

Title: A spacetime derivation of the Lorentzian OPE inversion formula

Abstract

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.

Authors:

Simmons-Duffin, David ^[1]; Stanford, Douglas ^[2]; Witten, Edward ^[2]

California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics; Inst. for Advanced Study, Princeton, NJ (United States)
Inst. for Advanced Study, Princeton, NJ (United States)

Publication Date:: 2018-07-12

Research Org.:: Inst. for Advanced Study, Princeton, NJ (United States)

Sponsoring Org.:: USDOE Office of Science (SC); Simons Foundation; National Science Foundation (NSF)

OSTI Identifier:: 1598252

Grant/Contract Number:: [SC0009988; 385600; 488657; PHY-1606531]

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: [Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 7]; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; field theory: conformal; dimension: 2; dimension: 1; n-point function: 4; operator product expansion; space-time; higher-dimensional; rotation; chaos; conformal field theory; conformal and W symmetry

Citation Formats


                    Simmons-Duffin, David, Stanford, Douglas, and Witten, Edward. A spacetime derivation of the Lorentzian OPE inversion formula.  United States: N. p., 2018. 
        Web.  doi:10.1007/JHEP07(2018)085.

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                    Simmons-Duffin, David, Stanford, Douglas, & Witten, Edward. A spacetime derivation of the Lorentzian OPE inversion formula.  United States.  doi:10.1007/JHEP07(2018)085.

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                    Simmons-Duffin, David, Stanford, Douglas, and Witten, Edward. Thu .  
        "A spacetime derivation of the Lorentzian OPE inversion formula".  United States.  doi:10.1007/JHEP07(2018)085.  https://www.osti.gov/servlets/purl/1598252.

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@article{osti_1598252,

  title        = {A spacetime derivation of the Lorentzian OPE inversion formula},

  author       = {Simmons-Duffin, David and Stanford, Douglas and Witten, Edward},

  abstractNote = {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.},

  doi          = {10.1007/JHEP07(2018)085},

  journal      = {Journal of High Energy Physics (Online)},

  number       = [7],

  volume       = [2018],

  place        = {United States},

  year         = {2018},

  month        = {7}

}

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DOI: 10.1007/JHEP07(2018)085

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Works referenced in this record:

Crossing symmetry in alpha space
journal, November 2017

Hogervorst, Matthijs; van Rees, Balt C.
Journal of High Energy Physics, Vol. 2017, Issue 11
DOI: 10.1007/JHEP11(2017)193

Analyticity in spin in conformal theories
journal, September 2017

Caron-Huot, Simon
Journal of High Energy Physics, Vol. 2017, Issue 9
DOI: 10.1007/JHEP09(2017)078

More on supersymmetric and 2d analogs of the SYK model
journal, August 2017

Murugan, Jeff; Stanford, Douglas; Witten, Edward
Journal of High Energy Physics, Vol. 2017, Issue 8
DOI: 10.1007/JHEP08(2017)146

Looking for a bulk point
journal, January 2017

Maldacena, Juan; Simmons-Duffin, David; Zhiboedov, Alexander
Journal of High Energy Physics, Vol. 2017, Issue 1
DOI: 10.1007/JHEP01(2017)013

Projectors, shadows, and conformal blocks
journal, April 2014

Simmons-Duffin, David
Journal of High Energy Physics, Vol. 2014, Issue 4
DOI: 10.1007/JHEP04(2014)146

Averaged null energy condition from causality
journal, July 2017

Hartman, Thomas; Kundu, Sandipan; Tajdini, Amirhossein
Journal of High Energy Physics, Vol. 2017, Issue 7
DOI: 10.1007/JHEP07(2017)066

A note on the SYK model with complex fermions
journal, December 2017

Bulycheva, Ksenia
Journal of High Energy Physics, Vol. 2017, Issue 12
DOI: 10.1007/JHEP12(2017)069

A bound on chaos
journal, August 2016

Maldacena, Juan; Shenker, Stephen H.; Stanford, Douglas
Journal of High Energy Physics, Vol. 2016, Issue 8
DOI: 10.1007/JHEP08(2016)106

Eikonal approximation in AdS/CFT: Conformal partial waves and finite N four-point functions
journal, April 2007

Cornalba, Lorenzo; Costa, Miguel S.; Penedones, João
Nuclear Physics B, Vol. 767, Issue 3
DOI: 10.1016/j.nuclphysb.2007.01.007

Conformal Regge theory
journal, December 2012

Costa, Miguel S.; Goncalves, Vasco; Penedones, João
Journal of High Energy Physics, Vol. 2012, Issue 12
DOI: 10.1007/JHEP12(2012)091

Convexity and liberation at large spin
journal, November 2013

Komargodski, Zohar; Zhiboedov, Alexander
Journal of High Energy Physics, Vol. 2013, Issue 11
DOI: 10.1007/JHEP11(2013)140

The analytic bootstrap and AdS superhorizon locality
journal, December 2013

Fitzpatrick, A. Liam; Kaplan, Jared; Poland, David
Journal of High Energy Physics, Vol. 2013, Issue 12
DOI: 10.1007/JHEP12(2013)004

Conformal bootstrap with slightly broken higher spin symmetry
journal, June 2016

Alday, Luis F.; Zhiboedov, Alexander
Journal of High Energy Physics, Vol. 2016, Issue 6
DOI: 10.1007/JHEP06(2016)091

An algebraic approach to the analytic bootstrap
journal, April 2017

Alday, Luis F.; Zhiboedov, Alexander
Journal of High Energy Physics, Vol. 2017, Issue 4
DOI: 10.1007/JHEP04(2017)157

Analytic bootstrap at large spin
journal, November 2015

Kaviraj, Apratim; Sen, Kallol; Sinha, Aninda
Journal of High Energy Physics, Vol. 2015, Issue 11
DOI: 10.1007/JHEP11(2015)083

Large Spin Perturbation Theory for Conformal Field Theories
journal, September 2017

Alday, Luis F.
Physical Review Letters, Vol. 119, Issue 11
DOI: 10.1103/PhysRevLett.119.111601

The lightcone bootstrap and the spectrum of the 3d Ising CFT
journal, March 2017

Simmons-Duffin, David
Journal of High Energy Physics, Vol. 2017, Issue 3
DOI: 10.1007/JHEP03(2017)086

Simplifying large spin bootstrap in Mellin space
journal, January 2018

Dey, Parijat; Ghosh, Kausik; Sinha, Aninda
Journal of High Energy Physics, Vol. 2018, Issue 1
DOI: 10.1007/JHEP01(2018)152

Positivity constraints for anomalous dimensions
journal, October 1973

Nachtmann, Otto
Nuclear Physics B, Vol. 63
DOI: 10.1016/0550-3213(73)90144-2

Bounds for OPE coefficients on the Regge trajectory
journal, October 2017

Costa, Miguel S.; Hansen, Tobias; Penedones, João
Journal of High Energy Physics, Vol. 2017, Issue 10
DOI: 10.1007/JHEP10(2017)197

Weight shifting operators and conformal blocks
journal, February 2018

Karateev, Denis; Kravchuk, Petr; Simmons-Duffin, David
Journal of High Energy Physics, Vol. 2018, Issue 2
DOI: 10.1007/JHEP02(2018)081

Casimir recursion relations for general conformal blocks
journal, February 2018

Kravchuk, Petr
Journal of High Energy Physics, Vol. 2018, Issue 2
DOI: 10.1007/JHEP02(2018)011

Conformal four point functions and the operator product expansion
journal, April 2001

Dolan, F. A.; Osborn, H.
Nuclear Physics B, Vol. 599, Issue 1-2
DOI: 10.1016/S0550-3213(01)00013-X

Bootstrapping mixed correlators in the 3D Ising model
journal, November 2014

Kos, Filip; Poland, David; Simmons-Duffin, David
Journal of High Energy Physics, Vol. 2014, Issue 11
DOI: 10.1007/JHEP11(2014)109

Recursion relations for conformal blocks
journal, September 2016

Penedones, João; Trevisani, Emilio; Yamazaki, Masahito
Journal of High Energy Physics, Vol. 2016, Issue 9
DOI: 10.1007/JHEP09(2016)070

On analytic properties of vertex parts in quantum field theory
journal, October 1959

Landau, L. D.
Nuclear Physics, Vol. 13, Issue 1
DOI: 10.1016/0029-5582(59)90154-3

Singularities in the physical region
journal, July 1965

Coleman, S.; Norton, R. E.
Il Nuovo Cimento, Vol. 38, Issue 1
DOI: 10.1007/BF02750472

Singularities and Discontinuities of Feynman Amplitudes
journal, September 1960

Cutkosky, R. E.
Journal of Mathematical Physics, Vol. 1, Issue 5
DOI: 10.1063/1.1703676

Causality constraints in conformal field theory
journal, May 2016

Hartman, Thomas; Jain, Sachin; Kundu, Sandipan
Journal of High Energy Physics, Vol. 2016, Issue 5
DOI: 10.1007/JHEP05(2016)099

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Title: A spacetime derivation of the Lorentzian OPE inversion formula

Abstract

Citation Formats

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Conformal Regge theory journal, December 2012

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Analytic bootstrap at large spin journal, November 2015

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Simplifying large spin bootstrap in Mellin space journal, January 2018

Positivity constraints for anomalous dimensions journal, October 1973

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Casimir recursion relations for general conformal blocks journal, February 2018

Conformal four point functions and the operator product expansion journal, April 2001

Bootstrapping mixed correlators in the 3D Ising model journal, November 2014

Recursion relations for conformal blocks journal, September 2016

On analytic properties of vertex parts in quantum field theory journal, October 1959

Singularities in the physical region journal, July 1965

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Causality constraints in conformal field theory journal, May 2016

Crossing symmetry in alpha space
journal, November 2017

Analyticity in spin in conformal theories
journal, September 2017

More on supersymmetric and 2d analogs of the SYK model
journal, August 2017

Looking for a bulk point
journal, January 2017

Projectors, shadows, and conformal blocks
journal, April 2014

Averaged null energy condition from causality
journal, July 2017

A note on the SYK model with complex fermions
journal, December 2017

A bound on chaos
journal, August 2016

Eikonal approximation in AdS/CFT: Conformal partial waves and finite N four-point functions
journal, April 2007

Conformal Regge theory
journal, December 2012

Convexity and liberation at large spin
journal, November 2013

The analytic bootstrap and AdS superhorizon locality
journal, December 2013

Conformal bootstrap with slightly broken higher spin symmetry
journal, June 2016

An algebraic approach to the analytic bootstrap
journal, April 2017

Analytic bootstrap at large spin
journal, November 2015

Large Spin Perturbation Theory for Conformal Field Theories
journal, September 2017

The lightcone bootstrap and the spectrum of the 3d Ising CFT
journal, March 2017

Simplifying large spin bootstrap in Mellin space
journal, January 2018

Positivity constraints for anomalous dimensions
journal, October 1973

Bounds for OPE coefficients on the Regge trajectory
journal, October 2017

Weight shifting operators and conformal blocks
journal, February 2018

Casimir recursion relations for general conformal blocks
journal, February 2018

Conformal four point functions and the operator product expansion
journal, April 2001

Bootstrapping mixed correlators in the 3D Ising model
journal, November 2014

Recursion relations for conformal blocks
journal, September 2016

On analytic properties of vertex parts in quantum field theory
journal, October 1959

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journal, July 1965

Singularities and Discontinuities of Feynman Amplitudes
journal, September 1960

Causality constraints in conformal field theory
journal, May 2016