Harmonic analysis and mean field theory

Karateev, Denis; Kravchuk, Petr; Simmons-Duffin, David

doi:10.1007/JHEP10(2019)217

Title: Harmonic analysis and mean field theory

Abstract

We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weight-shifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and “seed” operators in 4d, and MFT for currents and stress-tensors in 3d.

Authors:

Karateev, Denis ^[1];

^[2]; Simmons-Duffin, David ^[3]

Inst. of Physics (EPFL), Lausanne (Switzerland)
California Inst. of Technology (CalTech), Pasadena, CA (United States); Inst. for Advanced Study, Princeton, NJ (United States)
California Inst. of Technology (CalTech), Pasadena, CA (United States)

Publication Date:: Mon Oct 21 00:00:00 EDT 2019

Research Org.:: California Institute of Technology (CalTech), Pasadena, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), High Energy Physics (HEP); Swiss National Science Foundation (SNF)

OSTI Identifier:: 1594253

Report Number(s):: arXiv:1809.05111
Journal ID: ISSN 1029-8479; TRN: US2101742

Grant/Contract Number:: SC0019085; SC0009988

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: 2019; Journal Issue: 10; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Conformal and W Symmetry; Conformal Field Theory

Citation Formats


                    Karateev, Denis, Kravchuk, Petr, and Simmons-Duffin, David. Harmonic analysis and mean field theory.  United States: N. p., 2019. 
Web.  doi:10.1007/JHEP10(2019)217.

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                    Karateev, Denis, Kravchuk, Petr, & Simmons-Duffin, David. Harmonic analysis and mean field theory.  United States.  https://doi.org/10.1007/JHEP10(2019)217

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                    Karateev, Denis, Kravchuk, Petr, and Simmons-Duffin, David. Mon .  
"Harmonic analysis and mean field theory".  United States.  https://doi.org/10.1007/JHEP10(2019)217.  https://www.osti.gov/servlets/purl/1594253.

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@article{osti_1594253,

  title        = {Harmonic analysis and mean field theory},

  author       = {Karateev, Denis and Kravchuk, Petr and Simmons-Duffin, David},

  abstractNote = {We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weight-shifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and “seed” operators in 4d, and MFT for currents and stress-tensors in 3d.},

  doi          = {10.1007/JHEP10(2019)217},

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

  number       = 10,

  volume       = 2019,

  place        = {United States},

  year         = {Mon Oct 21 00:00:00 EDT 2019},

  month        = {Mon Oct 21 00:00:00 EDT 2019}

}

Copy to clipboard

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Similar Records

Title: Harmonic analysis and mean field theory

Abstract

Citation Formats

Comments on operators with large spin journal, November 2007

The analytic bootstrap and AdS superhorizon locality journal, December 2013

Casimir recursion relations for general conformal blocks journal, February 2018

Spinning conformal blocks journal, November 2011

Gauge theory correlators from non-critical string theory journal, May 1998

The spectrum in the Sachdev-Ye-Kitaev model journal, April 2016

Integrability of conformal fishnet theory journal, January 2018

Spinning conformal correlators journal, November 2011

Analyticity in spin in conformal theories journal, September 2017

Bootstrapping 3D fermions with global symmetries journal, January 2018

Anti de Sitter space and holography journal, January 1998

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

Factorization of Mellin amplitudes journal, October 2015

Seed conformal blocks in 4D CFT journal, February 2016

Light-ray operators in conformal field theory journal, November 2018

Counting conformal correlators journal, February 2018

The 3d stress-tensor bootstrap journal, February 2018

Recent developments in conformal invariant quantum field theory journal, August 1978

A generalization of Sachdev-Ye-Kitaev journal, February 2017

All unitary ray representations of the conformal group SU(2,2) with positive energy text, January 1975

A spacetime derivation of the Lorentzian OPE inversion formula journal, July 2018

Conformal partial waves and the operator product expansion journal, February 2004

Holography from conformal field theory journal, October 2009

Implications of conformal invariance in momentum space journal, March 2014

Bootstrapping 3D fermions journal, March 2016

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

Weight shifting operators and conformal blocks journal, February 2018

Interwining Operators for Semisimple Groups journal, May 1971

The Large-N Limit of Superconformal Field Theories and Supergravity journal, January 1999

Convexity and liberation at large spin journal, November 2013

Conformal correlators of mixed-symmetry tensors journal, February 2015

Bounding scalar operator dimensions in 4 D CFT journal, December 2008

Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory text, January 1976

Unitarity and the holographic S-Matrix journal, October 2012

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

Projectors, shadows, and conformal blocks journal, April 2014

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

d-dimensional SYK, AdS loops, and 6j symbols journal, March 2019

The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products journal, May 1972

Holography from Conformal Field Theory text, January 2009

Spinning Conformal Blocks text, January 2011

Projectors, Shadows, and Conformal Blocks text, January 2012

Factorization of Mellin amplitudes preprint, January 2014

Conformal correlators of mixed-symmetry tensors text, January 2014

Deconstructing Conformal Blocks in 4D CFT preprint, January 2015

Bootstrapping 3D Fermions text, January 2015

Seed Conformal Blocks in 4D CFT text, January 2016

The Spectrum in the Sachdev-Ye-Kitaev Model text, January 2016

A Generalization of Sachdev-Ye-Kitaev text, January 2016

Counting Conformal Correlators preprint, January 2016

Analyticity in Spin in Conformal Theories text, January 2017

Bootstrapping 3D Fermions with Global Symmetries text, January 2017

Weight Shifting Operators and Conformal Blocks text, January 2017

The 3d Stress-Tensor Bootstrap text, January 2017

Casimir recursion relations for general conformal blocks text, January 2017

Renormalised 3-point functions of stress tensors and conserved currents in CFT text, January 2017

Scalar-Fermion Analytic Bootstrap in 4D preprint, January 2017

Light-ray operators in conformal field theory text, January 2018

Renormalised CFT 3-point functions of scalars, currents and stress tensors text, January 2018

Conformal Partial Wave Expansions for N=4 Chiral Four Point Functions text, January 2004

Gauge Theory Correlators from Non-Critical String Theory text, January 1998

Anti De Sitter Space And Holography text, January 1998

Comments on operators with large spin
journal, November 2007

The analytic bootstrap and AdS superhorizon locality
journal, December 2013

Casimir recursion relations for general conformal blocks
journal, February 2018

Spinning conformal blocks
journal, November 2011

Gauge theory correlators from non-critical string theory
journal, May 1998

The spectrum in the Sachdev-Ye-Kitaev model
journal, April 2016

Integrability of conformal fishnet theory
journal, January 2018

Spinning conformal correlators
journal, November 2011

Analyticity in spin in conformal theories
journal, September 2017

Bootstrapping 3D fermions with global symmetries
journal, January 2018

Anti de Sitter space and holography
journal, January 1998

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

Factorization of Mellin amplitudes
journal, October 2015

Seed conformal blocks in 4D CFT
journal, February 2016

Light-ray operators in conformal field theory
journal, November 2018

Counting conformal correlators
journal, February 2018

The 3d stress-tensor bootstrap
journal, February 2018

Recent developments in conformal invariant quantum field theory
journal, August 1978

A generalization of Sachdev-Ye-Kitaev
journal, February 2017

All unitary ray representations of the conformal group SU(2,2) with positive energy
text, January 1975

A spacetime derivation of the Lorentzian OPE inversion formula
journal, July 2018

Conformal partial waves and the operator product expansion
journal, February 2004

Holography from conformal field theory
journal, October 2009

Implications of conformal invariance in momentum space
journal, March 2014

Bootstrapping 3D fermions
journal, March 2016

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

Weight shifting operators and conformal blocks
journal, February 2018

Interwining Operators for Semisimple Groups
journal, May 1971

The Large-N Limit of Superconformal Field Theories and Supergravity
journal, January 1999

Convexity and liberation at large spin
journal, November 2013

Conformal correlators of mixed-symmetry tensors
journal, February 2015

Bounding scalar operator dimensions in 4 D CFT
journal, December 2008

Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory
text, January 1976

Unitarity and the holographic S-Matrix
journal, October 2012

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

Projectors, shadows, and conformal blocks
journal, April 2014

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

d-dimensional SYK, AdS loops, and 6j symbols
journal, March 2019

The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products
journal, May 1972

Holography from Conformal Field Theory
text, January 2009

Spinning Conformal Blocks
text, January 2011

Projectors, Shadows, and Conformal Blocks
text, January 2012

Factorization of Mellin amplitudes
preprint, January 2014

Conformal correlators of mixed-symmetry tensors
text, January 2014

Deconstructing Conformal Blocks in 4D CFT
preprint, January 2015

Bootstrapping 3D Fermions
text, January 2015

Seed Conformal Blocks in 4D CFT
text, January 2016

The Spectrum in the Sachdev-Ye-Kitaev Model
text, January 2016

A Generalization of Sachdev-Ye-Kitaev
text, January 2016

Counting Conformal Correlators
preprint, January 2016

Analyticity in Spin in Conformal Theories
text, January 2017

Bootstrapping 3D Fermions with Global Symmetries
text, January 2017

Weight Shifting Operators and Conformal Blocks
text, January 2017

The 3d Stress-Tensor Bootstrap
text, January 2017

Casimir recursion relations for general conformal blocks
text, January 2017

Renormalised 3-point functions of stress tensors and conserved currents in CFT
text, January 2017

Scalar-Fermion Analytic Bootstrap in 4D
preprint, January 2017

Light-ray operators in conformal field theory
text, January 2018

Renormalised CFT 3-point functions of scalars, currents and stress tensors
text, January 2018

Conformal Partial Wave Expansions for N=4 Chiral Four Point Functions
text, January 2004

Gauge Theory Correlators from Non-Critical String Theory
text, January 1998

Anti De Sitter Space And Holography
text, January 1998