# 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:

- 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:

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

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); Swiss National Science Foundation (SNF)

- OSTI Identifier:
- 1594253

- Report Number(s):
- arXiv:1809.05111

Journal ID: ISSN 1029-8479

- 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.
```

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

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

```
@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 = {2019},

month = {10}

}

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