# 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:

- 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:

- 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.
```

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

```
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.
```

```
@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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