# Averaged null energy condition from causality

## Abstract

Unitary, Lorentz-invariant quantum field theories in at spacetime obey mi-crocausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, $$\int$$duT _{uu}, must be non-negative. This non-local operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to n-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an infinite family of new constraints of the form RduX uuu∙∙∙u ≥ 0. These lead to new inequalities for the coupling constants of spinning operators in conformal field theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, conformal collider methods, and relative entropy. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and information-theoretic inequalities in QFT.

- Authors:

- Cornell Univ., Ithaca, NY (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Cornell Univ., Ithaca, NY (United States)

- Sponsoring Org.:
- USDOE; National Science Foundation (NSF)

- OSTI Identifier:
- 1393403

- Grant/Contract Number:
- SC0014123; PHY-1316222

- Resource Type:
- Journal Article: Accepted Manuscript

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

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 7; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Conformal Field Theory; Field Theories in Higher Dimensions

### Citation Formats

```
Hartman, Thomas, Kundu, Sandipan, and Tajdini, Amirhossein.
```*Averaged null energy condition from causality*. United States: N. p., 2017.
Web. doi:10.1007/JHEP07(2017)066.

```
Hartman, Thomas, Kundu, Sandipan, & Tajdini, Amirhossein.
```*Averaged null energy condition from causality*. United States. doi:10.1007/JHEP07(2017)066.

```
Hartman, Thomas, Kundu, Sandipan, and Tajdini, Amirhossein. Fri .
"Averaged null energy condition from causality". United States.
doi:10.1007/JHEP07(2017)066. https://www.osti.gov/servlets/purl/1393403.
```

```
@article{osti_1393403,
```

title = {Averaged null energy condition from causality},

author = {Hartman, Thomas and Kundu, Sandipan and Tajdini, Amirhossein},

abstractNote = {Unitary, Lorentz-invariant quantum field theories in at spacetime obey mi-crocausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, $\int$duTuu, must be non-negative. This non-local operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to n-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an infinite family of new constraints of the form RduX uuu∙∙∙u ≥ 0. These lead to new inequalities for the coupling constants of spinning operators in conformal field theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, conformal collider methods, and relative entropy. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and information-theoretic inequalities in QFT.},

doi = {10.1007/JHEP07(2017)066},

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

number = 7,

volume = 2017,

place = {United States},

year = {Fri Jul 14 00:00:00 EDT 2017},

month = {Fri Jul 14 00:00:00 EDT 2017}

}

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