## Positive energy conditions in 4D conformal field theory

## Abstract

Here, we argue that all consistent 4D quantum field theories obey a spacetime-averaged weak energy inequality < T ^{00} > ≥ –C/L ^{4}, where L is the size of the smearing region, and C is a positive constant that depends on the theory. If this condition is violated, the theory has states that are indistinguishable from states of negative total energy by any local measurement, and we expect instabilities or other inconsistencies. We apply this condition to 4D conformal field theories, and find that it places constraints on the OPE coefficients of the theory. The constraints we find are weaker than the “conformal collider” constraints of Hofman and Maldacena. In _{3}D CFTs, the only constraint we find is equivalent to the positivity of 2-point function of the energy-momentum tensor, which follows from unitarity. Our calculations are performed using momentum-space Wightman functions, which are remarkably simple functions of momenta, and may be of interest in their own right.

- Authors:

- Univ. of California, Davis, CA (United States)

- Publication Date:

- Research Org.:
- Univ. of California, Davis, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1364045

- Grant/Contract Number:
- FG02-91ER40674

- 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: 2016; Journal Issue: 10; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; conformal and W symmetry; effective field theories; space-time symmetries

### Citation Formats

```
Farnsworth, Kara, Luty, Markus A., and Prilepina, Valentina. Positive energy conditions in 4D conformal field theory. United States: N. p., 2016.
Web. doi:10.1007/JHEP10(2016)001.
```

```
Farnsworth, Kara, Luty, Markus A., & Prilepina, Valentina. Positive energy conditions in 4D conformal field theory. United States. doi:10.1007/JHEP10(2016)001.
```

```
Farnsworth, Kara, Luty, Markus A., and Prilepina, Valentina. Mon .
"Positive energy conditions in 4D conformal field theory". United States. doi:10.1007/JHEP10(2016)001. https://www.osti.gov/servlets/purl/1364045.
```

```
@article{osti_1364045,
```

title = {Positive energy conditions in 4D conformal field theory},

author = {Farnsworth, Kara and Luty, Markus A. and Prilepina, Valentina},

abstractNote = {Here, we argue that all consistent 4D quantum field theories obey a spacetime-averaged weak energy inequality < T00 > ≥ –C/L4, where L is the size of the smearing region, and C is a positive constant that depends on the theory. If this condition is violated, the theory has states that are indistinguishable from states of negative total energy by any local measurement, and we expect instabilities or other inconsistencies. We apply this condition to 4D conformal field theories, and find that it places constraints on the OPE coefficients of the theory. The constraints we find are weaker than the “conformal collider” constraints of Hofman and Maldacena. In 3D CFTs, the only constraint we find is equivalent to the positivity of 2-point function of the energy-momentum tensor, which follows from unitarity. Our calculations are performed using momentum-space Wightman functions, which are remarkably simple functions of momenta, and may be of interest in their own right.},

doi = {10.1007/JHEP10(2016)001},

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

number = 10,

volume = 2016,

place = {United States},

year = {2016},

month = {10}

}

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