# The quantum Fokker-Planck equation of stochastic inflation

## Abstract

We derive the stochastic description of a massless, interacting scalar field in de Sitter space directly from the quantum theory. This is done by showing that the density matrix for the effective theory of the long wavelength fluctuations of the field obeys a quantum version of the Fokker-Planck equation. This equation has a simple connection with the standard Fokker-Planck equation of the classical stochastic theory, which can be generalised to any order in perturbation theory. We illustrate this formalism in detail for the theory of a massless scalar field with a quartic interaction.

- Authors:

- Carnegie Mellon Univ., Pittsburgh, PA (United States). Dept. of Physics
- Carnegie Mellon Univ., Pittsburgh, PA (United States). Dept. of Physics; Minerva Schools at KGI, San Francisco, CA (United States). College of Computational Sciences

- Publication Date:

- Research Org.:
- Carnegie Mellon Univ., Pittsburgh, PA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1507422

- Grant/Contract Number:
- FG02-91ER40682

- Resource Type:
- Journal Article: Accepted Manuscript

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

- Additional Journal Information:
- Journal Volume: 2017; Journal Issue: 11; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 79 ASTRONOMY AND ASTROPHYSICS; Cosmology of Theories beyond the SM; Effective Field Theories; Stochastic Processes

### Citation Formats

```
Collins, Hael, Holman, Richard, and Vardanyan, Tereza.
```*The quantum Fokker-Planck equation of stochastic inflation*. United States: N. p., 2017.
Web. doi:10.1007/jhep11(2017)065.

```
Collins, Hael, Holman, Richard, & Vardanyan, Tereza.
```*The quantum Fokker-Planck equation of stochastic inflation*. United States. doi:10.1007/jhep11(2017)065.

```
Collins, Hael, Holman, Richard, and Vardanyan, Tereza. Mon .
"The quantum Fokker-Planck equation of stochastic inflation". United States. doi:10.1007/jhep11(2017)065. https://www.osti.gov/servlets/purl/1507422.
```

```
@article{osti_1507422,
```

title = {The quantum Fokker-Planck equation of stochastic inflation},

author = {Collins, Hael and Holman, Richard and Vardanyan, Tereza},

abstractNote = {We derive the stochastic description of a massless, interacting scalar field in de Sitter space directly from the quantum theory. This is done by showing that the density matrix for the effective theory of the long wavelength fluctuations of the field obeys a quantum version of the Fokker-Planck equation. This equation has a simple connection with the standard Fokker-Planck equation of the classical stochastic theory, which can be generalised to any order in perturbation theory. We illustrate this formalism in detail for the theory of a massless scalar field with a quartic interaction.},

doi = {10.1007/jhep11(2017)065},

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

issn = {1029-8479},

number = 11,

volume = 2017,

place = {United States},

year = {2017},

month = {11}

}

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