# Massless scalar field vacuum in de Sitter spacetime

## Abstract

As a spacetime with compact spatial sections, de Sitter spacetime does not have a de Sitter-invariant ground state for a minimally-coupled massless scalar field that gives definite expectation values for any observables not invariant under constant shifts of the field. However, if one restricts to observables that are shift invariant, as the action is, then there is a unique vacuum state. Here we calculate the shift-invariant four-point function that is the vacuum expectation value of the product of the difference of the field values at one pair of points and of the difference of the field values at a second pair of points. We show that this vacuum expectation value obeys a cluster-decomposition property of vanishing in the limit that the one pair of points is moved arbitrarily far from the other pair. We also calculate the shift-invariant correlation of the gradient of the scalar field at two different points and show that it also obeys a cluster-decomposition property. Possible relevance to a putative de Sitter-invariant quantum state for gravity is discussed.

- Authors:

- Theoretical Physics Institute, Department of Physics, 4-183 CCIS, University of Alberta, Edmonton, AB T6G 2E1 Canada (Canada)

- Publication Date:

- OSTI Identifier:
- 22279854

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Cosmology and Astroparticle Physics

- Additional Journal Information:
- Journal Volume: 2012; Journal Issue: 11; Other Information: Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1475-7516

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ASTROPHYSICS; CORRELATIONS; COSMOLOGY; DE SITTER GROUP; DE SITTER SPACE; EXPECTATION VALUE; GRAVITATION; GROUND STATES; QUANTUM STATES; SCALAR FIELDS; SPACE-TIME; VACUUM STATES

### Citation Formats

```
Page, Don N., and Wu, Xing, E-mail: profdonpage@gmail.com, E-mail: xwu5@ualberta.ca.
```*Massless scalar field vacuum in de Sitter spacetime*. United States: N. p., 2012.
Web. doi:10.1088/1475-7516/2012/11/051.

```
Page, Don N., & Wu, Xing, E-mail: profdonpage@gmail.com, E-mail: xwu5@ualberta.ca.
```*Massless scalar field vacuum in de Sitter spacetime*. United States. doi:10.1088/1475-7516/2012/11/051.

```
Page, Don N., and Wu, Xing, E-mail: profdonpage@gmail.com, E-mail: xwu5@ualberta.ca. Thu .
"Massless scalar field vacuum in de Sitter spacetime". United States. doi:10.1088/1475-7516/2012/11/051.
```

```
@article{osti_22279854,
```

title = {Massless scalar field vacuum in de Sitter spacetime},

author = {Page, Don N. and Wu, Xing, E-mail: profdonpage@gmail.com, E-mail: xwu5@ualberta.ca},

abstractNote = {As a spacetime with compact spatial sections, de Sitter spacetime does not have a de Sitter-invariant ground state for a minimally-coupled massless scalar field that gives definite expectation values for any observables not invariant under constant shifts of the field. However, if one restricts to observables that are shift invariant, as the action is, then there is a unique vacuum state. Here we calculate the shift-invariant four-point function that is the vacuum expectation value of the product of the difference of the field values at one pair of points and of the difference of the field values at a second pair of points. We show that this vacuum expectation value obeys a cluster-decomposition property of vanishing in the limit that the one pair of points is moved arbitrarily far from the other pair. We also calculate the shift-invariant correlation of the gradient of the scalar field at two different points and show that it also obeys a cluster-decomposition property. Possible relevance to a putative de Sitter-invariant quantum state for gravity is discussed.},

doi = {10.1088/1475-7516/2012/11/051},

journal = {Journal of Cosmology and Astroparticle Physics},

issn = {1475-7516},

number = 11,

volume = 2012,

place = {United States},

year = {2012},

month = {11}

}