# A Brief Overview of Results about Uniqueness of the Quantization in Cosmology

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## Abstract

**:**

## 1. Introduction

## 2. Weyl Algebra and Standard Representations

## 3. Loop Quantum Cosmology

## 4. Fock Quantization in Non-Stationary Cosmological Settings

#### 4.1. Gowdy Models

#### 4.2. Quantum Field Theory in Cosmological Settings

- there exists a Fock representation defined by a state that is invariant under the symmetries of the metric ${h}_{ab}$ (or equivalently, a Fock representation with an invariant vacuum) and such that the classical dynamics can be unitarily implemented;
- that representation is unique, in the sense that any other Fock representation defined by an invariant state and allowing a unitary implementation of the dynamics is unitarily equivalent to the previous one.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Sketchof the Proof of Uniqueness of the Representation for the Scalar Field with Time-Dependent Mass in S^{1}

## Notes

1 | There are currently also LQC-inspired applications to inhomogeneous cosmologies. We will not consider them in the present work. |

2 | A state $\omega $ of a ⋆-algebra $\mathcal{A}$ is a linear functional such that $\omega \left(a{a}^{*}\right)\ge 0$, $\forall a\in \mathcal{A}$, and $\omega \left(\mathbf{1}\right)=1$, where $\mathbf{1}$ is the identity of the algebra and the symbol ${\phantom{\rule{0.166667em}{0ex}}}^{*}$ denotes the involution operation, e.g., complex conjugation in algebras of functions and adjointness in algebras of operators. |

3 | We also restrict attention to the more usual formulation of LQC, leaving aside the so-called Fleischhack approach, which is also considered in [10]. |

4 | Nevertheless, applications of LQC are effectively performed on a separable subspace of ${\mathcal{H}}_{\mathcal{P}}$. This can either be seen as a consistency requirement [21] or as a consequence of superselection [22], which, in any case, can be traced back to the fact that the LQC quantum Hamiltonian constraint is a difference operator of constant step. |

5 | |

6 | Alternatively, the system can be considered as an axially symmetric field propagating in a static (2+1)-dimensional spacetime with the spatial topology of a two-torus. |

7 | Only very mild technical conditions on $s\left(t\right)$ are required, see [50]. |

8 | Part of the techniques employed for fermions were already explored in the case of the scalar field in Bianchi I, in order to deal with the lack of conformal symmetry. A review of the range of different methods and improvements required to address the increasing degree of generalization encountered in the treatment of the scalar field can be found in [58]. |

9 | In order to provide a unitary representation of translations, measures are required to satisfy the technical condition of quasi-invariance, which is satisfied, e.g., by any Gaussian measure. |

10 | With the obvious adaptations, taking into account that the spatial manifold is now ${S}^{1}$ instead of ${\mathbb{R}}^{3}$. |

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Cortez, J.; Mena Marugán, G.A.; Velhinho, J.M.
A Brief Overview of Results about Uniqueness of the Quantization in Cosmology. *Universe* **2021**, *7*, 299.
https://doi.org/10.3390/universe7080299

**AMA Style**

Cortez J, Mena Marugán GA, Velhinho JM.
A Brief Overview of Results about Uniqueness of the Quantization in Cosmology. *Universe*. 2021; 7(8):299.
https://doi.org/10.3390/universe7080299

**Chicago/Turabian Style**

Cortez, Jerónimo, Guillermo A. Mena Marugán, and José M. Velhinho.
2021. "A Brief Overview of Results about Uniqueness of the Quantization in Cosmology" *Universe* 7, no. 8: 299.
https://doi.org/10.3390/universe7080299