# Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant

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## Abstract

**:**

## 1. Introduction

## 2. Extended Phase Space Formulation and Time-Dependent Canonical Transformations

#### 2.1. Time-Dependent Hamiltonians on Extended Phase Space

#### 2.2. Extended Canonical Transformations and Hamiltonian Flows

#### 2.3. The Reduced Phase Space Associated with ${T}^{*}M$ and the Infinitesimal Generator of $\Phi $

## 3. Quantization: One-Particle Hilbert Space

#### 3.1. Canonical Quantization of the Time-Dependent Canonical Transformation

#### 3.2. Baker-Campbell-Hausdorff Decomposition

#### 3.3. Time-Dependent Bogoliubov Maps

## 4. Implementing the Time-Dependent Canonical Transformation as a Unitary Map on the Bosonic Fock Space

#### Proposal of a Modified Map for the Infrared Modes: The Arnold Transformation

## 5. Relation of the Lewis-Riesenfeld Invariant Approach to the Bunch-Davies Vacuum and Adiabatic Vacua

## 6. Applications

#### 6.1. Solution of the Ermakov Equation on Quasi-de Sitter Spacetime

#### 6.2. Eigenstates of the Lewis-Riesenfeld Invariant

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | In general we could also take into account a time dependent mass in the Hamiltonian, however in the case of the Mukhanov-Sasaki equation it is sufficient to set the mass parameter m equal to $m=1$. |

2 | The symplectic form associated with the Poisson bracket ${\{.,.\}}_{\mathrm{ext}}$ on the extended phase space has the form $\Omega =d{\tilde{q}}^{a}\wedge d{\tilde{p}}_{a}+dt\wedge d{p}_{t}$. |

3 | Note that the roles of ${\widehat{a}}_{\mathbf{k}},{\widehat{a}}_{\mathbf{k}}^{\u2020}$ and ${\widehat{b}}_{\mathbf{k}},{\widehat{b}}_{\mathbf{k}}^{\u2020}$ are interchanged in comparison to the one-particle case considered in (48) for notational convenience, whereas the coefficients are named analogously. Here the first set of operators belongs to the Mukhanov-Sasaki Hamiltonian, whereas the second set is associated to the time-independent harmonic oscillator. In contrast, in (48) the operators $\widehat{B},{\widehat{B}}^{\u2020}$ belong to the time-dependent system, whereas $\widehat{A},{\widehat{A}}^{\u2020}$ are associated with the time-independent harmonic oscillator. |

**Figure 2.**Single-mode probability densities $|{\Psi}_{0}{(q,\eta )|}^{2}$ (upper line) and $|{\Psi}_{1}{(q,\eta )|}^{2}$ (lower line) according to the solutions in (89) and (90) on quasi-de Sitter for three different values of the effective slow-roll parameter $\nu $ from Equation (58), including de Sitter with $\nu =3/2$ at two different conformal times and with ${\omega}_{0}=k=1$. The used slow-roll parameters are to be understood as an example, consider Reference [33] for the allowed parameter space and constraints on them according to the Planck mission.

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Fahn, M.J.; Giesel, K.; Kobler, M.
Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant. *Universe* **2019**, *5*, 170.
https://doi.org/10.3390/universe5070170

**AMA Style**

Fahn MJ, Giesel K, Kobler M.
Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant. *Universe*. 2019; 5(7):170.
https://doi.org/10.3390/universe5070170

**Chicago/Turabian Style**

Fahn, Max Joseph, Kristina Giesel, and Michael Kobler.
2019. "Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant" *Universe* 5, no. 7: 170.
https://doi.org/10.3390/universe5070170