# Symmetries in Classical and Quantum Treatment of Einstein’s Cosmological Equations and Mini-Superspace Actions

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## Abstract

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## 1. Introduction

## 2. Classical Kinematics

- The first is to make the shift ${\tilde{N}}^{\alpha}$ zero. Then, the residual rigid “gauge” freedom described by ${\mathsf{\Lambda}}_{\beta}^{\alpha}=\mathrm{constant}$, ${P}^{\rho}=0$ provides us with Lie point symmetries, which can be used to reduce the order of the equations and ultimately acquire the entire solution space. In such a way, the general solution of Bianchi Types I–VII has been uncovered.
- The second is to simplify the scale factor matrix and then proceed to solve the reduced form of the equations. This option is more suitable for the case of Bianchi Types VIII and IX, since in this case, the time-dependent $\mathsf{\Lambda}$ suffices to diagonalize ${\gamma}_{\alpha \beta}$, and then, Equation (3b) enforces ${N}^{\alpha}=0$.

## 3. Bianchi Type I

#### 3.1. Three Unequal Real Eigenvalues

#### 3.2. Three Real with Two Equal Eigenvalues

#### 3.3. Case with One Real and Two Complex Conjugate Eigenvalues

#### 3.4. Case with Three Equal Eigenvalues

## 4. Diagonalizability of ${\gamma}_{\alpha \beta}$ for Types VIII–IX

**Case****IX**- Suppose $a\left(t\right)=b\left(t\right)$. Then, Equation (55) implies that ${N}^{1}={N}^{2}=0$ and ${N}^{3}\left(t\right)$ is unrestricted. However, there is an extra rotation in the plane (1–2), which has no effect on the form of ${\gamma}_{\alpha \beta}$. This particular matrix, being an automorphism, can be used to absorb the ${N}^{3}$; see Equations (9) and (54). The situation with $a\left(t\right)=c\left(t\right)$ or $b\left(t\right)=c\left(t\right)$ is exactly the same.
**Case****VIII**- For the case $b\left(t\right)=c\left(t\right)$, we follow exactly the same reasoning and arrive at zero shift, as well. Of particular interest and less known is the fact that the case $b\left(t\right)=a\left(t\right)$ or $c\left(t\right)=a\left(t\right)$ leads to the incompatibility of the resulting Einstein Equation (3c) (for the first time reported in [5]); they require $-\frac{2n{\left(t\right)}^{2}}{a\left(t\right)}=0$. This fact can, in view of the transformations (9), be understood as follows: while the transformation matrix that leaves the form of ${\gamma}_{\alpha \beta}$ invariant is still a rotation, the corresponding allowed one is a boost; thus, the form of ${\gamma}_{\alpha \beta}$ becomes block diagonal when the shift is zero, showing the incompatibility.

## 5. Reduced Dynamics

- The requirement for the operator to be scalar under coordinate transformations of the configuration space variables ${q}^{a}$.
- The requirement to contain up to second derivatives of ${G}_{\mu \nu}$ since the classical constraint is quadratic in momenta.
- The requirement to be covariant under conformal scalings of ${G}_{\mu \nu}$, since this is also a property of the classical system.

## 6. Massless Field in the FLRW Universe

#### 6.1. Classical Treatment

#### 6.2. Canonical Quantization and Semiclassical Analysis

#### 6.2.1. Subalgebra $({\widehat{Q}}_{1},{\widehat{Q}}_{2})$

#### 6.2.2. Subalgebra ${\widehat{Q}}_{3}$

## 7. Resume

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Christodoulakis, T.; Karagiorgos, A.; Zampeli, A.
Symmetries in Classical and Quantum Treatment of Einstein’s Cosmological Equations and Mini-Superspace Actions. *Symmetry* **2018**, *10*, 70.
https://doi.org/10.3390/sym10030070

**AMA Style**

Christodoulakis T, Karagiorgos A, Zampeli A.
Symmetries in Classical and Quantum Treatment of Einstein’s Cosmological Equations and Mini-Superspace Actions. *Symmetry*. 2018; 10(3):70.
https://doi.org/10.3390/sym10030070

**Chicago/Turabian Style**

Christodoulakis, Theodosios, Alexandros Karagiorgos, and Adamantia Zampeli.
2018. "Symmetries in Classical and Quantum Treatment of Einstein’s Cosmological Equations and Mini-Superspace Actions" *Symmetry* 10, no. 3: 70.
https://doi.org/10.3390/sym10030070