# Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’

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## Abstract

**:**

## 1. Introduction

## 2. The Flat FRW Model with Massless Scalar Field

#### 2.1. Classical Relational Dynamics and Internal Time Switches

- (i)
- On ${\mathcal{C}}_{\pm}^{t}\subset \mathcal{C}$, defined by ${C}_{\pm}^{t}=0$ and ${p}_{t}\ne 0$, we have$$\begin{array}{c}\hfill \frac{\mathrm{d}\phantom{\rule{0.166667em}{0ex}}\xb7}{\mathrm{d}s}=\{\xb7,{C}_{H}\}\approx \mp \phantom{\rule{0.166667em}{0ex}}2{s}_{t}\phantom{\rule{0.166667em}{0ex}}{h}_{e}\phantom{\rule{0.166667em}{0ex}}\{\xb7,{C}_{\pm}^{t}\}\phantom{\rule{0.166667em}{0ex}},\end{array}$$
- (ii)
- The set ${p}_{\alpha}={p}_{\varphi}=0$ is the shared boundary between ${\mathcal{C}}_{+}^{\varphi}$ and ${\mathcal{C}}_{-}^{\varphi}$, as well as between ${\mathcal{C}}_{+}^{\alpha}$ and ${\mathcal{C}}_{-}^{\alpha}$. Notice that orbits with ${p}_{\alpha}={p}_{\varphi}=0$ are just points in $\mathcal{C}$ so that the latter is stratified by gauge orbits of different dimension. Since $\mathrm{d}{C}_{H}=0$ for ${p}_{\alpha}={p}_{\varphi}=0$, no gauge-fixing surface can pierce every such gauge orbit once and only once.

#### 2.2. Reduced Quantization Relative to a Choice of Internal Time

#### 2.3. The Internal-Time-Neutral Dirac Quantization

#### 2.4. Quantum Reduction: From Dirac to Reduced Quantization

#### 2.5. Quantum Internal Time Switches

#### 2.6. Illustration in Concrete States

## 3. Perspective on the ‘Wave Function of the Universe’

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Hermiticity of the Relational Observable $\widehat{E}\left(\tau \right)$ on ${\mathcal{H}}_{\mathrm{phys}}$

## Appendix B. Changes of Internal Times in the Quantum Theory

## Appendix C. Continuity of the Quantum Relational Dynamics during a Switch

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1 | In fact, we have included a choice of lapse function $N={e}^{3\alpha}$. |

2 | For more details of this procedure in a different model, see [40]. |

3 | Please note that generally ${T}_{\pm}\left({\tau}_{e}^{i}\right)\ne {E}_{+}\left({\tau}_{t}^{f}\right)$, despite the form (12). |

4 | We set $\hslash =1$. |

5 | In the affine momentum representation, states are represented as ${|\psi \rangle}_{\pm}^{e\left(t\right)}={\int}_{-\infty}^{+\infty}\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{d}{p}_{e}}{|{p}_{e}|}\phantom{\rule{0.166667em}{0ex}}{\tilde{\psi}}_{\pm}^{e\left(t\right)}\left({p}_{e}\right)\phantom{\rule{0.166667em}{0ex}}{|{p}_{e}\rangle}_{\mathrm{aff}}$, where ${\tilde{\psi}}_{\pm}^{e\left(t\right)}=\sqrt{|{p}_{e}|}\phantom{\rule{0.166667em}{0ex}}{\psi}_{\pm}^{e\left(t\right)}$ and ${\langle {p}_{e}|{p}_{e}^{\prime}\rangle}_{\mathrm{aff}}=\left|{p}_{e}\right|\phantom{\rule{0.166667em}{0ex}}\delta ({p}_{e}-{p}_{e}^{\prime})$. The inner product then reads ${\langle \psi |\chi \rangle}_{\pm}^{e\left(t\right)}={\int}_{-\infty}^{+\infty}\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{d}{p}_{e}}{|{p}_{e}|}\phantom{\rule{0.166667em}{0ex}}{\left[{\tilde{\psi}}_{\pm}^{e\left(t\right)}\left({p}_{e}\right)\right]}^{*}\phantom{\rule{0.166667em}{0ex}}{\tilde{\chi}}_{\pm}^{e\left(t\right)}\left({p}_{e}\right)$ and the configuration observables are represented as $\widehat{E}\phantom{\rule{0.166667em}{0ex}}{\tilde{\psi}}_{\pm}^{e\left(t\right)}=i\phantom{\rule{0.166667em}{0ex}}{p}_{e}\phantom{\rule{0.166667em}{0ex}}{\partial}_{{p}_{e}}\phantom{\rule{0.166667em}{0ex}}{\tilde{\psi}}_{\pm}^{e\left(t\right)}$ and $\widehat{e}\phantom{\rule{0.166667em}{0ex}}{\tilde{\psi}}_{\pm}^{e\left(t\right)}=(i\phantom{\rule{0.166667em}{0ex}}{\partial}_{{p}_{e}}-\frac{i}{2{p}_{e}})\phantom{\rule{0.166667em}{0ex}}{\tilde{\psi}}_{\pm}^{e\left(t\right)}$. It is easy to check that this affine representation is equivalent to the canonical one above. |

**Figure 1.**Schematic representation of the constraint surface $\mathcal{C}$, defined by (1), as a ‘light cone’ in momentum space. Its four components have the following physical interpretation. Red: contracting universe, but $\varphi $ runs ‘backward’. Blue: contracting universe and $\varphi $ runs ‘forward’. Green: expanding universe and $\varphi $ runs ‘forward’. Purple: expanding universe, but $\varphi $ runs ‘backward’. At the intersection point (the origin) the dynamics is static.

**Figure 2.**Diagrammatic overview of the relation between Dirac and the four reduced quantizations. In a nutshell, the physical Hilbert space is mapped to any of the four (positive or negative frequency) reduced Hilbert spaces by first trivializing the Hamiltonian constraint via ${\mathcal{T}}_{\varphi}$ or ${\mathcal{T}}_{\alpha}$ to the corresponding choice of internal time variable and subsequently projecting onto the classical internal time gauge-fixing condition. (Details in the main text.)

**Figure 3.**Reduced probability distributions coming from the same physical state (defined through (20) and (46) and here $n=100$), but described relative to the choices of (

**a**) $\varphi $ and (

**b**) $\alpha $ as internal times in the ‘expanding-forward’ (green) quadrant of Figure 1. Recall that in the reduced theory the usual modulus square of the wave function is the probability distribution, see (15). Due to the symmetry of the model in $\alpha $ and $\varphi $, reduced probability distributions will always behave symmetrically.

**Figure 4.**Illustration of the quantum relational evolution given in (47) and (48) for an internal time switch from $\varphi $ to $\alpha $ at ${\tau}_{\varphi}^{f}={\tau}_{\alpha}^{i}=0$. The blue branch corresponds to the evolution of ${\u2329{\widehat{A}}_{-}\left({\tau}_{\varphi}\right)\u232a}_{-}^{\alpha \left(\varphi \right)}/{\langle {\widehat{p}}_{\alpha}\rangle}_{-}^{\alpha \left(\varphi \right)}$ in ${\tau}_{\varphi}$, while the golden branch depicts the evolution of ${\u2329{\widehat{\mathsf{\Phi}}}_{+}\left({\tau}_{\alpha}\right)\u232a}_{+}^{\varphi \left(\alpha \right)}/{\langle {\widehat{p}}_{\varphi}\rangle}_{+}^{\varphi \left(\alpha \right)}$ in ${\tau}_{\alpha}$. Together they trace out a continuous classical trajectory, describing an expanding universe.

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**MDPI and ACS Style**

Höhn, P.A.
Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’. *Universe* **2019**, *5*, 116.
https://doi.org/10.3390/universe5050116

**AMA Style**

Höhn PA.
Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’. *Universe*. 2019; 5(5):116.
https://doi.org/10.3390/universe5050116

**Chicago/Turabian Style**

Höhn, Philipp A.
2019. "Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’" *Universe* 5, no. 5: 116.
https://doi.org/10.3390/universe5050116