# A Universe that Does Not Know the Time

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## Abstract

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## 1. Introduction

## 2. Concepts of Time

- For homogeneous or nearly homogeneous Universes, in a spatially flat slicing, ${T}_{CS}$ measures the ratio of the Hubble volume to a co-moving volume (see the next section). That means that, when you can measure the Chern–Simons time over an interval, you learn the two most important facts which situates your moment in a cosmological history: how large the horizon is and whether it is growing or shrinking.Furthermore, it is immediately apparent that, if there are eras in the evolution of a quantum cosmology in which ${T}_{CS}$ is subject to large uncertainties or large quantum fluctuations, they are not going to fit into the existing catalogue of cosmological scenarios.
- In a very natural class of extensions of General Relativity, studied in [10], the cosmological constant $\mathrm{\Lambda}$ becomes a dynamical variable. In the quantum theory, $\widehat{\frac{1}{\mathrm{\Lambda}}}$ is in fact the operator conjugate to ${\widehat{T}}_{CS}$.
- In these models [10], the commutator of $\widehat{\mathrm{\Lambda}}$ and ${\widehat{T}}_{CS}$ is proportional to ${l}_{Pl}^{4}{\widehat{\mathrm{\Lambda}}}^{2}$, leading to uncertainty relation:$$\mathrm{\Delta}{\mathrm{\Lambda}}^{2}\mathrm{\Delta}{T}_{CS}^{2}\ge \frac{1}{4}{\left(\right)}^{\frac{16\pi \hslash G}{3}}2$$We see in these simple relations hints of novel scenarios for the evolution of the Universes through successions of classical and quantum phases. When, as now, $\hslash G\langle {\widehat{\mathrm{\Lambda}}}^{2}\rangle $ is small, $\mathrm{\Lambda}$ and ${T}_{CS}$ are simultaneously measurable and classical General Relativity suffices. However, in a quantum phase in which $\hslash G\langle {\widehat{\mathrm{\Lambda}}}^{2}\rangle $ is large, one or both of $\mathrm{\Lambda}$ and ${T}_{CS}$ are undeterminable and subject to large quantum fluctuations.

## 3. Chern–Simons Time and the Horizon Problem

**Firstly**, it would appear that the standard Big Bang Universe is going backwards in CS time. Indeed, this retrograde “motion” of time is nothing but a statement of the horizon problem of Big Bang cosmology. Conversely, any solution to the horizon problem (regardless of its nature) can therefore be reinterpreted as the statement that, in the “early” Universe, CS time moved forward enough so that it could rewind to now during the standard Big Bang phase.

**Secondly**, the fact that the Universe recently “started” accelerating suggests emphasizing the similarities between the two accelerating phases (“current” and “primordial”). Could the Universe be cyclic (in some variables) in the sense that it undergoes a pulsation of forwards and backwards CS time? It is tempting to envisage the Universe as a closed loop of ebb and flow in CS time. If so, the “recent” “start” of acceleration is nothing but a turning point of the tide4.

## 4. Quantum Time

## 5. A Model for Quantum Ebbing of Time

## 6. Cycling the Classical and Quantum Phases

- Classical phase. Localized time exists. Matter dominates Lambda. Time commutes with Lambda. We have the Big Bang Universe (BBU) as we know it. This may be represented as state:$$|BBU\rangle =\prod _{i=BB}^{\mathrm{\Lambda}\phantom{\rule{0.166667em}{0ex}}\mathrm{dom}}|\mathrm{\Lambda}\rangle \otimes |{\rho}_{i}\rangle \otimes |{T}_{i}\rangle ,$$
- As expansion dilutes matter and Lambda dominates, a quantum phase ensues, with delocalized time. The Universe finds itself in a squeezed state $|\alpha \zeta \rangle $, as in Equation (28), with highly localized Lambda and delocalized time. Therefore, a finite amplitude exists for a transition to a state with ${T}_{-}$:$$\langle {T}_{-}|\alpha \zeta \rangle =\frac{1}{{\left(2\pi {\mathrm{\Delta}}_{E}^{2}\right)}^{1/4}}exp\left(\right)open="("\; close=")">-\frac{{(T-{T}_{0})}^{2}}{4{\mathrm{\Delta}}_{T}^{2}}$$
- The state $|{T}_{-}\rangle $ itself can be written as:$$|{T}_{-}\rangle =\int d\mathrm{\Lambda}d\rho \langle \mathrm{\Lambda}\rho |{T}_{-}\rangle |\mathrm{\Lambda}\rho \rangle ,$$

## 7. Evolution without Time and the Emergence of Time

## 8. Conclusion and Outlook: A Universe that Lost the Plot

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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3 | We use the term “kinematic” to mean without the usual canonical dynamics, i.e., without a Hamiltonian over a classical phase space, from which a Poisson bracket or commutator structure (upon quantization) generates a time evolution. The point is that, since there may be situations in which there is no time in the conventional sense, we should also be prepared to eschew the conventional concept of dynamics. We do, however, postulate commutators, and these have kinematical implications, such as uncertainty relations and the form of eigenfunctions in terms of representations diagonalising the complementary variable. In this paper, we do not use the term kinematical in the sense it has acquired in a specific community—of something akin to “unphysical because it does not commute with the quantum constraints.” The commutation relations of our operators also do not come from classical Poisson bracket analogues. Note that this structure does not preclude the existence of a base space (where these kinematic quantities act as parameters, or measures) which is itself dynamical, and endowed with a Hamiltonian, such as in the constructions of [16]. The base space may or may not be quantized. |

4 | The strange grammar of this paragraph is a reminder that one should purge one’s thoughts from the unavoidable inconsistencies of our language, tied to the human experience of time. If ${T}_{CS}$ is to be the only definition of time that can be extrapolated to the whole life of the Universe then expressions such as “early” Universe, or “motion” in time are misleading and inappropriate. |

5 | In the above, we have assumed expanding Universes only. In bouncing/cyclic scenarios, the relation between the direction of flow of ${T}_{CS}$ and the horizon problem has to be revisited. As it happens, CS time continues to flow forward in the contracting phase, passing zero at the turnaround point. The ebbing of CS time happens fully during the bounce phase. We leave to a future publication a more complete discussion of this case. |

6 | Note that CS time is dimensionless, as is its conjugate “energy”, function of the dimensionless quantity ${l}_{P}^{2}\mathrm{\Lambda}$. |

7 | Pauli’s theorem is the statement that no Hermitian operator $\widehat{t}$ can be found satisfying $[\widehat{t},\widehat{H}]=i$, where $\widehat{H}$ is the Hamiltonian operator seen as a function of P and Q variables (as would be the case for $\widehat{t}$). A number of assumptions are made, not necessarily valid, even before we note that the “theorem” would not be applicable here. |

8 | Clearly “domination” must mean ${\mathrm{\Omega}}_{\mathrm{\Lambda}}$ much closer to 1 than 0.7 to comply with observations (but see a possible loophole to this in the speculations presented in the Conclusions). |

9 | For another approach to a quantum time, see [32]. |

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**MDPI and ACS Style**

Magueijo, J.; Smolin, L.
A Universe that Does Not Know the Time. *Universe* **2019**, *5*, 84.
https://doi.org/10.3390/universe5030084

**AMA Style**

Magueijo J, Smolin L.
A Universe that Does Not Know the Time. *Universe*. 2019; 5(3):84.
https://doi.org/10.3390/universe5030084

**Chicago/Turabian Style**

Magueijo, João, and Lee Smolin.
2019. "A Universe that Does Not Know the Time" *Universe* 5, no. 3: 84.
https://doi.org/10.3390/universe5030084