# A Wheeler–DeWitt Quantum Approach to the Branch-Cut Gravitation with Ordering Parameters

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## Abstract

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## 1. Introduction

- The action is defined, using the Hora$\stackrel{\u02c7}{\mathrm{v}}$a–Lifshitz theory of gravity, which is the General Relativity augmented by counter-terms to render the theory regularized. For more information, please consult Ref. [24]. The basic ingredients are now expressed in terms of $ln\left[\beta \right(t\left)\right]$, which substitutes the standard scale factor $a\left(t\right)$. In Section 2.1, the classical impulse variable is defined and the classical Hamiltonian constructed.
- A quantization procedure is applied, elevating the momentum operator and Hamiltonian to operators. As a result we obtain the Wheeler–DeWitt equation.
- Following this path, a parameter $\alpha $ appears which defines the ordering of the operators, as applied in the past to the Wheeler–DeWitt equation. This leaves us with three possible equations.
- These equations are solved using the Range–Kutta numerical analysis iterative method. Unlike the approaches usually found in the literature, in our calculations we do not use approximations. We then obtain new analytic solutions, depending on the boundary conditions based on the Bekenstein’s theorem, which provides an upper limit for the entropy. For more information, please consult [10,11,12,13,36].

## 2. Extended Class of the Branched Quantum Cosmological Solutions

#### 2.1. Branch-Cut Formulation of the Weeler-DeWitt Equation

## 3. Spacetime Topological Canonical Quantization

#### 3.1. Complex Conjugation of the Friedmann’s-Type Wave Equations

#### 3.2. Solutions and Boundary Conditions

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | Hawking and Hertog, in 2018, revisited the multiverse concept, conjecturing that the output of eternal inflation does not produce an infinite fractal-type multiverse, but is finite and reasonably smooth. |

2 | For simplicity the cosmological constant term has been suppressed. |

3 | We emphasize that these equations do not represent a direct parameterization or generalization of the conventional Friedmann equations described in a single-pole metric and likewise the new cosmic scale factor does not represent a simple parameterization of the standard theory scale factor. Due to the non-linearity of Einstein’s equations, such a direct generalization or parametrization would be inconsistent. For the details, see [7,8,9,12]. |

4 | The impossibility of packing energy and entropy according to the Bekenstein Criterion into a finite size makes the transition phase between contraction and expansion very peculiar, imposing a topology where space-time shapes itself topologically around a branch point. |

5 | The Hořava–Lifshitz (HL) formulation main goal is to get a renormalizable theory by means of higher spatial-derivative terms of the curvature which are added to the Einstein–Hilbert action [20]. A recurring problem addressed in the analysis of the Hořava–Lifshitz theory of gravity is related to the preservation of general diffeomorphism, a fundamental constraint of general relativity [22]. Although this is not the main topic of discussion, we would like to address that, in the case of restricted foliation preserving diffeomorphism invariance of the Hořava–Lifshitz theory, a well behaved Hamiltonian for gravity may be found [23]. |

6 | For an interesting discussion of this topic see Ref. [31]. |

7 | We emphasize once more that ${ln}^{-1}\left[\beta \left(t\right)\right]$ represents the reciprocal of $ln\left[\beta \right(t\left)\right]$ and $\beta \left(t\right)$ identifies the range and cuts of the helix-like cosmological factor in branched gravitation. ${ln}^{-1}\left[\beta \left(t\right)\right]$ characterizes complex topological leafs of singular foliations by means of Riemann surfaces. |

8 | $N\left(t\right)$ does note represent a dynamical quantity; in turn it denotes a pure gauge variable. |

9 | As is well know, there are several quantization methods, as for instance, the canonical quantization and the related Dirac scheme, Segal and Borel quantizations, geometric quantization, various ramifications of deformation quantization, Berezin and Berezin–Toeplitz quantizations, prime quantization and coherent state quantization. For a broad overview see [45]. The advantage of the canonical procedure to quantize a classical theory resides in the preservation of the original formal structure, symmetries and conservation laws. The denomination ‘spacetime topological canonical quantization’ is due to the combination of the conventional canonical quantization procedure applied to a variable, the helix-like complex cosmic scale factor of the branched gravitation, $u={ln}^{-1}\left[\beta \left(t\right)\right]$, raised to the category of quantum operator, which presents an intricate topology. |

10 | The condition $\mathcal{H}\Psi \left(t\right)=0$ excludes the multiplicative term $\frac{1}{2}\frac{N}{u\left(t\right)}$ in Equation (8). |

11 | Despite that we consider only the real part of the effective potential, the variable u is complex, and the solutions still have a broader scope, describing the behavior of the wave function of the Universe both for the contraction region, prior to the primordial singularity, and for the later expansion cosmological region. |

12 | The tunneling boundary condition of Vilenkin [51] in particular has two degrees of freedom: the scale factor and a homogeneous scalar field. A tunneling wave function then describes an ensemble of universes tunneling from ‘‘nothing’’ to a de Sitter space, and then evolving along the lines of an inflationary scenario and eventually collapsing to a singularity [51]. |

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**Figure 1.**On the two left figures, characteristic plots of the Riemann surface associated with the imaginary and the real parts of the function $ln\left[\beta \right(t\left)\right]$, the scaling in time of the branch-cut universe (the reciprocal of ${ln}^{-1}\left[\beta \left(t\right)\right]$). The plot of the imaginary part shows connected glued domains: the various branches of the function are glued along the copies of each upper half plane with their copies on the corresponding lower half plane in a suitable way to make $ln\left[\beta \right(t\left)\right]$ continuous. Each time the variable $\beta $ moves around the origin, $ln\left[\beta \right(t\left)\right]$ moves to a different branch, with its values, on each foliaition leaf, differing from its principal value by a multiple of $2\pi i$. A similar analysis apply to ${ln}^{-1}\left[\beta \left(t\right)\right]$. On the two right figures, characteristic plots of ${ln}^{-1}\left[\beta \left(t\right)\right]$.

**Figure 2.**Plot of the real part of the potential defined in Equation (13). In the top figure the coupling constants values are: ${\eta}_{r}=0.6$, ${\eta}_{m}=0.2855$, ${\eta}_{k}=1$, ${\eta}_{q}=0.7$, ${\eta}_{\Lambda}=1/3$, and ${\eta}_{s}=-0.03$. In the bottom figure the coupling constants values are: ${\eta}_{r}=0.6$, ${\eta}_{m}=0.2855$, ${\eta}_{k}=1$, ${\eta}_{q}=0.7$, ${\eta}_{\Lambda}=1/3$, and ${\eta}_{s}=+0.03$. Values of parameters taken from [43,46,47].

**Figure 3.**Similar plot of the previous figure. Coupling constants values in the top figure: ${\eta}_{r}=0.024$, ${\eta}_{m}=0.2855$, ${\eta}_{k}=1$, ${\eta}_{q}=0.7$, ${\eta}_{\Lambda}=1/3$, and ${\eta}_{s}=-0.468$. Coupling constants values in the bottom figure: ${\eta}_{r}=0.024$, ${\eta}_{m}=0.2855$, ${\eta}_{k}=1$, ${\eta}_{q}=0.7$, ${\eta}_{\Lambda}=1/3$, and ${\eta}_{s}=+0.468$. Values of parameters taken from [43,46,47].

**Figure 4.**Similar plot of the previous figure. Coupling constants values in the top figure: ${\eta}_{r}=0.0$, ${\eta}_{m}=0.2855$, ${\eta}_{k}=1$, ${\eta}_{q}=0.7$, ${\eta}_{\Lambda}=1/3$, and ${\eta}_{s}=-234.0$. Coupling constants values in the bottom figure: ${\eta}_{r}=0.0$, ${\eta}_{m}=0.2855$, ${\eta}_{k}=1$, ${\eta}_{q}=0.7$, ${\eta}_{\Lambda}=1/3$, and ${\eta}_{s}=+234.0$. Values of parameters taken from [43,46,47].

**Figure 5.**Similar plot of the previous figure. Coupling constants values in the top figure: ${\eta}_{r}=-1.22$, ${\eta}_{m}=0.2855$, ${\eta}_{k}=1$, ${\eta}_{q}=0.7$, ${\eta}_{\Lambda}=1/3$, and ${\eta}_{s}=0.15$. Coupling constants values in the bottom figure: ${\eta}_{r}=-0.5$, ${\eta}_{m}=0.2855$, ${\eta}_{k}=1$, ${\eta}_{q}=0.7$, ${\eta}_{\Lambda}=1/3$, and ${\eta}_{s}=0.05$. Values of parameters taken from [43,46,47].

**Figure 6.**Artistic representations of the cosmic contraction and expansion phases of the branch-cut universe evolution scenarios. On the left figure, the branch-cut universe evolves from negative to positive values of the imaginary cosmological time ${t}_{i}$, circumventing continuously the branch-cut and no primordial singularity occurs, only branch points. On the right figure the branch-cut and branch point disappear after the realisation of imaginary time by means of a Wick rotation, which is replaced here by the real and continuous thermal time (temperature), T. In this scenario, a mirrored parallel evolutionary universe, adjacent to ours, is nested in the structure of space and time, with its evolutionary process going backwards in the cosmological thermal time negative sector. Figures based on artistic impressions [48].

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## Share and Cite

**MDPI and ACS Style**

Bodmann, B.A.L.; Vasconcellos, C.A.Z.; Bechstedt, P.O.H.; de Freitas Pacheco, J.A.; Hadjimichef, D.; Razeira, M.; Degrazia, G.A.
A Wheeler–DeWitt Quantum Approach to the Branch-Cut Gravitation with Ordering Parameters. *Universe* **2023**, *9*, 278.
https://doi.org/10.3390/universe9060278

**AMA Style**

Bodmann BAL, Vasconcellos CAZ, Bechstedt POH, de Freitas Pacheco JA, Hadjimichef D, Razeira M, Degrazia GA.
A Wheeler–DeWitt Quantum Approach to the Branch-Cut Gravitation with Ordering Parameters. *Universe*. 2023; 9(6):278.
https://doi.org/10.3390/universe9060278

**Chicago/Turabian Style**

Bodmann, Benno August Ludwig, César Augusto Zen Vasconcellos, Peter Otto Hess Bechstedt, José Antonio de Freitas Pacheco, Dimiter Hadjimichef, Moisés Razeira, and Gervásio Annes Degrazia.
2023. "A Wheeler–DeWitt Quantum Approach to the Branch-Cut Gravitation with Ordering Parameters" *Universe* 9, no. 6: 278.
https://doi.org/10.3390/universe9060278