# The de Broglie–Bohm Quantum Theory and Its Application to Quantum Cosmology

## Abstract

**:**

## 1. Introduction

## 2. The de Broglie–Bohm Quantum Theory

## 3. The de Broglie–Bohm Theory Applied to Quantum Cosmology: Background and Perturbations

#### 3.1. Perfect Fluids

#### 3.2. The Canonical Scalar Field

## 4. Quantum Bouncing Backgrounds and Their Cosmological Perturbations

#### 4.1. The Perfect Fluid

#### 4.2. Canonical Scalar Field

- (A)
- A long classical dust contraction, which traverses a dark energy phase and realizes a stiff matter quantum bounce, directly expanding afterwards to an asymptotically dust matter expanding phase, without passing through a dark energy phase.
- (B)
- A long classical dust contraction, without traversing a dark energy phase, which realizes a stiff matter quantum bounce and expands to a dark energy phase, ending in an asymptotically dust expanding phase.

## 5. The Quantum-to-Classical Transition of Quantum Cosmological Perturbations

## 6. Discussion and Conclusions

- (1)
- The measurement problem is naturally solved, without the necessity of invoking the presence of an external agent outside the quantum physical system, which does not make sense when the physical system is the whole Universe.
- (2)
- The fact that the usual quantum equations for the wave function of the Universe that emerge from many approaches to quantum gravity do not present a Schrödinger form makes it difficult to physically interpret the wave function of the Universe, especially in probabilistic terms [65,110]. In the dBB theory, however, the wave function of the Universe $\Psi $ yields the guidance equations, which provide the time evolution of all the quantum particles and fields present in the Universe. Hence, one can assign a nomological interpretation to $\Psi $, as giving the laws of motion for the quantum degrees of freedom, in the same way as Hamitonians and Lagrangians do. There is no need to talk about probabilities at this level; hence, the quantum equations for $\Psi $ may have any form. When dealing with subsystems in the Universe, one can construct the conditional wave function to describe this subset of fields and particles, which may satisfy a Schrödinger-like equation under reasonable assumptions, and a natural probabilistic interpretation in terms of the Born rule emerges for this conditional wave function.
- (3)
- There is the so-called problem of time in quantum cosmology, as it seems that the quantum theory is timeless [110]. This issue is intimately connected with the second one. In the dBB quantum approach, the guidance equations yield a parametric evolution for the fields. Note, however, that the space-time structure that emerges from orderly stacking the fields along this parameter may be very contrived, but they can be calculated; see [111] for details. Additionally, when going to subsystems described by the conditional wave function, where a Schrödinger-like equation emerges, a time evolution for the subsystem quantum state emerges.
- (4)
- As, in the dBB theory, the Bohmian trajectories emerge, the characterization of quantum singularities becomes clear. For instance, in the quantum cosmological models discussed here, the background model is said to be non-singular if the Bohmian trajectory of the scale factor satisfies $a\left(t\right)\ne 0$ for all t.
- (5)
- The classical limit is easily obtained, either by the inspection of the quantum potential or by direct comparison between the classical and Bohmian trajectories.

- (i)
- Feature (1) yields a clear understanding of a long standing problem, which is the quantum-to-classical transition of quantum cosmological perturbations in inflation and bouncing models. This was discussed in Section 5.
- (ii)
- Feature (4) allows a simple identification of non-singular quantum models, as shown in Section 4. All of them present a regular bounce.
- (iii)
- All the features above yield simple equations for quantum perturbations in quantum backgrounds, which is not an easy task under other approaches [112]. These simple equations could be solved, providing sensible bouncing models with inhomogeneous perturbations, in which the presence of a dust fluid (dark matter?) yields an almost scale-invariant spectrum of perturbations, as observed, with the correct amplitudes. Dark energy can also be included, as in the scalar field model of Section 4. In this model, we have seen that a quantum cosmological effect becomes very relevant during the quantum bounce, leading to observable consequences which solve a conflict with observational results that cannot be solved in classical terms, rendering it a viable model to be developed.
- (iv)
- Feature (5) makes direct the evaluation of the parameter limits under which the standard classical Friedmann solution arises from a quantum Bohmian solution.

- (a)
- The angular power spectrum of the temperature–temperature correlation function, and the E and B polarization modes corresponding to the bouncing models described here, and other possibilities, must be calculated in great detail, and compared with the most recent CMB results [113], in order to differentiate these models among themselves and with inflation. Additionally, one could try to find typical fingerprints of a quantum cosmological effect, which cannot be found by other methods. One promising example is the scalar field model presented in Section 4.
- (b)
- In the analysis of more elaborate models, some new observables must be calculated. For instance, in the two-fluid model described in Section 5, one needs to calculate the entropy perturbations. In preliminary calculations [86], as the entropy effective sound velocity is given by$${c}_{e}^{2}=\frac{w({\rho}_{r}+{p}_{r})+({\rho}_{m}+{p}_{m})/3}{{\rho}_{T}+{p}_{T}},$$
- (c)
- The role of dark energy in bouncing models is very important to understand. In Section 5, I presented a possible solution to the issues raised in Section 4, but there are many other possibilities. In the case that dark energy is a cosmological constant, the problem becomes more contrived, with the possibility of observational consequences. Note that bouncing models offer a unique possibility to learn about dark energy through the primordial power spectrum of cosmological perturbations, which is not the case for inflation.
- (d)
- (e)
- The dBB quantum theory, in principle, allows probability distributions that do not obey the Born rule, that are away from quantum equilibrium. It is difficult to find ordinary physical systems in this situation. In cosmology, this may not be the case. For instance, long wavelength perturbations originated from a vacuum quantum state do not relax quickly to quantum equilibrium, yielding a possible departure from quantum mechanical predictions [116]. Additionally, one could relax the conditions imposed in the conditional wave function explained in Section 5, which would lead to corrections to the effective Schrödinger equation for the perturbations in the quantum background regime and a departure from quantum equilibrium, with possible observational consequences.

“To try to stop all attempts to pass beyond the present viewpoint of quantum physics could be very dangerous for the progress of science and would furthermore be contrary to the lessons we may learn from the history of science. This teaches us, in effect, that the actual state of our knowledge is always provisional and that there must be, beyond what is actually known, immense new regions to discover”.

## Funding

## Acknowledgments

## Conflicts of Interest

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1 | In this section, I will name the Mukhanov–Sasaki mode ${v}_{k}$ of Section 4${f}_{k}$, reserving the name ${v}_{k}$ for the Bohmian mode that we are now discussing. |

2 | In inflation, this result is direct, while for bouncing models, some care must be taken with the interchange between growing and decaying modes after the bounce, but in the end, the result is the same; see [103]. |

**Figure 1.**Comparison between evolution of Hubble radius and cosmological scales in inflation. The green straight lines are the perturbation scales; the black solid line is the Hubble radius. The horizontal axis depicts $log\left(a\right)$.

**Figure 2.**Qualitative comparison between evolution of the curvature scale, in blue, and cosmological scales, in red, in bouncing models. During classical evolution, the curvature scale coincides with the Hubble radius. The scale factor at the bounce is normalized to one; hence, the origin corresponds to the bounce, where the scale factor attains its minimal value. The negative and positive horizontal axis correspond to the contracting and expanding phases, respectively. In the plot, the transitions from dust to radiation domination, and the bounce itself, are qualitatively depicted by sharp transitions. In reality they are smooth, but it does not alter the physical conclusions presented in the text.

**Figure 3.**The figure shows a comparison of our results, labeled by ${\overline{\eta}}_{b}$ (the smaller this parameter, the bigger the energy scale of the bounce, and the value ${10}^{-30}$ is only two orders of magnitude away from the Planck scale) with experimental sensitivities of LIGO’s 5th run, Advanced LIGO, and the forthcoming LISA and Einstein Telescope, and a prediction of the upper limits on the spectrum of primordial gravitational waves generated in inflationary models.

**Figure 4.**Qualitative comparison between evolution of the curvature scale, in blue, and cosmological scales, in red, in bouncing models with a cosmological constant. During classical evolution, the curvature scale coincides with the Hubble radius. The scale factor at the bounce is normalized to one; hence, the origin corresponds to the bounce, where the scale factor attains its minimal value. The negative and positive horizontal axis correspond to the contracting and expanding phases, respectively. In the plot, the transitions from dust to radiation domination and dust to cosmological constant domination, and the bounce itself, are qualitatively depicted by sharp transitions. In reality, they are smooth, but it does not alter the physical conclusions presented in the text.

**Figure 5.**Numerical results for ${n}_{S}\left(k\right)$ in the presence of a cosmological constant. The solid line shows the result obtained using ${\Omega}_{\Lambda}=0.7$; the dashed line, that for ${\Omega}_{\Lambda}={10}^{-3}$; and the dotted line, that for ${\Omega}_{\Lambda}={10}^{-6}$. The oscillations become smaller for smaller ${\Omega}_{\Lambda}$, indicating that they arise because of the presence of the cosmological constant.

**Figure 6.**Phase space for the planar system defined by (85) and (86). The critical points are indicated by ${M}_{\pm}$ for a dust-type effective equation of state, and ${S}_{\pm}$ for a stiff-matter equation of state. Note that the region $y<0$ shows the contracting solutions, while the $y>0$ region presents the expanding solutions. Lower and upper quadrants are not physically connected, because there is a singularity in between.

**Figure 7.**Phase space for the quantum bounce [95]. The bounces in the figure connect regions around ${S}_{+}$ in the contracting phase with regions around ${S}_{-}$ in the expanding phase.

**Figure 8.**Possible Bohmian trajectories associated with the canonical scalar field with exponential potential. The trajectories yielding relevant amplification of scalar perturbations are set 1 and set 2. The bounces are not deep, but they are steep, with very small x.

**Figure 9.**Evolution of scalar and tensor perturbations in the background of case B. Scalar and tensor perturbations grow almost at the same rate during classical contraction, but at the quantum bounce, the scalar perturbations are enormously enhanced over the tensor perturbations due to the quantum effects (shown in the detail of the figure). After the bounce, the perturbations get frozen. The final amplitudes of both perturbations are compatible with observations. The indices a and b refer to the real and imaginary parts of the perturbation amplitudes.

x | y | z |
---|---|---|

$-1$ | 0 | 1 |

1 | 0 | 1 |

$\frac{\lambda}{\sqrt{6}}$ | $-\sqrt{1-\frac{{\lambda}^{2}}{6}}$ | $\frac{1}{3}\left({\lambda}^{2}-3\right)$ |

$\frac{\lambda}{\sqrt{6}}$ | $\sqrt{1-\frac{{\lambda}^{2}}{6}}$ | $\frac{1}{3}\left({\lambda}^{2}-3\right)$ |

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Pinto-Neto, N.
The de Broglie–Bohm Quantum Theory and Its Application to Quantum Cosmology. *Universe* **2021**, *7*, 134.
https://doi.org/10.3390/universe7050134

**AMA Style**

Pinto-Neto N.
The de Broglie–Bohm Quantum Theory and Its Application to Quantum Cosmology. *Universe*. 2021; 7(5):134.
https://doi.org/10.3390/universe7050134

**Chicago/Turabian Style**

Pinto-Neto, Nelson.
2021. "The de Broglie–Bohm Quantum Theory and Its Application to Quantum Cosmology" *Universe* 7, no. 5: 134.
https://doi.org/10.3390/universe7050134