Deformation quantization of cosmological models

The Weyl-Wigner-Groenewold-Moyal formalism of deformation quantization is applied to cosmological models in the minisuperspace. The quantization procedure is performed explicitly for quantum cosmology in a flat minisuperspace. The de Sitter cosmological model is worked out in detail and the computation of the Wigner functions for the Hartle-Hawking, Vilenkin and Linde wave functions are done numerically. The Wigner function is analytically calculated for the Kantowski-Sachs model in (non)commutative quantum cosmology and for string cosmology with dilaton exponential potential. Finally, baby universes solutions are described in this context and the Wigner function is obtained.

Figure 1

The Hartle-Hawking Wigner function ($\hslash =\lambda =1$). The figure shows many oscillations due to the interference between wave functions of expanding and contracting universes.

Figure 2

Hartle-Hawking Wigner function density projection. It can be observed that the classical trajectory does not coincide with the highest peaks of the Wigner function.

Figure 3

The Vilenkin Wigner function. It is observed a clear maximum and fewer oscillations compared with the Hartle-Hawking case.

Figure 4

The density projection of the Vilenkin Wigner function. The classical trajectory is at some parts on the maxima of the Wigner function and has only one branch.

Figure 5

Wigner function for the Linde wave function. The figure shows a reduction in the amplitude of the oscillations compared to the Hartle-Hawking.

Figure 6

The density projection of the Linde Wigner function. In this case, the classical trajectory coincides with the highest peak of its corresponding Wigner function.

Figure 7

The Wigner function for the Kantowski-Sachs wave function for $\nu =1$. The number of oscillations are maxima around $P_{\Omega }=0$.

Figure 8

Density projection of the Kantowski-Sachs Wigner function for $\nu =1$. It can be observed that classical trajectory is close to the exterior peaks of the oscillations.

Figure 9

The Kantowski-Sachs Wigner function for $\nu =0.5$. Few oscillations are present for this case with a clear maximum around $\Omega =6$ and $P_{\Omega }=0$.

Figure 10

Kantowski-Sachs Wigner function density projection for $\nu =0.5$. It can be observed that the value of the Wigner function is large in an ample area.

Figure 11

Kantowski-Sachs Wigner function for $\nu =4$. There is a considerable increase in the number of oscillations centered at $P_{\Omega }=0$.

Figure 12

Kantowski-Sachs Wigner function density projection for $\nu =4$. The classical trajectory (near the $P_{\Omega }$ axis at the upper and bottom part of the figure) is far away from the peaks of the Wigner function.

Figure 13

Kantowski-Sachs noncommutative Wigner function weighted by a Gaussian. Several peaks of different amplitudes in the Wigner function can be observed around the Gaussian form. These are interpreted as different universes connected by tunneling.

Figure 14

The density projection of the Kantowski-Sachs noncommutative Wigner function weighted by a Gaussian. Because of the noncommutativity, more peaks and oscillations appears around $P_{\Omega }=0$.

Figure 15

The Kantowski-Sachs Wigner function weighted by a Gaussian function. It can be observed the Gaussian form between $\Omega =1$ and $\Omega =2$ centered at $P_{\Omega }=0$ with some very smooth oscillations around $P_{\Omega }=0$.

Figure 16

Kantowski-Sachs Wigner function density projection weighted by a Gaussian. There is a clear maximum with some other transversal oscillations around $P_{\Omega }=0$.