Polymer quantization of a self-gravitating thin shell

We study the quantum mechanics of self-gravitating thin shell collapse by solving the polymerized Wheeler-DeWitt equation. We obtain the energy spectrum and solve the time-dependent equation using numerics. In contradistinction to the continuum theory, we are able to consistently quantize the theory for super-Planckian black holes, and find two choices of boundary conditions which conserve energy and probability, as opposed to one in the continuum theory. Another feature unique to the polymer theory is the existence of negative energy stationary states that disappear from the spectrum as the polymer scale goes to 0. In both theories the probability density is positive semidefinite only for the space of positive energy stationary states. Dynamically, we find that an initial Gaussian probability density develops regions of negative probability as the wave packet approaches $R=0$ and bounces. This implies that the bouncing state is a sum of both positive and negative eigenstates.

Figure 1

Probability and energy densities of the lowest four bound eigenfunctions for $m=0.9$ satisfying $C_0=0$ on the lattice with $\sigma =0$ and $\mu =0.1$. The corresponding energies are (0.8067, 0.8785, 0.8909, 0.8950). Notice in (c) and (d) that the energy density grows near the origin while the probability density goes to 0.

Figure 2

Probability and energy densities of the lowest four bound eigenfunctions for $m=1.1$ satisfying $C_{-1}=C_0$ on the lattice with $\sigma =\frac{1}{2}$ and $\mu =0.1$. The corresponding energies are (0.8995, 1.0468, 1.0764, 1.0868). Notice in (c) and (d) that there are cusps at $x=\sigma +\mu$ in the probability and energy densities due to the choice of boundary condition. The coefficients themselves do not have a cusp.

Figure 3

Polymer energy spectra for rest mass $m=0.9$ with different lattice spacings $\mu$ ranging from 0.05 to 0.5. The spectrum resulting from Dirac quantization is shown in blue filled circles at $\mu =0$ (the y axis).

Figure 4

Polymer energy spectra for rest mass $m=2$ with different lattice spacings $\mu$ ranging from 0.05 to 0.5. No such eigenvalues exist in the continuum theory.

Figure 5

Real and imaginary parts of the coefficients describing typical scattering solutions. Here $m=\frac{1}{2}$, $E=1$, and the lattice is $x=k\mu$. Notice that the real parts of $C_0$ are the same for both solutions, while the imaginary parts have the same magnitude but opposite sign.