Horizon quantum geometries and decoherence

TL;DR

This work investigates how quantum aspects of black-hole horizons, encoded via horizon area quantization and a minimal-length metric, modify the decoherence of a particle in spatial superposition. Building on prior results for classical horizons, it contrasts global and local descriptions of horizon-induced decoherence and then introduces a frequency cutoff determined by the horizon’s area gap, effectively restricting soft modes that can pierce the horizon. The key finding is that this horizon-geometry-induced cutoff turns the previously linear-in-time decoherence into a saturating function, with the saturation level ${\mathcal D}_{\rm sat}$ scaling as $\sim (d/D)^2 / \Omega_0$ (or equivalently with the area quantum $A_0$) and becoming negligible for typical Planck-scale area quanta. Consequently, quantum geometric features of the horizon can suppress decoherence to very small values, challenging the universality of horizon-induced decoherence and offering a potential resolution to causality–complementarity tensions in certain regimes. The analysis further lays groundwork for extending these ideas to gravitational fields and to holographic or near-horizon effective descriptions of quantum geometry.

Abstract

There is mounting theoretical evidence that black hole horizons induce decoherence on a quantum system, say a particle, put in a superposition of locations, with the decoherence functional, evaluated after closure of the superposition, increasing linearly with the time the superposition has been kept open. This phenomenon has been shown to owe its existence to soft modes, that is modes with very low frequencies, of the quantum fields -- sourced by the particle -- which pierce through the horizon, or also can be understood as coming from the interaction with the black hole described as a thermodynamic quantum system at Hawking's temperature. Here we investigate the effects of ensuing quantum aspects of the geometry itself of the horizon, in an effective perspective in which the quantum geometry of the horizon is captured by existence of a limit length or by horizon area quantisation. We show that the discreteness of the energy levels associated to the different geometric configurations might have strong impact on the results, in particular reducing the decoherence effects even to a negligible level in case of quanta of area $A_0 = \mathcal{O}(1) \, \, l_p^2$ or larger, with $l_p$ the Planck length.

Figures (6)

• Figure 1: Plot of the dipole moment $\epsilon_1(\tau)$ in equation \ref{['eq:dpmom']}.
• Figure 2: Visualization of the integrand $I_T(\Omega)$ defined in equation \ref{['eq:decfctC']} in units of $a$ for different values of $a \mathcal{T}_1$ and $a \mathcal{T}$ such that $\mathcal{T}\gg \mathcal{T}_1$. It can be seen in particular from the top left plot that the oscillations are determined by the enveloping function $E(\Omega)$ defined in equation \ref{['eq:envelfct']} and are approximately independent of $\mathcal{T}$ after the first zero. The different values of $\mathcal{T}$ manifest in particular in the value of $I_\mathcal{T}(\Omega)$ at $\Omega=0$, which is proportional to $\mathcal{T}^2$. As the enveloping function changes slowly for $\mathcal{T}\gg \frac{1}{a}$ compared to the oscillations, the difference of the integral, that is of the area of $I_\mathcal{T}(\Omega)$ for different values of $\mathcal{T}$ is mostly determined by the behavior before the first zero. With increasing $\mathcal{T}$, the oscillations become faster (i.e. the period in $\frac{\Omega}{a}$ reduces). In the top right plot one can see that $\mathcal{T}_1>0$ acts as a UV-regulator that damps the integrand for $\Omega \mathcal{T}_1 \gg 1$. The plots in the second line show the influence of changing $a\mathcal{T}_1$, where larger values in $\mathcal{T}_1$ lead to a quicker damping of the enveloping function.
• Figure 3: Dependence of the saturation value in equation \ref{['eq:Dsataprox']} on $\frac{D}{d}$ for different cutoff values $\Omega_0$. The lowest value $\Omega_0 = 5\cdot 10^{-6} a$ was chosen such that $\mathds{D}_{\rm sat} \approx 2$ for $\frac{D}{d}=10$.
• Figure 4: Decoherence functional as a function of $\cal T$ in the approximation of equation \ref{['eq:aprdecfctplt']} for different cutoff frequencies using $\frac{D}{d}=10$ (left figure) and $\frac{D}{d}=100$ (right figure). While the case of a classical horizon $\Omega_0=0$ depends linearly on $T$ for large times, for a quantum horizon a saturation is obtained after a certain time. For very small $\frac{\mathcal{T}}{\mathcal{T}_{\rm dec}}$ a quadratic increase can be seen (which can be shown to correspond to the first oscillation period extending well beyond the upper integration limit); there then follows a transition period into the linear increase which gets saturated after a certain time depending inversely on the size of the cutoff.
• Figure 5: Dependence of the saturation value on the chosen cutoff according to equation \ref{['eq:analresdeco']} for different values of $\frac{D}{d}$. For small values of the cutoff, the graph goes as $\Omega_0^{-1}$ while at $\Omega\sim a$ a strong decrease sets in. The vertical lines correspond from left to right to the choices $\Omega_0 = 5\cdot 10^{-6}a$, $\Omega_0 = \Omega_H = 0.175 a$, $\Omega_0 = \Omega_q = \frac{a}{2}$ and $\Omega_0 = \Omega_{BM} = a$.