Quantum Information Theory and Space-Time Emergence from Universal Cavitation

Free information which is an abstract mathematical entity not registered and bound information [34]. The latter, which is subject to the laws of thermodynamics [33], has as its fundamental unit for our purposes the bit and requires a minimum energy equal to $E = kT \ln(2)$ (3.1) to be registered, where $T$ represents the temperature and $k$ the Boltzmann constant equal to $8.617330 \times 10^{-5} \text{ eV K}^{-1}$. At the same time it is experimentally demonstrated that through the realization of a Szilard engine it is possible to transform information into energy [31, 32] through gaining for each bit of information the quantity $E = kT \ln(2)$ (3.2). From this point of view energy and information find perfect equivalence and entropy in information theory finds complete identity with thermodynamic entropy [33].

3.2 Space-time, holographic principle and information

A growing current of contemporary cosmological research is investigating the emergent nature of space-time in relation to information. The central focus of these investigations is concentrated on the role that quantum correlation (quantum entanglement) has on space-time. The starting point is generally that of black hole thermodynamics and in particular the Bekenstein-Hawking entropy formula for a black hole [37], $S = \frac{A}{4L_p^2}$ (3.3) where the entropy $S$ is linked to the area of the black hole horizon $A$ and $L_p$ is the Planck length defined as $L_p = \sqrt{\frac{\hbar G}{c^3}}$ (3.4). This formula was subsequently explained in its microscopic origin within string theory [38] and within the AdS/CFT correspondence [39]. In a subsequent context it was realized that the same identical formula, generalized as the Ryu-Takayanagi formula, regulates the quantity of quantum correlation entropy in the vacuum [36]. The fact that the quantum correlation entropy of empty space-time follows an area law has motivated the idea of representing space-time as a network of correlated quantum information [41]. This approach has led to highlighting the possibility of considering space-time as emerging from a series of microscopic units of quantum information whose short-range correlation originates the Bekenstein-Hawking area law and provides the microscopic link responsible for space-time connectivity. With the Bekenstein-Hawking area law it is then possible to derive Einstein's general relativity equations [42] starting from the thermodynamic relation $Q = TdS$ (3.5) which links heat $Q$, entropy $S$ through temperature $T$. These reflections generally constitute the starting point of research lines on gravity as an emergent force [36, 40, 14, 41].

4 Universal Cavitation

In 1991, in "La Vergine dell'Infinito" [1], A. Benassai hypothesized the formation of matter at both microscopic and macroscopic levels as deriving from a form of "Universal Cavitation" arising from the excitation of a space-time fluid by sound waves. In the case of a superfluid space-time, the equations governing the evolution of the system are the equations of quantum turbulence from which we can derive the formation of matter.

4.1 Vinen Equation

To study quantum turbulences composed of vortices it is necessary to introduce the vortex line density $\ell(t)$ and study its evolution. The most famous equation that regulates this evolution is the phenomenological equation found by Vinen [44, 45] already in 1957, i.e. $\frac{d\ell(t)}{dt} = A\ell(t)^{3/2} - B\ell(t)^2$ (4.1) where $A$ and $B$ are two system parameters to be found experimentally. Some recent studies [47] question this equation by proposing alternative solutions indistinguishable from a numerical solutions point of view, but for the purposes of this exposition we will consider this which remains the classical equation in the sector. Its generalization to a curved space-time leads to $\frac{1}{\sqrt{g}}\frac{d}{dt}(\sqrt{g}\ell(t)) = A\ell(t)^{3/2} - B\ell(t)^2$ (4.2) where $g$ represents the absolute value of the determinant of the space metric. In the case of the Robertson-Walker metric the equation becomes $\dot{\ell} = -3H\ell + A\ell^{3/2} - B\ell^2$ (4.3). Indicating the energy density of the vortical ensemble as $\rho_v = \rho_0 \ell$ (4.4) with $\rho_0$ indicating the energy per unit length of the vortex, we obtain that the Vinen equation becomes $\dot{\rho_v} = -3H\rho_v + \alpha \rho_v^{3/2} - \beta \rho_v^2$ (4.5) with $\alpha$, $\beta$ constants depending on the system to which we can add a parameter $\lambda_0$ that links the vortex energy density to its average velocity according to $v^2 = \lambda_0 \rho_v$ (4.6) which is extremely important given that as indicated.