The logical structure of quantum gravity is addressed in the framework of the so-called manifestly covariant approach. This permits to display its close analogy with the logics of quantum mechanics. More precisely, in QG the conventional 2-way principle of non-contradiction holding in Classical Mechanics is shown to be replaced by a 3-way principle. The third state of logical truth corresponds to quantum indeterminacy/undecidability, i.e., the occurrence of quantum observables with infinite standard deviation. The same principle coincides, incidentally, with the earlier (...) one shown to hold in Part I, in analogous circumstances, for QM. However, this conclusion is found to apply only provided a well-defined manifestly-covariant theory of the gravitational field is adopted both at the classical and quantum levels. Such a choice is crucial. In fact it makes possible the canonical quantization of the underlying unconstrained Hamiltonian structure of general relativity, according to an approach recently developed by Cremaschini and Tessarotto. Remarkably, in the semiclassical limit of the theory, Classical Logic is proved to be correctly restored, together with the validity of the conventional 2-way principle. (shrink)

The Madelung equations map the non-relativistic time-dependent Schrödinger equation into hydrodynamic equations of a virtual fluid. While the von Neumann entropy remains constant, we demonstrate that an increase of the Shannon entropy, associated with this Madelung fluid, is proportional to the expectation value of its velocity divergence. Hence, the Shannon entropy may grow due to an expansion of the Madelung fluid. These effects result from the interference between solutions of the Schrödinger equation. Growth of the Shannon entropy due to expansion (...) is common in diffusive processes. However, in the latter the process is irreversible while the processes in the Madelung fluid are always reversible. The relations between interference, compressibility and variation of the Shannon entropy are then examined in several simple examples. Furthermore, we demonstrate that for classical diffusive processes, the “force” accelerating diffusion has the form of the positive gradient of the quantum Bohm potential. Expressing then the diffusion coefficient in terms of the Planck constant reveals the lower bound given by the Heisenberg uncertainty principle in terms of the product between the gas mean free path and the Brownian momentum. (shrink)

In an attempt to explore further the Madelung fluid-like representation of quantum mechanics, we derive the small perturbation equations of the fluid with respect to its basic states. The latter are obtained from the Madelung transform of the Schrödinger equation eigenstates. The fundamental eigenstates of de Broglie monochromatic matter waves are then shown to be mapped into the simple basic states of a fluid with constant density and velocity, where the latter is the de Broglie group velocity. The normal modes (...) with respect to these basic states are derived and found to also satisfy the de Broglie dispersion relation. Despite being dispersive waves, their propagation mechanism is equivalent to that of sound waves in a classical ideal adiabatic gas. We discuss the physical interpretation of these results. (shrink)

The Cramér–Rao bound, satisfied by classical Fisher information, a key quantity in information theory, has been shown in different contexts to give rise to the Heisenberg uncertainty principle of quantum mechanics. In this paper, we show that the identification of the mean quantum potential, an important notion in Bohmian mechanics, with the Fisher information, leads, through the Cramér–Rao bound, to an uncertainty principle which is stronger, in general, than both Heisenberg and Robertson–Schrödinger uncertainty relations, allowing to experimentally test the validity (...) of such an identification. (shrink)