  A century after the advent of Quantum Mechanics and General Relativity, both
theories enjoy incredible empirical success, constituting the cornerstones of
modern physics. Yet, paradoxically, they suffer from deep-rooted, so-far
intractable, conflicts. Motivations for violations of the notion of
relativistic locality include the Bell's inequalities for hidden variable
theories, the cosmological horizon problem, and Lorentz-violating approaches to
quantum geometrodynamics, such as Horava-Lifshitz gravity. Here, we explore a
recent proposal for a "real ensemble" non-local description of quantum
mechanics, in which "particles" can copy each others' observable values AND
phases, independent of their spatial separation. We first specify the exact
theory, ensuring that it is consistent and has (ordinary) quantum mechanics as
a fixed point, where all particles with the same values for a given observable
have the same phases. We then study the stability of this fixed point
numerically, and analytically, for simple models. We provide evidence that most
systems (in our study) are locally stable to small deviations from quantum
mechanics, and furthermore, the phase variance per value of the observable, as
well as systematic deviations from quantum mechanics, decay as $\sim$
(Energy$\times$Time)$^{-2n}$, where $n \geq 1$. Interestingly, this convergence
is controlled by the absolute value of energy (and not energy difference),
suggesting a possible connection to gravitational physics. Finally, we discuss
different issues related to this theory, as well as potential novel
applications for the spectrum of primordial cosmological perturbations and the
cosmological constant problem.
