Do we live in an eigenstate of the "fundamental constants" operators?
  We propose that the constants of Nature we observe (which appear as
parameters in the classical action) are quantum observables in a kinematical
Hilbert space. When all of these observables commute, our proposal differs
little from the treatment given to classical parameters in quantum information
theory, at least if we were to inhabit a constants' eigenstate.
Non-commutativity introduces novelties, due to its associated uncertainty and
complementarity principles, and it may even preclude hamiltonian evolution. The
system typically evolves as a quantum superposition of hamiltonian evolutions
resulting from a diagonalization process, and these are usually quite distinct
from the original one (defined in terms of the non-commuting constants). We
present several examples targeting $G$, $c$ and $\Lambda $, and the dynamics of
homogeneous and isotropic Universes. If we base our construction on the
Heisenberg algebra and the quantum harmonic oscillator, the alternative
dynamics tends to silence matter (effectively setting $G$ to zero), and make
curvature and the cosmological constant act as if their signs are reversed.
Thus, the early Universe expands as a quantum superposition of different Milne
or de Sitter expansions for all material equations of state, even though matter
nominally dominates the density, $\rho $, because of the negligible influence
of $G\rho $ on the dynamics. A superposition of Einstein static universes can
also be obtained. We also investigate the results of basing our construction on
the algebra of $SU(2)$, into which we insert information about the sign of a
constant of Nature, or whether its action is switched on or off. In this case
we find examples displaying quantum superpositions of bounces at the initial
state for the Universe.
