  Despite its importance in general relativity, a quantum notion of general
covariance has not yet been established in quantum gravity and cosmology,
where, given the a priori absence of coordinates, it is necessary to replace
classical frames with dynamical quantum reference systems. As such, quantum
general covariance bears on the ability to consistently switch between the
descriptions of the same physics relative to arbitrary choices of quantum
reference system. Recently, a systematic approach for such switches has been
developed (arXiv:1809.00556, 1809.05093, 1810.04153). It links the descriptions
relative to different choices of quantum reference system, identified as the
correspondingly reduced quantum theories, via the reference-system-neutral
Dirac quantization, in analogy to coordinate changes on a manifold. In this
work, we apply this method to a simple cosmological model to demonstrate how to
consistently switch between different internal time choices in quantum
cosmology. We substantiate the argument that the conjunction of Dirac and
reduced quantized versions of the theory defines a complete relational quantum
theory that not only admits a quantum general covariance, but, we argue, also
suggests a new perspective on the 'wave function of the universe'. It assumes
the role of a perspective-neutral global state, without immediate physical
interpretation, that, however, encodes all the descriptions of the universe
relative to all possible choices of reference system at once and constitutes
the crucial link between these internal perspectives. While, for simplicity, we
use the Wheeler-DeWitt formulation, the method and arguments might be also
adaptable to loop quantum cosmology.
