  We consider the background-independent quantization of a general
scale-invariant theory of gravity with matter, which supports a conserved Weyl
current recently suggested as a natural flow of time. For scalar-metric
systems, a conformal class of Ashtekar-Barbero connection variables is
constructed, which can be quantized using spin networks. Crucially, the quantum
states become separable into the eigen states of the generator of the scale
transformation and spin-network states in the Einstein frame. The eigen values
consist of additional quantum numbers including a new type of fundamental
frequency $\omega$ and energy $E = \hbar \omega$ with respect to a new local
time $\tau$ carried by every spin-network vertex. The discretely distributed
$\tau$ values as the "quanta of time" correspond to a functional time related
to the integrated Weyl current in the classical theory. The Immirzi ambiguity
of loop quantum geometry is removed by scale symmetry. To probe the quantum
behaviour of the early Universe, the new formalism is applied to a
scale-invariant homogenous and isotropic cosmological model coupled to a
doublet of scalars with illustrative numerical simulations. The Einstein-frame
volume is quantized in recently improved and regularized polymer
representations with an arbitrary Immirzi parameter. The resulting unitary
evolution of the quantum state of an expanding universe has a positive energy
spectrum. A rescaling of the Immirzi parameter is equivalent to a translation
in time without changing dynamics. The big bang can be identified in the past
time limit when the expectation values of the Jordan-frame volume tend to zero.
Remarkably, the quantized big bang is not replaced by a big bounce---a
prevalent scenario in present loop quantum cosmology.
