  We analyze the loop quantization of the family of vacuum Bianchi I
spacetimes, a gravitational system whose classical solutions describe
homogeneous anisotropic cosmologies. We rigorously construct the operator that
represents the Hamiltonian constraint, showing that the states of zero volume
completely decouple from the rest of quantum states. This fact ensures that the
classical cosmological singularity is resolved in the quantum theory. In
addition, this allows us to adopt an equivalent quantum description in terms of
a well defined densitized Hamiltonian constraint. This latter constraint can be
regarded in a certain sense as a difference evolution equation in an internal
time provided by one of the triad components, which is polymerically quantized.
Generically, this evolution equation is a relation between the projection of
the quantum states in three different sections of constant internal time.
Nevertheless, around the initial singularity the equation involves only the two
closest sections with the same orientation of the triad. This has a double
effect: on the one hand, physical states are determined just by the data on one
section, on the other hand, the evolution defined in this way never crosses the
singularity, without the need of any special boundary condition. Finally, we
provide these physical states with a Hilbert structure, completing the
quantization.
