  In this paper canonical quantum gravity on noncommutative space-time is
considered. The corresponding generalized classical theory is formulated by
using the moyal star product, which enables the representation of the field
quantities depending on noncommuting coordinates by generalized quantities
depending on usual coordinates. But not only the classical theory has to be
generalized in analogy to other field theories. Besides, the necessity arises
to replace the commutator between the gravitational field operator and its
canonical conjugated quantity by a corresponding generalized expression on
noncommutative space-time. Accordingly the transition to the quantum theory has
also to be performed in a generalized way and leads to extended representations
of the quantum theoretical operators. If the generalized representations of the
operators are inserted to the generalized constraints, one obtains the
corresponding generalized quantum constraints including the Hamiltonian
constraint as dynamical constraint. After considering quantum geometrodynamics
under incorporation of a coupling to matter fields, the theory is transferred
to the Ashtekar formalism. The holonomy representation of the gravitational
field as it is used in loop quantum gravity opens the possibility to calculate
the corresponding generalized area operator.
