  A finite dimensional system with a quadratic Hamiltonian constraint is Dirac
quantized in holomorphic, antiholomorphic and mixed representations. A unique
inner product is found by imposing Hermitian conjugacy relations on an operator
algebra. The different representations yield drastically different Hilbert
spaces. In particular, all the spaces obtained in the antiholomorphic
representation violate classical expectations for the spectra of certain
operators, whereas no such violation occurs in the holomorphic representation.
A subset of these Hilbert spaces is also recovered in a configuration space
representation. A propagation amplitude obtained from an (anti)holomorphic path
integral is shown to give the matrix elements of the identity operators in the
relevant Hilbert spaces with respect to an overcomplete basis of
representation-dependent generalized coherent states. Relation to quantization
of spatially homogeneous cosmologies is discussed in view of the no-boundary
proposal of Hartle and Hawking and the new variables of Ashtekar.
