HOLOMORPHIC QUANTUM-MECHANICS WITH A QUADRATIC HAMILTONIAN CONSTRAINT

A finite-dimensional system with a quadratic Hamiltonian constraint is Dirac quantized in holomorphic, antiholomorphic, and mixed representa tions. 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 o btained in the antiholomorphic representation violate classical expect ations for the spectra of certain operators, whereas no such violation occurs in the holomorphic representation. A subset of these Hilbert s paces is alw recovered in a configuration space representation. A prop agation amplitude obtained from an (anti)holomorphic path integral is shown to give the matrix elements of the identity operators in the rel evant Hilbert spaces with respect to an overcomplete basis of represen tation-dependent generalized coherent states. The relation to quantiza tion of spatially homogeneous cosmologies is discussed in view of the no-boundary proposal of Hartle and Hawking and the new variables of As htekar.