  A proof is given for the Fourier transform for functions in a quantum
mechanical Hilbert space on a non-compact manifold in general relativity. In
the (configuration space) Newton-Wigner representation we discuss the spectral
decomposition of the canonical operators and give a proof of the
Parseval-Plancherel relation and the Born rule for linear superposition. We
then discuss the representations of pure quantum states and their dual vectors,
and construct the Fock space and the associated quantum field theory for
Bose-Einstein and Fermi-Dirac statistics.
