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

Fourier Transform, Quantum Mechanics and Quantum Field Theory on the Manifold of General Relativity