Abstract: A rigorous proof of quantum parallelism cannot be based on the Heisenberg inequality because the standard deviation of self- adjoint operators in complex Hilbert space is insensitive to fine structures of the collective stationary interference distribution generated by a Mach-Zehnder interferometer from a coherent primary beam. Actually Niels Bohr's indeterminacy principle of spatio-temporal quantum electrodynamics cannot be based on any of the known uncertainty principles. It is shown how the holographic transform allows to circumvent the difficulties with the standard deviation by using a group theoretical implementation of the canonical commutation relations of quantum electrodynamics. The geometric quantization approach combined with the emitter- absorber transaction model of quantum dynamics on the whole real line R allows one to describe by a Liouville density the flow and counter-flow of single optical photons in split fan-in/fan-out coherent photonic channels. It makes the heuristic arguments concerning quantum parallelism rigorous by considering the collective stationary interference distributions of coherent wavepacket densities as symplectic spinors over the linear symplectic manifold modelled on the hologram plane. Consequently the symplectic spinorial organizational form governs photonic holograms. It implies the existence of single-photon holograms and includes the standard uncertainty inequality as a special case.!99
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