The uncertainty inequality in quantum holography, Bohr’s indeterminacy principle, and synchronized neural networks

Abstract

A rigorous proof of quantum parallelism cannot be based on the Heisenberg inequality because the standard deviation is insensitive to fine structures of the interference pattern generated by a Mach-Zehnder interferometer. Actually Niels Bohr’s indeterminacy principle 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 relation of quantum electrodynamics. The geometric quantization approach makes the heuristic arguments concerning quantum parallelism rigorous by considering wave packets as symplectic spinors on the hologram plane. It includes the standard uncertainty inequality as a special case.

As an application to the area of quantum computers, we study synchronized neural networks and their photonic implementations as suggested by recent experiments in brain research. As another result of the geometric quantization approach to holography, the uncertainty minimizing Gabor wavelets arise which provide useful wavelet expansions for image analysis, segmentation, and compression.

Some ten years after writing his fundamental papers on optics, Hamilton made a startling observation: that the same formalism applies to mechanics of point particles. Replace the optical axis by the time. Then the transformation from initial position and momenta to final position and momenta is always symplectic. This discovery led to remarkable progress in the nineteenth century. In the 1920s — almost a century later — Hamilton’s analogy between optics and mechanics served as one of the major clues in the discovery of quantum mechanics.

Shlomo Sternberg (1988)

Great progress was possible thanks to the wonderful tool in atomic physics which is the laser.

Alain Aspect (1986)

The Schrödinger equation preserves the norm of the wave function and thus the number of particles. Now it is well known that in the presence of sources, photons can be absorbed or emitted. Thus one cannot introduce a Schrödinger equation for a single photon in the presence of sources. In fact, the electromagnetic field itself must be quantized, and photons then occur as elementary excitations of the quantized field.

Claude Cohen-Tannoudji (1989)

Each photon bears information about the entire system.

H. John Caulfield (1991)