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.
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.
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. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract Real Hilbert space is a Euclidean space of infinite dimensionality. Complex Hilbert space is less amenable to elementary geometrical intuitions, but these do serve to give structure to the present argument. The familiar process of finding, relative to a vector x, the expectation Ā of a self-adjoint linear operator A, and then using the deviation A — Ā to obtain a dispersion ΔA, may be interpreted as a projection leading to a unit vector e orthogonal to x. If we have n self-adjoint operators Ar, we thus form n unit vectors er orthogonal to x. From a basic property of Hilbert space, the vector c1e1 + c2e2 + ... + cnen has a non-negative norm for arbitrary complex coefficients. This demands the positive-indefinite character of a real quadratic form in 2n variables; equivalently, all its eigenvalues must be non-negative. This leads to inequalities on the dispersions ΔAr, the expectations of the commutators i[Ar, As] and other expectations. For n = 2, this process leads to Schrödinger’s improvement of the Heisenberg inequality. The case n = 3 is here worked out explicitly.
We propose a geometric formulation of quantum indeterminacy based on polar duality between convex bodies representing position and momentum data. A pair (X,P) of centrally symmetric convex bodies is said to satisfy the indeterminacy condition when the h-polar of X is included in P. We show that this condition naturally arises from the geometry of quantum blobs and is closely related to Hardy's uncertainty principle and the Donoho--Stark inequalities. Using symplectic capacities and John ellipsoids, we associate a canonical covariance ellipsoid with every quantum polar pair and prove that it satisfies the quantum condition. The Robertson--Schrödinger inequalities follow as a consequence. This provides a non-statistical formulation of quantum indeterminacy from which the usual uncertainty relations emerge. which the usual uncertainty relations emerge.
Quantum holography is a well established theory of mathematical physics based on harmonic analysis on the Heisenberg Lie group G. The geometric quantization is performed by projectivization of the complexified coadjoint orbit picture of the unitary dual Ĝ of G in order to achieve a geometric adjustment of the quantum scenario to special relativity theory. It admits applications to various imaging modalities such as synthetic aperture radar (SAR) in the microwave range, and, most importantly for the field of non-invasive medical diagnosis, to the clinical imaging modality of magnetic resonance tomography (MRI) in the radio frequency range. Quantum holography explains the quantum teleportation phenomemon through Einstein-Podolsky-Rosen (EPR) channels which is a consequence of the non-locality of phase coherent quantum field theory, the concept of absolute simultaneity of special relativity theory which provides the Einstein equivalence of energy and Fitzgerald-Lorentz dilated mass, and the perfect quantum holographic replication process of molecular genetic information processing. It specifically reveals what was before unobservable in quantum optics, namely the interference phenomena of matter wavelets of Bose-Einstein condensates, and what was before unobservable in special relativity, namely the light in flight (LIF) recording processing by ultrafast laser pulse trains. Finally, it provides a Lie group theoretical approach to the Kruskal coordinatized Schwarzschild manifold of relativistic cosmology with large scale applications to general relativity theory such as gravitational instanton symmetries and the theory of black holes. The direct spinorial detection of gravitational wavelets emitted by the binary radio pulsar PSR 1913+16 and known only by anticipatory system computation so far will also be based on the principles of quantum holography applied to very large array (VLA) radio interferometers.
Coherent wavelets form a unified basis of the multichannel perfect reconstruction analysis-synthesis filter bank of high resolution radar imaging and clinical magnetic resonance imaging (MRI). The filter bank construction is performed by the Kepplerian temporospatial phase detection strategy which allows for the stroboscopic and synchronous cross sectional quadrature filtering of phase histories in local frequency encoding multichannels with respect to the rotating coordinate frame of reference. The Kepplerian strategy and the associated filter bank construction take place in symplectic affine planes which are immersed as coadjoint orbits of the Heisenberg two-step nilpotent Lie group G into the foliated three-dimensional real projective space $\mbox{\bf P}\bigl(\mbox{\bf R}\times\mbox{\rm Lie}(G)^{\star}\bigr)$. Due to the factorization of transvections into affine dilations of opposite ratio, the Heisenberg group G under its natural sub-Riemannian metric acts on the line bundle realizing the projective space $\mbox{\bf P}\bigl(\mbox{\bf R}\times \mbox{\rm Lie}(G)^{\star}\bigr)$. Its elliptic non-Euclidean geometry without absolute quadric, associated to the unitary dual G^, governs the design of the coils inside the bore of the MRI scanner system. It determines the distributional reproducing kernel of the tracial read-out process of quantum holograms excited and coexisting in the MRI scanner system. Thus the pathway of this paper leads from Keppler's approach to projective geometry to the Heisenberg approach to the sub-Riemannian geometry of quantum physics, and finally to the enormously appealing topic of ensemble quantum computing.
The notion of polarity between sets, well-known from convex geometry, is a geometric version of the Fourier transform. We exploit this analogy to propose a new simple definition of quantum indeterminacy, using what we call "hbar-polar quantum pairs", which can be viewed as pairs of position-momentum indeterminacy with minimum spread. The existence of such pairs is guaranteed by the usual uncertainty principle, but is at the same time more general. We use recent advances in symplectic topology to show that this quantum indeterminacy can be measured using a particular symplectic capacity related to action and which reduces to area in the case of one degree of freedom. We show in addition that polar quantum pairs are closely related to Hardy's uncertainty principle about the localization of a function and its Fourier transform.