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Revista de la Academia Colombiana de Ciencias Exactas Físicas y Naturales

Geometric uncertainty in non-paraxial interference

Román Castañeda

Published Dec 12, 2023• 0 citations
Publisher

Abstract

In this article, a novel meaning for the notion of uncertainty is discussed, within the framework of the non-paraxial interference theory based on confinement in geometric states of space. This novel meaning refers to the fact that, for any set of space states whose vertices are distributed in an arbitrary array of size less than λ 10, both the excitation provided by the geometric potential and the positions of the vertices of the states are completely uncertain, such that the complete set is represented by the Lorentzian well of an individual ground state of space, with vertex at any of the points of the array, even if the set is under the maximum prepared non-locality (i.e., under a strong geometric potential). It is shown that the geometric uncertainty is different but compatible with the Heisenberg uncertainty principle. In fact, geometrical uncertainty establishes both the upper limit of momentum uncertainty and the lower limit of position uncertainty in the Heisenberg principle.

DOI
10.18257/raccefyn.1952

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A new type of indetermination is discussed in the framework of the exact (non-paraxial) and generalized description of interference in ordinary space for light, photons and matter particles. This description is based on the novel phenomenological principle of confinement in geometric states of space. It is shown that space states with vertices within a sphere of diameter no greater than a fixed value must be ground states of space, which can be represented by a single ground state with vertex at any point of this sphere. This geometric uncertainty differs from Heisenberg's uncertainty which was established for matter particles and deals with dynamical quantities. However, it is shown that geometric uncertainty establishes limitations for Heisenberg's uncertainty.
Abstract Non-paraxial interference has recently been explained as light energy confinement in spatially structured Lorentzian wells. This theory describes, in the same way, single particle interference in ordinary space too. Such an alternative thought is discussed here in the framework of a quantum formalism, which introduces confinement operators and density matrices of space states for the first time. Therefore, Lorentzian wells are characterized as the base states of space, which are excited by the geometric potential to produce interference. Geometrical uncertainty and space entanglement are discussed as novel features that approach the proposed theory in the quantum context. This formalism provides insight into the role of space in interference.
Non-paraxial interference in ordinary space has been described as the confinement of light, photons, and matter particles in geometric states of space, which are realized as spatially structured Lorentzian wells in the setup volume. Michelson interference is discussed in this framework by regarding the space states dependence from lengths of interferometer arms. It is optically equivalent to considering the space state vertices distributed in a three-dimensional region before the detector plane. This description is potentially relevant in devices where light comes from three-dimensional distributions of points, for instance, the outputs of waveguide arrays in optical or integrated photonics elements.
An exact and exhaustive phenomenology of interference in ordinary space describes it in terms of confinement in geometric states of space. Under high prepared non-locality, such space states exhibit spatial entanglement, a behavior not reported in the standard interference formalism of optics and quantum mechanics. In this paper, the fundamental theoretical features of the spatial entanglement of geometric states of space are discussed and illustrated with numerical simulations of non-paraxial Young interference.
As a foundation of quantum physics, uncertainty relations describe ultimate limit for the measurement uncertainty of incompatible observables. Traditionally, uncertainty relations are formulated by mathematical bounds for a specific state. Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables, ranging over all possible input states. We find that for the pair of position and momentum operators, Heisenberg’s uncertainty principle points exactly to the attainable area of the variances of position and momentum. Moreover, for finite-dimensional systems, we prove that the corresponding area is necessarily semialgebraic; in other words, this set can be represented via finite polynomial equations and inequalities, or any finite union of such sets. In particular, we give the analytical characterization of the areas of variances of (a) a pair of one-qubit observables and (b) a pair of projective observables for arbitrary dimension, and give the first experimental observation of such areas in a photonic system.
This paper presents a geometric reinterpretation of Heisenberg’s uncertainty principle, arguing that the inequality does not represent fundamental randomness in nature but a structural constraint imposed by finite closed surfaces.Momentum uncertainty is shown to arise from continuous rotation of the momentum vector along a closed surface, not from fluctuations in its magnitude.Position uncertainty, in turn, reflects the impossibility of collapsing a closed surface into a mathematical point; below a minimal geometric scale, position is not uncertain but undefined. By identifying the minimal spatial scale with the Bohr radius and expressing momentum variation as Δp≈mc\Delta p \approx mcΔp≈mc, the numerical form of Heisenberg’s inequality is recovered without invoking probability.The uncertainty relation thus emerges as a consequence of geometric closure and phase continuity, rather than probabilistic indeterminacy. This closed-surface formulation restores determinacy and uniqueness to physical reality, rejects point-particle ontology, and reframes quantum mechanics as a theory of constrained geometry rather than inherent randomness.