The bicomplex algebra B = C ⊗R C has recently been identified as a minimal commutative and associative extension of the complex scalar algebra underlying quantum mechanics. This structure contains a proper ideal J ∼= C that admits a complex-like exponential representation and leads to an additive phase relation between the complex and ideal sectors. The present work explores the consequences of this algebraic phase coupling under the hypothesis that it may become physically effective in suitable experimental conditions. In particular, we consider an optical setup involving a nonlinear medium, where mixed terms between complex and ideal components may arise. Within this framework, a specific, testable prediction is derived: a lateral displacement of the source in a double-slit experiment could lead to an amplified fringe shift, whose magnitude depends on the geometric parameters of the setup. The predicted scaling differs significantly from the standard geometric expectation of conventional wave optics. The analysis is explicitly conditional on the assumed physical realization of the bicomplex phase coupling. The proposed experiment therefore provides a direct way to test whether the additional algebraic structure has observable consequences or remains purely formal.
The bicomplex algebra B = C ⊗R C has recently been identified as a minimal commutative and associative extension of the complex scalar algebra underlying quantum mechanics. This structure contains a proper ideal J ∼= C that admits a complex-like exponential representation and leads to an additive phase relation between the complex and ideal sectors. The present work explores the consequences of this algebraic phase coupling under the hypothesis that it may become physically effective in suitable experimental conditions. In particular, we consider an optical setup involving a nonlinear medium, where mixed terms between complex and ideal components may arise. Within this framework, a specific, testable prediction is derived: a lateral displacement of the source in a double-slit experiment could lead to an amplified fringe shift, whose magnitude depends on the geometric parameters of the setup. The predicted scaling differs significantly from the standard geometric expectation of conventional wave optics. The analysis is explicitly conditional on the assumed physical realization of the bicomplex phase coupling. The proposed experiment therefore provides a direct way to test whether the additional algebraic structure has observable consequences or remains purely formal.
The bicomplex extension of quantum mechanics introduces an ideal sector J1 whose phases are algebraically coupled to the complex sector C1 via the algebraic identity e^(iα)*k*e^(jβ) = k*e^j(α+β). In an earlier work, a double-slit test proposal for a possible physical realization of this phase coupling was formulated, which contained a calculation error. The present work refines this proposal and presents the corrected theoretical interpretation. The starting point is a bicomplex state Y = S + R, where S represents the complex component and R the ideal component, which is phenomenologically associated with a monochromatic laser source. In a nonlinear, frequency-doubling crystal, the state is phenomenologically mapped to Y^2 = S^2 + 2SR + R^2. The component S^2 lies in the complex sector, while the mixing term SR according to the above equation lies in the ideal sector and carries phase information transferred by the algebraic phase coupling. For a measurement at a detector or screen, this ideal mixing term must be transferred into a complex measurable channel with an effective coupling strength ηeff . This detector coupling is a phenomenological assumption and does not follow from the algebra alone. The mathematical evaluation of the double-slit interference, taking into account the given field structure (superposition of two slit contributions, cosine profile), shows that within the used Fraunhofer effective model the principal maxima follow the classical geometric shift to first approximation. An effectively coupled SR term at the detector primarily generates contrast and brightness modulations of the interference pattern. The double slit is interesting as a spatial structure test because a brightness modulation of the interference fringes depending on the position of the source is at least surprising.
Standard quantum mechanics takes the complex numbers as its scalar algebra. We ask whether this complex structure is algebraically fundamental or may instead form a distinguished embedded sector of a larger commutative and associative scalar algebra. Requiring the ordinary complex sector to remain intact while admitting an additional stable complex-like analytic sector turns the latter into a proper ideal, makes zero divisors unavoidable in finite dimension, and implies a lower bound of four real dimensions. The bicomplex algebra realizes this minimum, and under an additional compatibility assumption the four-dimensional realization is unique up to isomorphism. The resulting algebra exhibits two distinct decompositions: the canonical decomposition into orthogonal idempotent ideals and a decomposition adapted to the embedded complex sector and the additional ideal-valued sector. The ordinary complex scalars therefore lie diagonally across the canonical idempotent decomposition. The ideal sector supports Taylor series, oscillatory exponentials, Fourier-type kernels, and Schrödinger-type expressions, and carries a formal ideal-valued eigenvalue structure. Restricting the Born rule to the embedded complex sector recovers its standard form. Whether this dual scalar architecture has physical significance beyond the algebraic framework remains an open question.
Quantum mechanics is formulated on complex Hilbert spaces, where the complex numbers constitute the underlying scalar algebra. In this work, the structural question is investigated whether this scalar structure is algebraically minimal or whether a consistent extension exists that preserves the analytical infrastructure of the theory. We formulate this problem as a minimality question for finite-dimensional real, commutative and associative algebras that contain the complex numbers as a subalgebra and at the same time possess a nontrivial ideal that is isomorphic to C as a real algebra and carries a complex-like exponential structure. It is shown that such structures necessarily require at least four real dimensions. The bicomplex algebra B=C ⊗R C realizes these requirements in minimal dimension and is unique up to isomorphism under the stated conditions. For the extended state space M=H ⊗C B one obtains a canonical decomposition M≅H⊕H, so that the bicomplex scalar extension structurally corresponds to two parallel complex quantum sectors. The Born rule remains unchanged. The present work provides a structural analysis of possible extensions of the scalar algebra underlying quantum mechanics.
This paper develops a phase-first reconstruction of quantum interference within standard quantum mechanics and wave optics. Its central idea is that interference patterns become conceptually clearer when the complex phase is treated as the organizing structure and probability or intensity density is treated as the visible statistical expression of that structure. The guiding principle is: the phase structures the pattern; the density renders it visible. The paper shows how this principle applies to single-slit diffraction, double-slit interference, Airy patterns, Fourier optics, Fresnel propagation, probability current, the Madelung representation, which-way marking, reduced coherence, and quantum eraser experiments. In each case, the observed density pattern is interpreted as the visible result of coherent phase relations among amplitudes. The reconstruction is strictly conservative. It does not modify quantum mechanics, the Born rule, unitary dynamics, or standard wave optics. It introduces no new postulates and makes no new empirical prediction. Its purpose is conceptual and pedagogical: to clarify the architecture of quantum interference by placing relative phase at the center of the analysis. This work is Part I of the series The Geometry of Phase. It establishes the phase-geometric foundation for later parts of the series, where the same perspective is extended to de Broglie matter waves, proper-time phase, relativistic action, and possible curvature-coupled phase extensions.
This manuscript presents a deterministic boundary-phase model of the double-slit experiment. In this framework, particles follow continuous, definite trajectories while interference arises from structured curvature induced by slit boundary microstructure. The wavefunction is treated as a mathematical representation of a real boundary-generated field rather than a physical matter wave.The model preserves established interference results while introducing a falsifiable experimental prediction: coherent fringe displacement scaling with engineered slit roughness. For a baseline electron configuration (100 eV beam energy, 1 m screen distance, 1 µm slit separation), a measurable 2 µm fringe shift requires a relative phase offset of approximately 0.10 radians, corresponding to the coupling condition β_eff(δ/r) ≈ 0.10.A quantitative δ-sweep prediction is provided along with explicit experimental control requirements necessary to distinguish coherent boundary-phase displacement from conventional scattering or electrostatic effects. If the predicted scaling is observed under controlled conditions, the double-slit paradox can be resolved deterministically without invoking intrinsic randomness or wavefunction collapse.This work is presented as a falsifiable experimental proposal in quantum foundations and electron interferometry.
This paper proposes a double-slit experiment designed to test a hypothetical extension ofthe de Broglie–Bohm theory. The hypothesis investigated is that a component associated withthe wave function, referred to here as a “wave component”, could be guided by external opticaldevices independently of the photon itself. This contribution is modeled as a modification ofthe wave-function input entering Bohm’s guiding equation. To the best of my knowledge, noexperimental setup explicitly investigates a mechanism of this type involving the capture andreinjection of a wave-function-related contribution through an external optical system.