Classical and quantum chemistry routinely treat electrons as permanently existing particles whose number and spatial distribution define chemical structure and reactivity. At the same time, quantum field theory, polarization theory, and modern electronic-structure methods demonstrate that many experimentally observed charge phenomena occur without literal particle transport or permanent localization. In this work, I introduce the Bansal-Chem (B-Chem) framework, a projection-based formulation in which the chemically relevant electron is not a primitive particle but an emergent, geometry-and field-constrained projection event. The electron is modeled as a structured B-fiber object with stratified components-real, complex, dual (nilpotent), and hypercomplex-corresponding respectively to charge density, phase coherence, transient/tunneling behavior, and spin-relativistic structure. Standard electron models used across chemistry and condensed-matter physics are shown to arise as operational projections of this single underlying object, selected by experimental context and environmental coupling. Avogadro's number and Faraday relations are reinterpreted as projection-scaling constants rather than as evidence of permanent microscopic particle inventories. Measurement-induced collapse and decoherence are reformulated as contextual projection and dynamical suppression of inaccessible fiber components, without modifying Schrödinger or Standard Model dynamics. The framework is fully compatible with quantum electrodynamics and density functional theory, while providing a unified structural explanation for the coexistence of particles, waves, densities, quasiparticles, and effective electrons in chemical theory. B-Chem thus offers a geometry-and field-driven ontology for electrons that preserves established calculations while clarifying their conceptual foundations.
We use the variational quantum Monte Carlo (VMC) method to study the wire-width
(b) and electron-density
(rs) dependences of the ground-state properties of quasi-one-dimensional paramagnetic electron fluids. The onset of a quasi-Wigner crystal phase is known to depend on electron density and the crossover occurs in the low density regime. We study the effect of wire width on the crossover of the dominant peak in the static structure factor from
k=2kF to
k=4kF. It is found that, for a fixed electron density, in the charge structure factor the crossover from the dominant peak occurring at
2kF to
4kF occurs as the wire width decreases. Our study suggests that the crossover is due to the interplay of both
rs and
b<rs. The finite wire-width correlation effect is reflected in the peak height of the charge and spin structure factors. We fit the dominant peaks of the charge and spin structure factors assuming fit functions based on our finite wire-width theory and clues from bosonization, resulting in a good fit of the VMC data. The pronounced peaks in the charge and spin structure factors at
4kF and
2kF, respectively, indicate the complete decoupling of the charge and spin degrees of freedom. Furthermore, the wire-width dependence of the electron correlation energy and the Tomonaga-Luttinger parameter ${K}_{\ensuremath{\rho}}$ is found to be significant.
Modern quantum chemistry is built on a conceptual picture of atoms and molecules developed over the past century, yet key elements of this picture especially the structure of the nucleus and the nature of the electron were never fully reconciled with the wave‑based formalism of quantum mechanics. The author’s previous work on quantum chemistry (https://doi.org/10.5281/zenodo.13336683) revisited these foundations, showing that many conceptual inconsistencies arise from treating electrons as particles rather than as extended wavefields. That reconstruction begins with the nucleus. Rather than a point charge orbited by discrete electrons, the nucleus defines a field environment in which the electron exists as an extended wave. Its spatial distribution, energy levels, and interactions follow from this field structure, not from the motion of a particle. This field‑centric view allows neutrality, shielding, and core–valence structure to be understood as continuous wave effects rather than as discrete electron “occupancy.” From this basis, the earlier work introduced a distinction between two operational modes of electronic behavior. In stable atomic and molecular settings, electrons appear as extended wave‑like configurations shaped by the surrounding nuclei; these determine bonding, structure, and charge distribution. The familiar particle‑like electron arises only during interaction events measurement, scattering, ionization, or chemical reaction when the extended wave becomes localized. These are not two kinds of electrons but two manifestations of the same quantum object. The original exposition presented this distinction clearly but without a formal mathematical framework. The insights were qualitative and chemically motivated, yet lacked the Hilbert‑space and field‑operator machinery needed to integrate them into mainstream quantum theory. The present paper provides the rigorous foundation that this conceptual distinction requires.