Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics

Neutrosophic Sets and Systems, Vol. 98, 2026 University of New Mexico Florentin Smarandache, Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics Florentin Smarandache1,* 1 Department of Mathematics & Sciences, University of New Mexico, Gallup, NM 87301, USA * Correspondence: [email protected] Abstract: The article’s goal is to compare the "infinitesimally punctured wave" with the major interpretations of the fundamental philosophical differences in how physicists view reality at the quantum level. Keywords: Infinitesimally Punctured Wave, Duality, Copenhagen Interpretation (Many Worlds Interpretation), De Broglie-Bohm Theory (Pilot-Wave Theory - Hidden Variables), Smarandache Algebraic Structures, Transactional Interpretation, General Relativity, Neutrosophic Quantum Theory, Quantum Gravity, Kaluza-Klein Theory, Born-Infeld Model, Non-Linear Schrödinger Equation, Non-Classical Soliton, Non-Linear Electrodynamics. 1. Interpretation of the wave-particle duality as an infinitesimally punctured wave Here there are comparisons of these interpretations of wave-particle duality and the wavefunction. The interpretation of wave-particle duality as an infinitesimally punctured wave is a specific conceptual model, largely associated with the work of physicist Florentin Smarandache who defined it in 2019 {[3], [4]} and explored alternative foundational frameworks for quantum mechanics. This model attempts to reconcile the seemingly contradictory wave and particle descriptions by viewing the quantum entity not as two mutually exclusive things, but as a single entity with an inherent structure that incorporates both aspects. 2. The Infinitesimally Punctured Wave Model In this interpretation, the quantum entity (like an electron or photon) is fundamentally a wave, but its identity as a particle arises from a structural singularity or defect within the wave itself. 2.1. The Wave Component The primary structure is the wave (described by the wavefunction ψ): • It is non-localized and extends through space. • It is responsible for phenomena like diffraction and interference. • This wave carries the energy and momentum of the particle over a region of space. 2.2. The Puncture (The Particle Component) The particle aspect is represented by an infinitesimally small puncture, singularity, or structural defect within the wave: Neutrosophic Sets and Systems, Vol. 98, 2026 86 Florentin Smarandache, Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics • This puncture is a point of extreme localization or a singularity within the field. • It is where the wave's energy and momentum are compressed or concentrated into a single, measurable location. • This singularity is responsible for the localized impacts observed during measurement, such as a flash on a detector screen or a discrete hit on a photographic plate. 3. Mathematical and Conceptual Implications 3.1. Resolution of Collapse This model offers a distinct perspective on the collapse of the wavefunction during measurement: • Standard View (Copenhagen): The wave is a description of probability, which instantaneously collapses to a localized particle upon measurement. • Punctured Wave View: The wave never truly "collapses." The measurement process simply isolates and registers the presence of the fixed singularity (the particle) that was already embedded within the extended wave structure. The wave guides the singularity, but the singularity itself is what interacts locally. 3.2. Connection to Smarandache Structures This interpretation aligns philosophically with Smarandache Algebraic Structures because it involves an entity where: • The weak (extensive) structure is the smooth, continuous wave. • The strong (localized) structure is the discrete, singular particle point (the "puncture") existing as a proper subset (or singularity) within the whole wave field. 3.3. Comparison to de Broglie-Bohm Theory This concept shares philosophical ground with the de Broglie–Bohm pilot wave theory (or ontological interpretation), where: • A real, physical wave exists (the pilot wave). • A real particle is carried along by the wave, guided by it but maintaining its discrete identity. The "punctured wave" model is similar in spirit but more radical, suggesting that the particle is the singularity of the wave itself, rather than a separate entity merely guided by the wave. 4. Comparison of Quantum Interpretations Feature Infinitesimally Punctured Wave Copenhagen Interpretation (CI) Many-Worlds Interpretation (MWI) Wavefunctio n (ψ) Represents a real, physical field that extends through space. Represents knowledge or probabi lity amplitude; not physically real. Represents a real, physical field (like the Punctured Wave), evolving deterministically. The Particle A physical singularity or defect (puncture) embedded within the wave structure. Exists only as a definite entity upon measurement; prior to measurement, only probabilities exist. Is a localized manifestation arising from the ψ field; all possible states are physically real in parallel universes. WaveParticle Duality Structural Unity: The particle is the singularity of the wave; the two are Complementarity: The particle and wave aspects are mutually Wave Only: Everything is a wave; the appearance Neutrosophic Sets and Systems, Vol. 98, 2026 87 Florentin Smarandache, Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics inseparable components of a single entity. exclusive and revealed depending on the experimental setup. of a particle (localization) is due to decoherence. Measureme nt Problem No true collapse. Measurement simply reveals the fixed location of the singularity already present within the wave structure. Wavefunction Collapse: The ψ instantaneously and non-locally "collapses" from a superposition of possibilities to a single outcome. No Collapse. Measureme nt causes the observer and system to become entangled, splitting the universe into noncommunicating branches for every possible outcome. Causality Ontological/Deterministic: T he underlying wave and singularity follow definite, albeit complex, laws. NonDeterministic/Probabilistic: The act of collapse is fundamentally random (non-causal). Deterministic: The evolution of the total wavefunction is smooth and deterministic. 5. Key Distinctions 5.1. Focus on Reality (Ontology) • The Punctured Wave and MWI are ontological interpretations—they assert that the wavefunction is physically real. • The CI is an epistemological interpretation—it asserts that the wavefunction is a tool representing our knowledge or probability, not a physical entity. 5.2. Handling the "Collapse" • The Punctured Wave solves the collapse problem by proposing that the particle state never truly collapses; it was always localized as the singularity, and the wave merely defined its potential location. • The MWI solves the collapse problem by eliminating it entirely; all possibilities occur in separate, branching universes. • The CI accepts collapse as a fundamental, unexplained feature of the measurement process. 5.3. Superposition • In the Punctured Wave model, a superposition means the wave is simultaneously spread out in configuration space, but the singularity (particle) is still definite, but its location is probabilistically determined by the wave's amplitude. • In the CI, the particle literally exists in all possible states simultaneously until observed. • In the MWI, all states in the superposition actually exist simultaneously in parallel universes. Comparisons between infinitesimally punctured wave and other theories and phenomena that may be related to it The "infinitesimally punctured wave" (IPW) interpretation can be compared and contrasted with several other theories and phenomena in physics, particularly those dealing with the foundations of quantum mechanics, wave-particle duality, and the structure of space-time. Neutrosophic Sets and Systems, Vol. 98, 2026 88 Florentin Smarandache, Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics 6. Interpretations of Quantum Mechanics The IPW theory, which posits a real wave containing a singular "puncture" (the particle), directly relates to the central debates in quantum foundations: 6.1. De Broglie-Bohm Theory (Pilot-Wave Theory) Comparison Point Infinitesimally Punctured Wave (IPW) De Broglie-Bohm Theory (DBB) Nature of Wave Real, physical field (ψ). Real, physical field (Pilot Wave, guiding wave). Nature of Particle The particle is the singularity/puncture of the wave itself. The particle is a separate, definite entity always having a precise position. Relationship Unity/Singularity: Particle is structurally part of the wave. Guidance: Particle is guided by the separate wave field. Causality Deterministic/Ontological: Both follow definite laws. Deterministic/Ontological: Known for being a deterministic theory. Relation: DBB is the closest cousin. Both are ontological interpretations (claiming the wave is real and deterministic) and use a physical wave to explain quantum phenomena. The key difference is the IPW's assertion that the particle is made of the wave (the singularity), whereas DBB posits a particle that is separate from the wave. 6.2. Transactional Interpretation (TI) The TI interprets quantum interactions as a transaction (a handshake) between retarded (forward-in-time) and advanced (backward-in-time) waves. • Contrast: TI deals with the process of interaction and measurement using timesymmetric waves. IPW focuses on the structure of the individual quantum entity (the wave-singularity pair). • Similarity: Both are highly conceptual, non-mainstream theories that seek to provide a concrete physical picture underlying the mathematical formalism of quantum mechanics. 7. Theories of Space-Time and Singularity The "puncture" concept relates to how singularities are treated in general relativity and field theories. 7.1. General Relativity (GR) and Point Singularities • Relation: The IPW models the particle as a point singularity in a field (the wave). In GR, singularities (like those at the center of black holes) are points where the traditional laws of physics break down, often associated with infinite density or curvature. • Interpretation: The IPW applies this singular point concept to quantum particles, suggesting that the particle's mass and charge are localized features stemming from this puncture, much like a mass point curves space-time in GR. 8. Classical Field Theories (Electromagnetism) • Contrast: In classical electromagnetism, a particle (like an electron) is modeled as a localized charge that generates an external field. • IPW Shift: The IPW reverses this. It posits that the particle is a feature of the field itself (a puncture), rather than a separate source generating a field. It attempts to achieve a unified description where the particle and field are fundamentally one. Neutrosophic Sets and Systems, Vol. 98, 2026 89 Florentin Smarandache, Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics 8.1. Related Phenomena (Structural Duality) The IPW concept is a specific mathematical realization of the philosophical idea of structural duality or hybridity, which links it to other phenomena and theories: 8.2. Smarandache Structures (S-Structures) • Relation: The IPW is a manifestation of the Smarandache concept applied to physics. • Structural Hybridity: The entity exhibits structural duality: it is both a continuous, extended wave (the "weak" structure) and a localized, singular particle (the "stronger" substructure/defect) existing simultaneously. This structural inconsistency is the core theme of S-Structures in algebra: https://fs.unm.edu/SmarandacheStrongStructures.htm https://fs.unm.edu/SmarandacheWeakStructures.htm https://fs.unm.edu/SmarandacheStrong-WeakStructures.htm 8.3. Solitons and Kinks • Relation: Solitons are stable, localized wave packets that retain their shape while propagating. They behave almost like particles (localized energy). • Comparison: While a soliton is a localized excitation of a field, the IPW's particle is a more radical concept: a true singularity or defect within the field, not just a localized, temporary solution to the wave equation. Solitons are highly non-linear but maintain continuity; the IPW explicitly involves a discontinuity (the puncture). 8.4. More theories, or fields related to IPW The "Infinitesimally Punctured Wave" (IPW) interpretation relates to several other theories, especially those dealing with the non-linear nature of quantum fields and alternative approaches to the particle concept. 9. Non-Linear and Alternative Field Theories The IPW model, by treating the particle as a singularity or non-linear defect within a field, connects to theories that try to derive particles from field structure rather than treating them as fundamental, point-like entities. 9.1. Quantum Field Theory (QFT) and Renormalization • Relation: QFT treats particles as excitations of quantum fields. However, QFT initially faces the problem of infinities (singularities) arising from point-like particle interactions, which must be managed through the complex process of renormalization. • IPW Connection: The IPW concept is a more explicit, geometric attempt to deal with the singularity. Instead of managing the infinite self-energy of a point particle mathematically (as in renormalization), the IPW suggests the singularity is a fundamental, perhaps bounded, structural feature of the physical wave field itself, aiming to inherently avoid the infinities associated with a purely mathematical pointparticle assumption. 9.2. Kaluza-Klein Theory and Extra Dimensions • Relation: Kaluza-Klein theory, and later String Theory, propose that particles are manifestations of vibrations or geometrical configurations in curled-up, extra spatial dimensions. Neutrosophic Sets and Systems, Vol. 98, 2026 90 Florentin Smarandache, Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics • IPW Connection: In both cases, the particle's properties (mass, charge) are derived from the geometry of the background space or field. The IPW shares this geometrical philosophy: the particle is a localized, non-trivial geometrical feature (the puncture/singularity) of the wave field, rather than being fundamentally distinct from it. 9.3. Non-Classical Solitons and Non-Linear Electrodynamics The IPW is strongly linked to non-linear theories that create stable, localized structures from waves. 10. Non-Linear Electrodynamics (NLE) • Relation: Theories like the Born-Infeld model of NLE attempt to modify Maxwell's equations at very high field strengths to give the electron a finite radius and finite selfenergy, thus avoiding the classical singularity problem. • IPW Connection: Both NLE and IPW seek to "desingularize" the electron. NLE achieves this by modifying the field equations themselves. IPW achieves this conceptually by declaring the particle as the singularity of the wave (or field) which might imply that the wave equation itself is non-linear and prevents the puncture from becoming a mathematical infinite point. 11. Non-Linear Schrödinger Equation (NLSE) and Solitons • Relation: The NLSE is a modification of the standard Schrödinger equation that is used to model non-linear wave phenomena. In certain NLSE systems, solutions exist in the form of solitons (localized, stable waves that maintain their shape). • IPW Connection: Solitons are perhaps the clearest physical analog to the IPW concept. They show how a wave can self-localize and maintain a particle-like identity due to non-linear interactions. The IPW essentially posits that the fundamental quantum particle is a specific type of non-linear, singularity-containing soliton solution to the ultimate field equation. These related fields demonstrate that the core idea of the IPW—deriving the particle from the structure of the wave/field—is a recurring and active theme in physics seeking a deeper, less contradictory foundation for quantum mechanics. 12. Applications of IPW The Infinitesimally Punctured Wave (IPW) interpretation is a conceptual model, not a mainstream theory, so it doesn't have established, practical engineering applications. Its possible "applications" are purely theoretical and foundational, serving to advance the philosophical and structural understanding of quantum mechanics. The potential applications lie in providing a new framework for solving long-standing problems in physics: 12.1. Applications in Foundational Physics 12.1.1. Solving the Measurement Problem (No-Collapse Model) The primary application is to offer an alternative resolution to the wavefunction collapse issue. • Goal: To describe the transition from the quantum world (superposition) to the classical world (definite outcome) without invoking the non-physical, instantaneous "collapse" postulate of the Copenhagen Interpretation. Neutrosophic Sets and Systems, Vol. 98, 2026 91 Florentin Smarandache, Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics • IPW Solution: By defining the particle as a pre-existing singularity (puncture) within a real, extended wave, the model suggests that measurement is simply the detection of the singularity's location, not the cause of the wave's collapse. This leads to a smoother, deterministic picture of quantum dynamics, similar to the de Broglie-Bohm theory. 12.1.2. Unifying Particle and Field Conceptions The IPW provides a model for structural unity between the two fundamental entities of physics: • Goal: To move beyond the dualism where particles and fields are treated as separate, distinct concepts. • IPW Solution: The particle is literally a geometric defect (the singularity) of the wavefield itself. This offers a conceptual framework for a unified field theory where mass, charge, and energy are all derived from the localized geometry or topological structure of a single underlying quantum field. 12.1.3. Addressing the Singularity Problem in Quantum Field Theory (QFT) The concept of a singular structure relates to the mathematical challenges in QFT: • Goal: To find a physical description of the electron that avoids the infinite self-energy (singularities) that arise when treating the particle as a mathematical point source. • IPW Connection: By defining the particle as a "punctured wave," the model suggests that the fundamental wave equation must be non-linear in a way that naturally prevents the singularity from reaching a true mathematical infinity, perhaps by defining a finite core structure for the "puncture." 13. Applications in Conceptual Frameworks 13.1. Neutrosophic Quantum Theory The IPW model, developed by Florentin Smarandache, fits directly into the structural framework of Neutrosophic Theory: • Goal: To model systems where indeterminacy and inconsistency are inherent parts of the structure. • IPW Connection: The IPW describes a structure that is partially wave (continuous, extended) and partially particle (discrete, singular). This structural hybridity is a classic example of a Smarandache structure, allowing researchers to apply the algebraic tools of Neutrosophic Logic to analyze the indeterminacy of the quantum state. 14. Alternative Models for Quantum Gravity This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation as well as the experimental conclusions that can be drawn. The connection between the particle's singularity and the geometry of its wave field opens avenues for alternative gravitational models: • Goal: To find a way to incorporate quantum effects into General Relativity, especially concerning the nature of matter that warps space-time. • IPW Connection: If the particle is a space-time singularity embedded in a wave field, it could suggest a mechanism where the wave's extended nature influences the geometry of space-time, providing new starting points for non-local or non-linear theories of quantum gravity. Neutrosophic Sets and Systems, Vol. 98, 2026 92 Florentin Smarandache, Comparisons of Infinitesimally Punctured Wave with Copenhagen and De Broglie-Bohm Interpretations, Neutrosophic Quantum Mechanics, and Non-Linear Electromagnetics 16. Conclusions The Infinitesimally Punctured Wave IPW) — in which a quantum object is visualized as an aggregation of infinitely many infinitesimally spaced particles. When these particles are densely packed, the ensemble appears as a continuous wave; when a measurement isolates a single constituent, particle-like behavior emerges. The model is situated alongside established alternative interpretations (e.g., De Broglie–Bohm pilot wave theory, wave packet descriptions) and linked to Neutrosophic Quantum Theory, which supplies a logical framework for handling indeterminacy.” [4] The article’s goal is to compare the infinitesimally punctured wave (IPW) with the major interpretations of the fundamental philosophical differences in how physicists view reality at the quantum level. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. References 1. Omnès, Roland (1999). “The Copenhagen Interpretation”. Understanding Quantum Mechanics. Princeton University Press. pp. 41–54. DOI: 10.2307/j.ctv173f2pm.9. S2CID 203390914 2. de Broglie, Louis (1987). “Interpretation of quantum mechanics by the double solution theory”. Annales de la Fondation. 12 (4): 399–421. ISSN 0182-4295. Archived from the original (PDF) on 19 January 2012. Retrieved 7 December 2016. 3. Smarandache, Florentin (2019). “Wave Particle Duality as an Infinitesimally Punctured Wave.” In: Nidus Idearum. Scilogs, vol. 4, 2019, p. 122. https://fs.unm.edu/NidusIdearum4-v2.pdf 4. Smarandache Florentin (2026). “The Infinitesimally Punctured Wave: A Corpuscular Visualization of Wave-Particle Duality.” Neutrosophic Sets and Systems, 97, 704-708. https://fs.unm.edu/nss8/index.php/111/article/view/7472/3331 Received: July 16, 2025. Accepted: Dec 19, 2025