A mathematical model for the slit experiments in the heart of quantum mechanics is developed to gain insight into quantum theory. The proposed system-theoretical model is entirely based on commutative mathematics, i.e. convolution, and integral transformations, and starts with spacetime functions with inherent energy-based cause and effect relations of the state-function Ѱ in the complex Hilbert space. The benefits of his approach are as 1-Invariance in time reversal. 2-Deterministic result functions in the model in line with the outcome of slit experiments. 3- Separation of causality and cross-correlations of attained states. 4- Disappearance of a posteriori probability of quantum states. 5- Quantum a priori fixed states after causality interactions have ended, (even) when quanta are (light-years) separated. The model predicts the patterns in the experiments with mathematical functions of the energy distributions. The quantum mechanical counterpart description of the physical reality of slit experiments thus may be considered complete in A. Einstein’s definition. The patterns in double slit experiments are found to be an effect of energy (amplitude-) modulation. An equivalent double-slit pattern can be retrieved from an input modulated 1-slit experiment excluding interference interpretations. The system-theoretical model uses generic properties of quanta and evolves into determinism in quantum mechanics slit experiments. The mathematics in the model handles beables by treatment of momentum p in system theoretical I/O relations of the transformed functions and allows the proposed description by the avoidance of a direct addressing of the individual quanta through variables. The following method yields exact, non-probabilistic results.
A mathematical model for the slit experiments in the heart of quantum mechanics is developed to gain insight into quantum theory. The proposed system-theoretical model is entirely based on commutative mathematics, i.e. convolution, and integral transformations, and starts with spacetime functions with inherent energy-based cause and effect relations of the state-function Ѱ in the complex Hilbert space. The benefits of his approach are as 1-Invariance in time reversal. 2-Deterministic result functions in the model in line with the outcome of slit experiments. 3- Separation of causality and cross-correlations of attained states. 4- Disappearance of a posteriori probability of quantum states. 5- Quantum a priori fixed states after causality interactions have ended, (even) when quanta are (light-years) separated. The model predicts the patterns in the experiments with mathematical functions of the energy distributions. The quantum mechanical counterpart description of the physical reality of slit experiments thus may be considered complete in A. Einstein’s definition. The patterns in double slit experiments are found to be an effect of energy (amplitude-) modulation. An equivalent double-slit pattern can be retrieved from an input modulated 1-slit experiment excluding interference interpretations. The system-theoretical model uses generic properties of quanta and evolves into determinism in quantum mechanics slit experiments. The mathematics in the model handles beables by treatment of momentum p in system theoretical I/O relations of the transformed functions and allows the proposed description by the avoidance of a direct addressing of the individual quanta through variables. The following method yields exact, non-probabilistic results.
The fundamental deterministic nature of quantum mechanics (QM) is mathematically demonstrated by the modeling of the quantum (multiple) slit experiments; hence this paper is written from a deterministic perspective (DP) in quantum mechanics. Due to approximately 90 years of indeterminism and probabilistic statistical results, the theory consists of many interpretations and is often regarded as counterintuitive. The latter is in the deterministic perspective not the case and is illustrated with some examples. After a brief historic perspective, ‘invisible’ i.e. invisible entities of reality in mathematical treatment, are introduced. These entities are handled by mathematics indirectly i.e. are described in a transformed domain without variables violating the Heisenberg relation. In nature, i.e. on micro and macro levels, causality is fully ‘entangled’ with energy, information - in the meaning of ordering or coding - as well as time. In contrast with the macro scale with many forms and types of memory functions, without a memory property of quanta, causality is the bearer of information symmetry on the quantum scale. Further on in this DP paper: mathematical and philosophical consequences and influences in several paragraphs, regarding subjects such as ‘free will’ and expected ‘threats to science’, the Bell inequalities, Alice and Bob & entanglement, causality, information, retro-causality, spooky action, encryption and computing, teleportation, and in general interpretations rooted in indeterminacy in QM as well as associated topics on independence, fine-tuning. The last paragraph outlines the mathematical treatment of QM more extensively and may clarify references to the above-mentioned topics of discussion in QM further.
A Set-Based Alternative to the Hilbert-Space Formalism: Two-Slit Interference in the CEGS Framework This work presents a deterministic reformulation of the two-slit experiment within the Calculus of Element Generated Sets (CEGS), a set-based framework for modeling physical systems without assuming Hilbert space as a fundamental structure. Instead of representing quantum states as vectors in a complex Hilbert space, physical states are described as structured sets of phase-generating elements subject to intrinsic constraint operations prior to aggregation. Within this framework, observable quantities arise only after a coherent aggregation map acts on constrained generative sets. The standard interference pattern of the two-slit experiment is recovered exactly at the level of observable predictions, preserving empirical equivalence with conventional quantum mechanics. A central conceptual shift is that complex amplitudes are not treated as fundamental entities. Instead, they emerge as effective compressed representations of underlying distributed phase structures under observer-dependent projection. In this view, Hilbert space is interpreted as an emergent effective description rather than a primary ontological structure. The formulation provides a deterministic and structural alternative perspective on quantum interference, contributing to ongoing discussions in quantum foundations, interpretation of quantum mechanics, and set-based or non-vectorial reformulations of physical theory. The analysis is restricted to the two-slit experiment as a minimal test case, while maintaining consistency with standard quantum predictions.
The double-slit experiment has remained one of the most profound and conceptually challenging demonstrations in physics. While the mathematical framework of Quantum Mechanics - superposition, wave function, and state collapse - accurately predicts outcomes, it fails to provide a clear causal description of how potential quantum states transform into actual, registered events. This paper presents the Event-Chain Model, a new conceptual framework that represents every quantum process as a causal sequence of connected physical events. Each event reflects an irreversible registration, transforming quantum potentials into factual reality. The model introduces three core components: the Potential Pattern, the Irreversible Event, and the Event Chain that constitutes reality itself. Using this framework, the paper explains interference, decoherence, quantum erasure, and the EPR-eraser experiments without invoking many-worlds or retrocausality. Superposition is defined as a Potential Pattern - a structured map of possibilities - while collapse is redefined as a natural, physical registration process that generates an event in the chain of reality. Interference patterns thus emerge not as physical waves but as cumulative records of uncollapsed potentials. The Event-Chain Model preserves all results of Quantum Mechanics while adding a transparent causal layer, linking mathematical formalism with physical reality. It provides a unified interpretation of how potential, event, and actuality co-exist and evolve within a consistent quantum framework.
The double-slit interference experiment serves as a fundamental experiment in quantum mechanics. Core theories including wave-particle duality, quantum superposition and measurement collapse are established on the basis of this phenomenon. The mainstream Copenhagen interpretation and Bohmian pilot-wave theory rely heavily on mathematical statistical fitting to interpret experimental outcomes, yet they fail to reveal the real physical evolution mechanism of photons at the microscale. Moreover, they inherently suffer from symmetry breaking, causal discontinuity and logical inconsistency. The symmetry-breaking analogy widely adopted in academic circles contains dimensional mismatch and conceptual substitution, representing a cognitive bias that places mathematical tools above physical mechanisms. Based on the symmetry principle of classical physics and the rule of continuous causal evolution, this paper systematically identifies the core defects of mainstream quantum interpretations and points out the paradigm limitation of quantum mechanics that prioritizes mathematical fitting over physical ontology. A spatial particle resonance propagation model is proposed. This model explains that photons are essentially the resonance manifestation of space particles. When resonance passes through double slits, the resonance body splits symmetrically, and then recombines after passing through the slits to form interference. Measurement will break the symmetry of resonance, resulting in the disappearance of interference fringes. This model maintains continuous causality and geometric symmetry, and provides a new physical mechanism explanation for quantum uncertainty and measurement effects in double-slit experiments.
We present a focused formulation of Dynamic State-Space Theory (DSST) applied exclusively to the double-slit experiment. The theory addresses a single question: why are individual detection events localized and unpredictable, while an ensemble of events yields a stable interference pattern? An energetic process (source, slits, propagation, environment and detector) is treated as generating and evolving a relational state-space S of dynamically accessible configurations. A measurement interaction is modelled as a physical sampling of S that produces one definite, localized outcome and a subsequent state. Individual outcomes remain intrinsically probabilistic; the Born distribution and the interference term are recovered exactly from the geometry of S. The framework is phenomenological: it does not modify the Schr¨odinger equation or the Born rule, and it makes no claim of experimental departure from standard quantum mechanics. Its contribution is a precise conceptual and mathematical language for the origin of randomness in the double-slit setting.
## AbstractA deterministic interpretation of the double-slit experiment is proposed, in which the electron is treated as a localized particle at all times. The observed interference pattern is attributed to a real physical energy wave in the electromagnetic field, generated by the electron's accelerated motion through the field. This energy wave reaches the slit barrier, where the slit geometry creates coherent waves that merge to form a constructive interference wave. This wave redistributes energy in the field, creating an Electron-EMF coupled pathway that guides the particle to the detector. The apparent randomness of individual impacts arises from incomplete knowledge of the initial conditions, not from intrinsic indeterminism. The measurement effect is modelled as a disturbance of the coherent energy waves, either preventing the formation of the coupled pathway or destroying it after formation. The model is extended to the quantum eraser experiment, where the restoration of coherence via a polarizer naturally explains the reappearance of interference. A further extension to the delayed-choice quantum eraser is proposed, introducing the Entangled Constructive Interference Wave (ECIW) as a physical mechanism for entanglement within the model. The mechanism is shown to apply across geometries beyond the double slit. Predictions include a correlation between landing position and flight time, offering a testable distinction from standard quantum mechanical interpretations. A preliminary, illustrative mathematical framework is provided, situating the model within the pilot-wave tradition. **Keywords: ** Double-slit experiment, pilot wave, electromagnetic field, quantum interpretation, determinism, hidden variables, measurement problem, energy wave, coupled pathway, quantum eraser, delayed-choice quantum eraser, entanglement.