Author: Jan Šági
sagiphp@gmail.com
The Scalar Field Interaction Theory offers a transformative perspective on the wave-like properties of photons, no longer attributing them to an intrinsic duality but rather to interactions with an oscillatory scalar field. By treating the scalar field as a locally oscillating entity with zero macroscopic mean, this theory demonstrates how photons acquire wave-like characteristics during their journey through regions of scalar field fluctuations. This innovative framework not only redefines classical and quantum interpretations of light but also provides unparalleled accuracy in explaining interference and diffraction patterns. Recent simulations show that the scalar field model achieves error rates up to 99.99% lower than traditional quantum mechanical approaches, establishing it as a groundbreaking alternative in modern physics.
For over a century, the wave-particle duality of photons has stood as a central tenet of quantum mechanics, fundamentally shaping our understanding of light. Yet, this duality has sparked intense debates due to its probabilistic nature and reliance on intrinsic randomness. The Scalar Field Interaction Theory disrupts this paradigm by presenting a deterministic framework: photons are purely particle-like entities, and their wave-like behavior emerges solely from interactions with a scalar field.
This scalar field, defined by localized oscillations with a macroscopic average of zero (⟨ϕ⟩=0), serves as the medium through which wave-like phenomena, such as interference and diffraction, manifest. Unlike traditional quantum models, this approach eliminates the need for probabilistic interpretations, offering a deterministic and highly precise explanation of observed phenomena. Recent comparisons reveal that the scalar field model achieves near-perfect alignment with experimental data, with RMS errors reduced by orders of magnitude compared to quantum mechanical baselines.
By bridging the gap between particle-based and wave-based descriptions, the Scalar Field Interaction Theory introduces a unified framework that challenges longstanding assumptions. This paper delves into the theoretical foundations, derived parameters, and empirical validations of this approach, shedding light on its potential to redefine our understanding of one of the most fundamental aspects of physics.
In this framework, the following assumptions are made:
The scalar field is characterized by local fluctuations over a finite volume V.
The average value of the field is zero, i.e. ⟨ϕ⟩=0; the field oscillates such that in some regions it is positive while in others it is negative.
The field interacts weakly with matter and light, permitting the use of linear approximations for its fluctuations.
The field exhibits an exponential decay profile, ϕ(r)=ϕ0e−mr, on macroscopic scales.
Quantum fluctuations ξ(x,t) are modeled as Gaussian noise with zero mean.
These assumptions facilitate the analytical derivation of key parameters while capturing the local, oscillatory behavior of the field.
The Planck length ℓp represents the smallest physically meaningful scale:
ℓp=ℏGc3
Numerically, this is approximately:
ℓp≈1.616×10−35m
The mass parameter m sets the scale of the field’s spatial variation. Initially defined as:
m=1rscale,
we now confine this to a finite volume by writing:
m=1λℓp,
with λ being a dimensionless parameter that adjusts the effective extent of the field.
The self-interaction parameter α quantifies the field’s nonlinearity:
α=m2ϕ02.
The scalar field modifies the effective speed of light via local interactions. Considering the local value and spatial gradient of the field, we define:
ceff(r)=c0(1+κ(r)ϕ2(r)).
To reflect local fluctuations, the interaction parameter is given by:
κ(r)=ceff(r)−c0c0(ϕ2(r)+(∇ϕ)2m2).
Instead of assuming a uniform field, the energy is now computed by integrating the energy density over a finite volume V:
E=∫V(12(∇δϕ)2+12m2⟨(δϕ)2⟩+α4⟨(δϕ)4⟩)dV.
Here, δϕ(x,t) represents the local fluctuation of the field, with the constraint ⟨ϕ⟩=0.
The characteristic amplitude ϕ0 is initially derived via
ϕ0=ℏλℓpc,
but with the inclusion of local fluctuations (and ⟨ϕ⟩=0), ϕ0 represents the scale of the oscillatory deviations rather than a constant background value.
The dynamics of the scalar field are governed by a modified Klein–Gordon equation. In order to incorporate the local fluctuations, the total field is written as:
ϕ(x,t)=ϕ0+δϕ(x,t),
with the stipulation that ϕ0=0 (i.e. ⟨ϕ⟩=0), so the evolution is entirely in the fluctuation δϕ(x,t). The governing equation becomes:
◻δϕ(x,t)−m2δϕ(x,t)+α[δϕ(x,t)]3=ξ(x,t).
Moreover, to explicitly include the temporal oscillations of the field, the fluctuation is decomposed as:
δϕ(x,t)=ϕ1(x)cos(ωt)+ϕ2(x)sin(ωt),
where the oscillation frequency is given by ω=k2+m2.
In this section, we demonstrate how the scalar field parameters can be derived using fundamental Planck constants and associated physical quantities. The calculations are based on the following definitions:
The Planck length represents the smallest meaningful length scale in nature and is given by:
ℓp=ℏGc3
Numerically:
ℓp≈1.616×10−35m.
The mass parameter determines the characteristic range of the scalar field and is expressed as:
m=1λℓp,
where λ is a dimensionless scaling factor. For λ=1010:
m≈6.187×1024m−1.
The scalar field amplitude ϕ0 is calculated as:
ϕ0=ℏλℓpc,
Substituting the known values:
ℏ=1.054×10−34J·s
c=3.0×108m/s
ℓp=1.616×10−35m
λ=1010
The resulting value is:
ϕ0≈2.177×10−18(dimensionless).
ParameterFormulaCalculated ValuePlanck Length (ℓp)ℏGc31.616×10−35mMass Parameter (m)1λℓp6.195×1024m−1Amplitude (ϕ0)ℏλℓpc2.177×10−18(dimensionless)
These calculations form the basis for connecting the scalar field's theoretical properties with fundamental physical constants.
In our deterministic scalar field model, gravity emerges not solely from the intrinsic mass of objects but from the spatial variations (i.e., gradients) and amplitude differences of an infinite scalar field—a field that existed even before the Big Bang. This section provides a detailed derivation of the gravitational constant G from our model, explaining each step and discussing the role of the field gradient.
We begin by assuming that the scalar field ϕ(r) obeys a modified Klein–Gordon equation with a matter source:
∇2ϕ−mϕ2ϕ=−γρ(r),
where the mass parameter mϕ is defined asmϕ=λℓp,with λ being a dimensionless scaling factor and ℓp the Planck length. The coupling constant γ dictates how the matter density ρ(r) sources the field.
For a spherically symmetric point source of mass M, the solution to this equation is given by:
ϕ(r)=γM4πe−mϕrr.
In the long-range limit (mϕr≪1), the exponential factor approaches unity, so we have:
ϕ(r)≈γM4πr.
We next define the gravitational potential Φ(r) to be directly proportional to the local scalar field amplitude:
Φ(r)=−δϕ(r).
The negative sign indicates an attractive potential. For Newtonian gravity, the potential is
ΦN(r)=−GMr.
Equating our expression for Φ(r) in the low-mϕr limit with ΦN(r), we have:
−δγM4πr=−GMr.
Canceling common terms (with M and r arbitrary) leads to the relation:
G=δγ4π.
To maintain determinism in our model—with all parameters fixed by fundamental constants—we choose
δ=ℓp2andγ=4πc3ℏ,
where c is the speed of light and ℏ is the reduced Planck constant. Substituting these into the above relation gives:
G=ℓp2(4πc3/ℏ)4π=ℓp2c3ℏ.
Now, let us calculate the numerical value of G. Using the fundamental constants:
ℓp=1.616×10−35m
c=2.9979×108m/s
ℏ=1.0546×10−34J·s
First, compute the square of the Planck length:
ℓp2=(1.616×10−35)2≈2.61×10−70m2.
Next, compute c3:
c3≈(2.9979×108)3≈2.70×1025m3/s3.
Multiplying these, we obtain:
ℓp2c3≈2.61×10−70m2×2.70×1025m3/s3≈7.05×10−45m5/s3.
Dividing by ℏ yields:
G=7.05×10−45m5/s31.0546×10−34J·s≈6.68×10−11m3kg−1s−2.
This value is in excellent agreement with the experimentally measured gravitational constant:
Gexp≈6.67×10−11m3kg−1s−2.
In our model, the gravitational interaction is ultimately interpreted as arising from the spatial gradient of the scalar field—regions where ϕ changes rapidly (i.e., has a steep gradient) correspond to stronger gravitational effects. The derivation above shows that by fixing all parameters deterministically via fundamental constants, we can derive a unique expression for G that matches observation. This approach highlights the central role played by the field gradient in generating the gravitational potential and confirms that the emergent gravity from such a field is consistent with Newtonian gravity.
The key takeaway is that in the deterministic scalar field framework, the gravitational constant is not an arbitrarily set parameter but a calculated value:
G=ℓp2c3ℏ,
a result that is fully determined by the Planck length, the speed of light, and the reduced Planck constant. The near-perfect match of the computed G with the measured value supports the potential of this model to offer a new, unified understanding of gravitational interactions.
Datasets Used: Three experimental datasets (DataExfig3a, DataExfig3b, DataExfig3c) derived from quantum interference measurements on GaAs quantum dots.
Source: https://zenodo.org/records/6371310
In our simulation, we compare the predictions of a Scalar Field model against a simplified Quantum Baseline model. The script automatically processes each dataset, applies scalar field parameters derived from theoretical considerations (phi_0, m_scalar), and calculates the Root Mean Square (RMS) and Akaike Information Criterion (AIC) for both models.
For details on how to run this simulation, install dependencies, or adjust the parameters, please refer to:
Simulation Script: simulation-theory.py
Documentation (README): See Scalar Field Simulation: README section.
The following table summarizes the RMS values (lower is better) for the Scalar Field and Quantum Baseline models on each dataset. The Improvement column indicates how much lower the RMS is (in percent) for the scalar field model, relative to the quantum model.
DatasetRMS (Scalar Field)RMS (Quantum Model)Improvement (%)AIC (Scalar Field)AIC (Quantum Model)DataExfig3a3.12e-051.02e-01~99.97%-203.45-150.31DataExfig3b2.13e-219.54e-12~100.00%-490.77-310.56DataExfig3c1.83e-202.19e-11~100.00%-512.10-341.44
All numerical values, including RMS and AIC, are automatically calculated by the script and stored in all_data_summary.csv located in Output_Combined.
The following figures show the observed data (black markers) versus the simulated curves for the Scalar Field (blue dashed lines) and Quantum Baseline (red dotted lines). Each image is generated and saved by the script:
DataExfig3a: Delay (s) vs. Normalized g2
DataExfig3b: Frequency (Hz) vs. PSD
DataExfig3c: Frequency (Hz) vs. PSD_offres
In addition to the summary, the script produces per-dataset CSV files containing the original data columns (e.g., delay(s) or freq(Hz)) alongside the simulated ScalarField and QuantumBaseline values:
DataExfig3a_detailed.csv
Columns: delay(s), Observed_g2, ScalarField, QuantumBaseline
DataExfig3b_detailed.csv
Columns: freq(Hz), Observed_PSD, ScalarField, QuantumBaseline
DataExfig3c_detailed.csv
Columns: freq(Hz), Observed_offres, ScalarField, QuantumBaseline
You can download the original input data, the final simulation outputs, and all generated plots below:
Input Datasets:
Simulation Outputs (zip):
Download the full simulation: simulation-theory.zip
To further validate our scalar field approach, we tested it on tunnel diode data obtained from this Kaggle dataset. Our script (simulation.py) computes diode currents using both a standard quantum baseline model and our deterministically derived scalar field model. In the simplest quantum (Shockley-like) picture, the diode current follows:
IQM(V)=I0[exp(qVkBT)−1],
but in true tunneling regimes, one often invokes a barrier penetration expression or a Fowler–Nordheim-like form. For instance, a simplified tunneling current can appear as:
Itun(V)∝V2exp(−βΦ3/2V),
where Φ is the barrier height and β is a constant depending on effective mass, charge, and Planck’s constant. In our Scalar Field approach, we modify the barrier by a deterministic term κϕ2(V) derived from fundamental constants. Specifically, the barrier Vb is replaced byVeff(V)=Vb+κϕ2(V).The scalar field ϕ(V) itself is governed by dimensionless parameters (like λ) along with ϕ0≈2.18×10−18 and m≈6.19×1024m−1, all derived from the Planck length and Planck constants.
The resulting plots show a clear advantage for the scalar field approach, with RMS errors reduced by about 63% compared to the quantum baseline:
Quantum Baseline RMS: 1.78 × 105
Scalar Field RMS: 6.55 × 104
Improvement: ~63%
We likewise observed improvements in other metrics (MAE, AIC, BIC), underscoring the consistent benefit of a purely deterministic modification of the tunneling barrier. The detailed CSV (scalar_vs_quantum_results_noMAPE_R2.csv) and the generated plot can be found in the Output_Detailed folder. An example of the output plot is shown below:
All simulation scripts and data-processing code are available in this downloadable archive. Researchers can thus replicate and refine the analysis—tuning parameters such as ϕ0 or m within the deterministically constrained Planck-based approach—and confirm that the Scalar Field Interaction Theory not only applies to photonic interference but also provides a precise, non-probabilistic description of tunneling phenomena in semiconductor diodes.
A key question arising from our deterministic Scalar Field Interaction Theory is whether the modifications applied to describe quantum tunneling in diodes remain fully consistent with the field’s other governing equations—particularly the modified Klein–Gordon equation (Section 4) and the fundamental parameters derived from Planck constants (Section 5).
In essence, the standard quantum model for diode tunneling typically starts with a baseline current:
IQM(V)=I0[exp(qVkBT)−1],
supplemented by a barrier-penetration term (such as the Fowler–Nordheim or standard tunneling exponent). In our scalar field extension, the electron’s effective potential barrier, Vb, is deterministically shifted by κϕ2(V). This shift is directly consistent with the field’s exponential decay form ϕ(r)=ϕ0e−mr (Section 3.2) and does not require any probabilistic adjustment. Instead, the field amplitude ϕ0 and mass parameter m (Section 5.2–5.3) govern how strongly (and how far) the barrier is modified.
Since ϕ0 and m are derived from Planck scales, and the field itself obeys the modified Klein–Gordon equation
◻δϕ−m2δϕ+α[δϕ]3=ξ(x,t),
the same parameters m and ϕ0 naturally carry over into the tunneling formalism without contradiction. Indeed, the local fluctuations δϕ (consistent with ⟨ϕ⟩=0) become the mechanism by which the barrier is “modulated,” rendering quantum tunneling a deterministic phenomenon under the scalar field’s influence. The measured diode current
ISF(V)=I0[exp(qVkBT)exp(−VeffVb)−1],whereVeff=Vb+κϕ2(V),
thus maintains fidelity with the exponential decay profile, the local field strength ϕ(V), and the mass parameter m. This ensures that the tunneling current model is fully aligned with the rest of the theory—no additional parameters or probabilistic corrections are introduced.
Numerical simulations confirm that this approach yields improved RMS, MAE, and AIC metrics relative to the unmodified quantum baseline, all while respecting the scalar field’s fundamental equations described in Sections 2–5. Hence, quantum tunneling under the Scalar Field Interaction Theory remains in complete agreement with the deterministic formalism that governs photonic interference and other wave-like effects, further validating the theory’s self-consistency across multiple physical domains.
In summary, the Scalar Field Interaction Theory provides a new deterministic framework that reinterprets the wave-like behavior of photons as an emergent phenomenon resulting from local, oscillatory fluctuations in a scalar field. By rigorously deriving key parameters – such as the effective mass parameter, characteristic amplitude, self-interaction coefficient, and interaction parameter – directly from fundamental constants and integrating the field’s energy over a finite volume, our approach not only replicates the established predictions of quantum mechanics but also offers a path toward substantially reduced RMS errors in fitting experimental data.
Our analysis demonstrates that:
The scalar field oscillates locally with a zero macroscopic average, ensuring that the positive and negative fluctuations cancel out on a large scale while still producing measurable interference effects.
The energy associated with these fluctuations is consistently computed by integrating both the gradient and potential contributions over a limited volume, thereby grounding the theory in physical realism.
The local definition of the interaction parameter κ captures both the field’s strength and its spatial gradients, which is crucial for accurately reproducing phenomena like interference and tunneling.
All relevant parameters – including ϕ0, m, α, κ, and the integrated field energy E – are deterministically recalculated to account for the finite extent and intrinsic fluctuations of the field, leading to results that closely match experimental observations.
Ultimately, our work suggests that the deterministic scalar field model not only challenges the conventional reliance on probabilistic interpretations in quantum mechanics but also opens up new avenues for achieving unprecedented precision in theoretical predictions. Future research will focus on further refining the model, extending its application to other quantum phenomena, and conducting detailed experimental validations to fully establish its advantages over traditional approaches.
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