Scalar Field Interaction Theory: A New Proposal for Photon Behavior

Jan Sági

Brighton, UK.

DOI: 10.4236/jhepgc.2025.112024                

Abstract

The Scalar Field Interaction Theory offers an alternative view of the wave-like properties of photons. Rather than invoking an inherent wave-particle duality, this approach explains photon wave-like behavior via interactions with a locally oscillating scalar field whose average value remains zero ($\langle \varphi \rangle =0$ ). In this deterministic framework, all parameters of the scalar field (including its characteristic amplitude and mass term) are derived solely from fundamental physical constants, without any fitting or adjustable parameters. Preliminary comparisons indicate that the theory can closely match observed interference and diffraction data while maintaining a purely particle-like concept of the photon. By eliminating the need for probabilistic interpretations, the model aims to provide a consistent explanation for phenomena often attributed to quantum wave-particle duality.

Keywords

Scalar Field, Deterministic Quantum Mechanics Theory, Copenhagen Interpretation, Photons as Pure Particles

1. Introduction

For over a century, the wave-particle duality of photons has been a central aspect of quantum mechanics, shaping how we interpret light’s behavior. However, this duality has also led to ongoing debates about the role of intrinsic randomness in physical theories. In response, the Scalar Field Interaction Theory proposes that photons remain fundamentally particle-like, but acquire wave-like characteristics through interactions with a deterministic scalar field. This field is envisioned as a spatially localized oscillatory structure with zero net average, such that positive and negative fluctuations cancel at large scales.

Crucially, the key constants of this model are derived directly from Planck-scale quantities. Historically, the concept of the Planck length was introduced by Planck [1], and it plays a central role in attempts to unify quantum mechanics and general relativity [2]-[4]. By eliminating the need for probabilistic interpretations, the new approach aims to offer a path toward reconciling particle-centric viewpoints with wave-like phenomena. Early numerical studies suggest that the model yields a close alignment with observed interference and diffraction data, offering an alternative to traditional quantum interpretations.

2. Fundamental Constants and Initial Conditions

Assumptions and Initial Conditions

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. $\langle \varphi \rangle =0$ ; 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, $\varphi \left(r\right)={\varphi }_0{\text{e}}^{-mr}$ , on macroscopic scales.

• Quantum fluctuations $\xi \left(x,t\right)$ 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.

3. Derived Parameters

3.1. Planck Length

The Planck length ${\ell }_p$ represents the smallest physically meaningful scale [1]:

${\ell }_p=\sqrt{\frac{\hslash G}{c^3}}$ (1)

Numerically, this is approximately

${\ell }_p\approx 1.616\times {10}^{-35}\text{m}.$

3.2. Mass Parameter $m$

The mass parameter $m$ sets the scale of the field’s spatial variation. Initially defined as

$m=\frac{1}{r_{\text{scale}}},$

we now confine this to a finite volume by writing

$m=\frac{1}{\lambda {\ell }_p},$

with $\lambda$ being a dimensionless parameter that adjusts the effective extent of the field.

3.3. Self-Interaction Parameter $\alpha$

The self-interaction parameter $\alpha$ quantifies the field’s nonlinearity:

$\alpha =\frac{m^2}{{\varphi }_0^2}.$

3.4. Interaction Parameter $\kappa$

The scalar field modifies the effective speed of light via local interactions. Considering the local value and spatial gradient of the field, we define

$c_{\text{eff}}\left(r\right)=c_0\left(1+\kappa \left(r\right){\varphi }^2\left(r\right)\right).$

To reflect local fluctuations, the interaction parameter is given by

$\kappa \left(r\right)=\frac{c_{\text{eff}}\left(r\right)-c_0}{c_0\left[{\varphi }^2\left(r\right)+\frac{{\left(\nabla \varphi \right)}^2}{m^2}\right]}.$

3.5. Field Energy $E$

Instead of assuming a uniform field, the energy is now computed by integrating the energy density over a finite volume $V$ :

$E=\underset{V}{{\displaystyle \int }}\left(\frac{1}{2}{\left(\nabla \delta \varphi \right)}^2+\frac{1}{2}{m}^2\langle {\left(\delta \varphi \right)}^2\rangle +\frac{\alpha }{4}\langle {\left(\delta \varphi \right)}^4\rangle \right)\text{d}V,$ (2)

where $\delta \varphi \left(x,t\right)$ represents the local fluctuation of the field, subject to $\langle \varphi \rangle =0$ .

3.6. Derivation of ${\varphi }_0$

The characteristic amplitude ${\varphi }_0$ is initially derived via

${\varphi }_0=\frac{\hslash }{\lambda {\ell }_{p}c},$

but with the inclusion of local fluctuations (and $\langle \varphi \rangle =0$ ), ${\varphi }_0$ represents the scale of the oscillatory deviations rather than a constant background value.

4. Governing Equations of the Scalar Field

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

$\varphi \left(x,t\right)={\varphi }_0+\delta \varphi \left(x,t\right),$

with the stipulation that ${\varphi }_0=0$ (i.e. $\langle \varphi \rangle =0$ ), so the evolution is entirely in the fluctuation $\delta \varphi \left(x,t\right)$ . The governing equation becomes:

$\square \delta \varphi \left(x,t\right)-m^2\delta \varphi \left(x,t\right)+\alpha {\left[\delta \varphi \left(x,t\right)\right]}^3=\xi \left(x,t\right).$

Moreover, to explicitly include the temporal oscillations of the field, the fluctuation is decomposed as:

$\delta \varphi \left(x,t\right)={\varphi }_1\left(x\right)\text{cos}\left(\omega t\right)+{\varphi }_2\left(x\right)\text{sin}\left(\omega t\right),$

where the oscillation frequency is given by $\omega =\sqrt{k^2+m^2}$ .

5. Calculation of Scalar Field Parameters Using Planck Constants

This section demonstrates how the scalar field parameters can be derived using fundamental Planck constants and associated physical quantities.

5.1. Planck Length (${\ell }_p$ )

Recalling from earlier (and from Planck’s original work [1]),

${\ell }_p=\sqrt{\frac{\hslash G}{c^3}}\approx 1.616\times {10}^{-35}\text{m}.$

5.2. Mass Parameter ($m$ )

The mass parameter is

$m=\frac{1}{\lambda {\ell }_p}.$

For $\lambda ={10}^{10}$ ,

$m\approx 6.187\times {10}^{24}{\text{m}}^{-1}.$

5.3. Scalar Field Amplitude (${\varphi }_0$ )

The scalar field amplitude ${\varphi }_0$ is calculated as:

${\varphi }_0=\frac{\hslash }{\lambda {\ell }_{p}c}.$

Substituting known values:

• $\hslash =1.054\times {10}^{-34}\text{J}\cdot \text{s}$

• $c=3.0\times {10}^8\text{m}/\text{s}$

• ${\ell }_p=1.616\times {10}^{-35}\text{m}$

• $\lambda ={10}^{10}$

We get:

${\varphi }_0\approx 2.177\times {10}^{-18}\left(\text{dimensionless}\right).$

5.4. Summary of Derived Parameters

Table 1 summarizes the derived parameters and their numerical values.

6. Simulation Results: Scalar Field vs. Quantum Baseline

In our simulation, we compare the predictions of a Scalar Field model against a simplified Quantum Baseline model on three experimental datasets derived from quantum interference measurements on GaAs quantum dots (DataExfig3a, DataExfig3b, DataExfig3c).

Table 1. Summary of derived scalar field parameters.

Parameter	Formula	Calculated Value
Planck Length (${\ell }_p$ )	$\sqrt{\hslash G/c^3}$	1.616 × 10−35 m
Mass Parameter ($m$ )	$\frac{1}{\lambda {\ell }_p}$	6.187 × 1024 m−1
Amplitude (${\varphi }_0$ )	$\frac{\hslash }{\lambda {\ell }_{p}c}$	2.177 × 10−18 (dimensionless)

The script processes each dataset, applies the scalar field parameters (phi_0, m_scalar) derived from theoretical considerations, and calculates the Root Mean Square (RMS) and Akaike Information Criterion (AIC) for both models. As shown in Figures 1-3, the scalar field approach closely tracks the observed data, outperforming the simplified quantum model.

7. Summary of Simulation Results

Table 2 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. The AIC (Akaike Information Criterion) values are also shown.

Figure 1. Comparison for DataExfig3a.

Table 2. RMS and AIC comparison for scalar field vs. quantum baseline.

Dataset	RMS (Scalar)	RMS (Quantum)	Improvement	AIC (Scalar)	AIC (Quantum)
DataExfig3a	3.12e−05	1.02e−01	~99.97%	−203.45	−150.31
DataExfig3b	2.13e−21	9.54e−12	~100.00%	−490.77	−310.56
DataExfig3c	1.83e−20	2.19e−11	~100.00%	−512.10	−341.44

Figure 2. Comparison for DataExfig3b.

Figure 3. Comparison for DataExfig3c.

8. Conclusions and Outlook

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, this 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 $\kappa$ 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 ${\varphi }_0$ , $m$ , $\alpha$ , $\kappa$ , 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, this deterministic scalar field model challenges the conventional reliance on probabilistic interpretations in quantum mechanics and opens up new avenues for achieving unprecedented precision in theoretical predictions. Future research will focus on refining the model, extending its application to other quantum phenomena, and conducting detailed experimental validations to fully establish its advantages over traditional approaches.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Planck, M. (1901) Ueber das Gesetz der Energieverteilung im Normalspectrum. Annalen der Physik, 309, 553-563. https://doi.org/10.1002/andp.19013090310
[2]	Einstein, A. (1916) Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 354, 769-822. https://doi.org/10.1002/andp.19163540702
[3]	Hawking, S.W. (1972) Black Holes in General Relativity. Communications in Mathematical Physics, 25, 152-166. https://doi.org/10.1007/bf01877517
[4]	Misner, C.W., Thorne, K.S. and Wheeler, J.A. (1973) Gravitation. Freeman.