A Sub-Quantum Theory: Theoretical and Experimental Proposal

1. Reinterpreting Locality: Malus’s Law and Bell’s Inequalities

1.1. The Standard Quantum Postulate

Quantum physics interprets the violation of Bell’s inequalities [4,6] as the irrefutable proof of non-locality non-locality [2]. According to this view, the existence of a correlation stronger than what classical mechanics would allow implies that particles must “communicate” instantaneously (faster than the speed of light) or be “non-local states” to coordinate their measurement outcomes, even when separated by great distances.

1.2. The Locality Argument (Malus’s Law)

The argument is based on the fact that the standard interpretation ignored the full scope of local probabilistic laws, notably Malus’s Law, when applied to pairs of particles produced with an identical orientation (polarization). Experiments on Bell’s inequalities can be reinterpreted as only demonstrating the reality of Malus’s Law and that we can create identically oriented (polarized or anti-polarized) particles simultaneously, but without any long-distance interaction.

This perspective finds strong formal support in the work of K. Wódkiewicz [18], who demonstrated that the quantum Malus’s law shares the exact mathematical structure of a local hidden-variable theory. However, whereas Wódkiewicz is constrained to introduce negative quasi-probabilities to fit the standard formalism, our approach resolves this contradiction. By establishing a physical equivalence between parallel and sequential measurements via the Statistical Identity of doublets, it allows for the direct application of the strictly positive classical Malus’s law ($0\le p\le 1$), reproducing the exact quantum correlations without any mathematical anomalies.

1.3. Experimental Scenario (4 Detectors: $a_1,b_1$ and $a_2,b_2$)

Consider non-polarized source modeled by a uniform distribution $\rho \left(\lambda \right)={\displaystyle \frac{1}{2\pi }}$, emitting two identical particles photons (identical responses to the same measurements but strictly local, without any distance-related influence) sent to two measurement arms. Photon 1 passes through $a_1$ then $b_1$. Photon 2 passes through $b_2$ then $a_2$. $\mathrm{The}\mathrm{polarizer}\mathrm{angle}{a}_1=a_2\mathrm{and}\mathrm{the}\mathrm{polarizer}\mathrm{angle}{b}_1=b_2\mathrm{throughout}\mathrm{the}\mathrm{experiment}.$ We assume we remove detector $a_2$ and $b_1$.

For Photon 1 (passing through $a_1$ then $b_1$), the local probabilities are:

• Probability to pass $a_1$:

  $$P(a_1+)={\int }_0^{2\pi }{cos}^2(\lambda -a_1)\rho \left(\lambda \right)d\lambda ={\displaystyle \frac{1}{2}}$$
• Probability to pass $b_1$ given that it passed $a_1$ (Malus’s Law):

  $$P(b_1+|a_1+)={cos}^2(\Delta {\theta }_{b_1,a_1})$$
• Joint probability to pass both (sequential $a_1$ then $b_1$):

  $$P(a_1+,b_1+)=P(a_1+)\times P(b_1+|a_1+)={\displaystyle \frac{1}{2}}{cos}^2(\Delta {\theta }_{b_1,a_1})$$
• Joint probability to pass $a_1$ but be blocked by $b_1$:

$$P(a_1+,b_1-)=P(a_1+)\times P(b_1-|a_1+)={\displaystyle \frac{1}{2}}{sin}^2(\Delta {\theta }_{b_1,a_1})$$

Note: We apply Malus’s law photon by photon to avoid confusion between global transmission probabilities and single-photon passage probabilities. The passage of a photon through $a_1$ and $b_1$ (in any sequence) thus reveals the pre-existing passage probabilities for its twin through $b_2$ and $a_2$ (also in any sequence), without any possibility to influence it.

For Photon 2 (passing through $b_2$ then $a_2$), the local probabilities are:

$$P(b_2+)={\displaystyle \frac{1}{2}},P(b_2-)={\displaystyle \frac{1}{2}},P(b_2+|a_2+)={cos}^2(\Delta {\theta }_{a_2,b_2}),P(b_2+|a_2-)={sin}^2(\Delta {\theta }_{a_2,b_2})$$

$$P(b_2+,a_2+)={\displaystyle \frac{1}{2}}{cos}^2(\Delta {\theta }_{a_2,b_2}),P(b_2+,a_2-)={\displaystyle \frac{1}{2}}{sin}^2(\Delta {\theta }_{a_2,b_2})$$

Inter-Photon Correlation ($a_1$ vs $b_2$): The calculation of the correlation $\mathbf{E}({`}_{{\mathbf{a}}_{\mathbf{1}}},{`}_{{\mathbf{b}}_{\mathbf{2}}})$ depends only on the settings of the first detector of Photon 1 (${\mathbf{a}}_{\mathbf{1}}$) and the first detector of Photon 2 (${\mathbf{b}}_{\mathbf{2}}$), and the identical nature of their initial state $\lambda$. We have the trigonometric identities:

$${cos}^2({\theta }_{b_2}-{\theta }_{a_2})={cos}^2({\theta }_{a_2}-{\theta }_{b_2}),{sin}^2({\theta }_{b_2}-{\theta }_{a_2})={sin}^2({\theta }_{a_2}-{\theta }_{b_2}).$$

Numerically, the probability values are therefore the same regardless of polarizer order $(a_1=a_2,b_1=b_2)$ :

$$P(a_1+,b_1+)=P(a_1+,b_2+)=P(b_2+,a_2+)=P(b_2+,a_1+)={\displaystyle \frac{1}{2}}{cos}^2(\Delta \theta )\mathrm{and}$$

$$P(a_1+,b_1-)=P(a_1+,b_2-)=P(b_2+,a_2-)=P(b_2+,a_1-)={\displaystyle \frac{1}{2}}{sin}^2(\Delta \theta ).$$

so we have : $P(a_1+,b_2-)=P(a_2+,b_2-)=P(b_2-,a_2+)={\displaystyle \frac{1}{2}}{sin}^2(\Delta \theta ),andP(b_2-)=1/2,$ $soP(b_2-,a_2-)={\displaystyle \frac{1}{2}}{cos}^2(\Delta \theta )$

Thus, we have the equalities for identical particles:

$$P(a_1+,b_2+)=P(a_1-,b_2-)={\displaystyle \frac{1}{2}}{cos}^2(\Delta \theta )P(a_1+,b_2-)=P(a_1-,b_2+)={\displaystyle \frac{1}{2}}{sin}^2(\Delta \theta )$$

Correlation Calculation:

$$E({\theta }_{a_1},{\theta }_{b_2})=p(++)+p(--)-p(+-)-p(-+)$$

$$=\left({\displaystyle \frac{1}{2}}{cos}^2\Delta \theta +{\displaystyle \frac{1}{2}}{cos}^2\Delta \theta \right)-\left({\displaystyle \frac{1}{2}}{sin}^2\Delta \theta +{\displaystyle \frac{1}{2}}{sin}^2\Delta \theta \right)={cos}^2\Delta \theta -{sin}^2\Delta \theta$$

$$\begin{array}{|c|}\hline E({\theta }_{a_1},{\theta }_{b_2})=cos(2\Delta \theta )\\ \hline\end{array}$$

With the standard CHSH combination: $S=E(a,b)+E(a,b^{\prime })+E(a^{\prime },b)-E(a^{\prime },b^{\prime })$

and the angles $a=0$, $a^{\prime }={\displaystyle \frac{\pi }{4}}$, $b={\displaystyle \frac{\pi }{8}}$, $b^{\prime }={\displaystyle \frac{7\pi }{8}}$: $\begin{array}{|c|}\hline \left|S\right|=2\sqrt{2}>2\\ \hline\end{array}$

Removing the two duplicate polarizers from $b_1$ and $a_2$ that are behind the first polarizers in the paths will not create non-locality, and it will preserve realism and setting independence, while preserving the same probability of Bell’s violation. This observation fundamentally calls into question all interpretations that Bell’s violations could have proven, such as entanglement (non-locality) or non-realism.

Conclusion: any system with a single particle or twin particles with statistical identity satisfying perfect marginal equilibrium must inevitably lead to CHSH violations.

1.4. Statistical Identity

Our theoretical framework unifies quantum phenomena under the principle of statistical identity, with profound implications for understanding EPR correlations:

- Bell violations prove statistical identity, not non-locality (Entanglement).

- Quantum correlations emerge from classical geometric constraints.

Statistical identity means that two particles share exactly the same probabilistic properties, without any influence at a distance:

- Same probability distribution $\rho \left(\lambda \right)$ - Same response $A(\theta ,\lambda )$ to any measurement $\theta$ - Same realization of $\lambda$ for each pair

$${\rho }_1\left(\lambda \right)={\rho }_2\left(\lambda \right)=\rho \left(\lambda \right),A^{\left(1\right)}(\theta ,\lambda )=A^{\left(2\right)}(\theta ,\lambda )=A(\theta ,\lambda )$$

Emissions possibility: Unpolarized emitters ($P(A+)=P(B+)=0.5$):

• Random behavior: (null correlations)
• Correlated behavior: $P(++)=P(--)=\frac{1}{2}{cos}^2(\Delta \theta )$, $P(+-)=P(-+)=\frac{1}{2}{sin}^2(\Delta \theta )$ (correlations $cos(2\Delta \theta )$)
• Anti-correlated behavior: $P(+-)=P(-+)=\frac{1}{2}{cos}^2(\Delta \theta )$, $P(++)=P(--)=\frac{1}{2}{sin}^2(\Delta \theta )$ (correlations $-cos(2\Delta \theta )$)

This result shows that quantum physics predictions are perfectly consistent with locality. If this reinterpretation of Malus’s Law is adopted. Particles do not need to communicate at any distance.

2. Reinterpreting of Superposition: The Interferometer Experiment

2.1. The Standard Quantum Postulate

Standard quantum mechanics interprets the operation of an interferometer (Mach-Zehnder type) by the principle of state superposition. It postulates that a particle entering the device simultaneously places itself in two distinct states (or paths) (e.g., $|\psi \rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\right|0\rangle +|1\rangle )$). The interference fringes at the output are said to result from the interaction of these two superimposed realities when they recombine (the famous "collapse").

2.2. The Realistic Argument (Geometric Transformation)

The realistic approach demonstrates that the interferometer is not a creator of ontological superpositions, but a device performing a geometric rotation of the polarization (or phase) angle, simply transforming the probabilistic distributions according to Malus’s Law.

2.3. Probabilistic Transformation by Geometric Rotation

Consider a polarized source at angle ${\lambda }_0$ and a detector oriented at angle a and $x={\lambda }_0-a$. According to Malus’ law:

$$P(a+)={cos}^2({\lambda }_0-a)={cos}^2\left(x\right)\mathrm{and}P(a-)={sin}^2({\lambda }_0-a)={sin}^2\left(x\right)$$

2.4. Deterministic and Balanced States

The analysis of angles reveals distinct regimes:

• Deterministic States ($x=k\pi$): $P(a+)=1,P(a-)=0$. Maximum tendency toward positive detection.
• Opposite States ($x=(2k+1)\pi /2$): $P(a-)=1,P(a+)=0$. Maximum tendency toward negative detection.
• Balanced States ($x=(2k+1)\pi /4$): $P(a+)=P(a-)=1/2$. Equal probabilities between both tendencies.

Intermediate states exhibit a probabilistic advantage ($P(+)>P(-)$ or $P(-)>P(+)$) without absolute determinism.

2.5. The Interferometer Mechanism

The interferometer physically performs a $\pi /4$ rotation on the angle ${\lambda }_0$, transforming probabilistic distributions:

• Transformation from Extreme States to Balanced Marginals:

  $$\mathrm{Positive}\mathrm{state}(x=k\pi )\stackrel{\pi /4}{\to }x=(2k+1)\pi /4\Rightarrow \mathbf{P}(+)=\mathbf{P}(-)=\mathbf{1}/\mathbf{2}$$

  $$\mathrm{Negative}\mathrm{state}(x=(2k+1)\pi /2)\stackrel{\pi /4}{\to }x=(2k+1)\pi /4\Rightarrow \mathbf{P}(+)=\mathbf{P}(-)=\mathbf{1}/\mathbf{2}$$
• Transformation from Balanced Marginals to Extreme States:

  $$\mathrm{Balanced}(x=(2k+1)\pi /4)\stackrel{+\pi /4}{\to }x=k\pi \Rightarrow \mathrm{Output}+\left(\mathrm{Constructive}\right)$$

  $$\mathrm{Balanced}(x=(2k+1)\pi /4)\stackrel{-\pi /4}{\to }x=(2k+1)\pi /2\Rightarrow \mathrm{Output}\text{-}\left(\mathrm{Destructive}\right)$$

2.6. Reproduction of Quantum Statistics

This geometric rotation mechanism perfectly explains interferometric experiment results: - No state superposition necessary, Purely geometric transformations of probabilistic parameters, and Exact reproduction of predicted quantum statistics.

The interferometer thus appears as a device of probabilistic transformation through rotation, rather than a revealer of ontological superpositions, preserving a totally realistic and local interpretation of quantum phenomena.

3. Reinterpreting Wave-Particle Duality (The Young’s Slits)

ℓ : distance from extended source → plane of the slits, L : distance slits → screen, a : effective width of a slit, b : distance between the centers of the two slits, $\lambda$ : wavelength (or de Broglie wavelength), x : transverse coordinate on the screen.

3.1. Standard quantum-optical derivation (reference model)

This section recalls the standard quantum-optical (or classical wave) derivation of the interference formula, based on the simultaneous superposition of two fields and explicit temporal averaging. It serves as a reference framework against which the event-by-event formulation introduced later will be compared.

We start from two real time-dependent contributions coming from the two “paths”:

$$E_1\left(t\right)=E_{01}cos(\omega t+{\varphi }_1),E_2\left(t\right)=E_{02}cos(\omega t+{\varphi }_2),$$

where ${\varphi }_1,{\varphi }_2$ include the spatial dependence (position x) and the path difference $\Delta \varphi ={\varphi }_2-{\varphi }_1$.

The detector measures a power averaged over a time window long compared with the period $2\pi /\omega$. We denote temporal averaging by ${\langle ·\rangle }_t$. The instantaneous total intensity (real field) is

$$I_{\mathrm{tot}}\left(t\right)={(E_1\left(t\right)+E_2\left(t\right))}^2.$$

Averaging gives:

$$\begin{array}{cc}\hfill I\left(x\right)& ={\langle I_{\mathrm{tot}}\left(t\right)\rangle }_t={〈E_1^2\left(t\right)〉}_t+{〈E_2^2\left(t\right)〉}_t+2{〈E_1\left(t\right)E_2\left(t\right)〉}_t.\hfill \end{array}$$

Computation of the terms (time average) and for the cross term, using the identity $cosAcosB=\frac{1}{2}\left[cos(A-B)+cos(A+B)\right]$ and the vanishing of the fast oscillating terms:

$${〈E_{0i}^2{cos}^2(\omega t+{\varphi }_i)〉}_t={\displaystyle \frac{E_{0i}^2}{2}}\equiv I_i\left(x\right),i=1,2,{〈E_1\left(t\right)E_2\left(t\right)〉}_t={\displaystyle \frac{E_{01}{E}_{02}}{2}}cos({\varphi }_1-{\varphi }_2)=\sqrt{I_1{I}_2}cos\Delta \varphi .$$

Collecting terms:

$$\begin{array}{|c|}\hline I\left(x\right)=I_1\left(x\right)+I_2\left(x\right)+2\sqrt{I_1\left(x\right)I_2\left(x\right)}cos\Delta \varphi \left(x\right)\\ \hline\end{array}$$

To account for partial coherence (extended source, fluctuations, etc.) one introduces the complex degree of coherence ${\gamma }_{12}\left(x\right)$ (with modulus $\le 1$), hence the commonly used form:

$$\begin{array}{|c|}\hline I\left(x\right)=I_1\left(x\right)+I_2\left(x\right)+2{\gamma }_{12}\left(x\right)\sqrt{I_1\left(x\right)I_2\left(x\right)}cos\Delta \varphi \left(x\right)\\ \hline\end{array}$$

Practical remarks:

• the temporal average ${\langle ·\rangle }_t$ removes terms like $cos(2\omega t+\cdots )$;
• for a uniform source at large distance, Van Cittert–Zernike gives $V\left(d\right)=\left|\mathrm{sinc}\left(\frac{\pi bd}{\ell \lambda }\right)\right|=\left|{\gamma }_{12}\left(x\right)\right|$;
• experimentally V is measured by $V=(I_{max}-I_{min})/(I_{max}+I_{min})$.

3.2. Physical Limitations of the Simultaneity Postulate

It should be noted that the temporal average ${\langle ·\rangle }_t$ only removes terms such as $cos(2\omega t+\dots )$ if one assumes that the events occur simultaneously—meaning the probability field passes through both slits at the exact same time. Under this postulate, the formula $I_{\mathrm{tot}}\left(t\right)={(E_1\left(t\right)+E_2\left(t\right))}^2$ is not an independent proof matching experimental results, but rather the logical consequence of the assumption that fields traversing the slits at the same moment must interfere; this is a petitio principii (circular reasoning). While it is logical under this postulate that one cannot simply use $I_{\mathrm{tot}}\left(t\right)=E_1{\left(t\right)}^2+E_2{\left(t\right)}^2$, the situation changes if we assume that passages occur at different times: $E_1\left(t_1\right)=E_{01}cos(\omega t_1+{\varphi }_1)$ and $E_2\left(t_2\right)=E_{02}cos(\omega t_2+{\varphi }_2)$. In such a case, the standard identity ${\langle E_{0i}^2{cos}^2(\omega t+{\varphi }_i)\rangle }_t=E_{0i}^2/2\equiv I_i\left(x\right)$ would no longer be applicable, as the averaging process would involve distinct temporal variables $t_i$ :

$$\begin{array}{|c|}\hline {\langle E_{0i}^2{cos}^2(\omega t+{\varphi }_i)\rangle }_t\ne \frac{E_{0i}^2}{2}\\ \hline\end{array}$$

In the standard postulat, the simultaneity assumption is precisely what allows the fast-oscillating terms to vanish, leaving only the static interference pattern. Consequently, a reformulation is necessary, particularly since recent advancements in “Time-Domain Young’s Slit” experiments e.g., [16] have demonstrated that the temporal delay between paths is a determining physical factor in the formation of the pattern, directly challenging the validity of simple temporal averaging.

3.3. New model: event-by-event

Unlike the standard derivation recalled above, the model developed in this paper does not assume simultaneous coexisting fields nor an a priori temporal averaging. The following sections present an alternative event-by-event formulation where modulation arises from temporal correlations of individual emissions rather than from an $E_1{E}_2$ cross-term.

No temporal averaging is introduced at this stage. The description is strictly event by event.

The contributions associated with the two slits are not assumed simultaneous. We consider two distinct emissions reaching the screen at different times $t_1$ and $t_2$.

$$\begin{array}{c}\hfill E_1\left(t_1\right)=E_{01}cos(\omega t_1+{\varphi }_1),E_2\left(t_2\right)=E_{02}cos(\omega t_2+{\varphi }_2),\end{array}$$

We square each contribution separately at its respective arrival time:

$$\begin{array}{cc}\hfill E_1^2\left(t_1\right)& =E_{01}^2{cos}^2(\omega t_1+{\varphi }_1)={\displaystyle \frac{E_{01}^2}{2}}\left[1+cos(2\omega t_1+2{\varphi }_1)\right],\hfill \end{array}$$

We define the base intensities:

$$\begin{array}{c}\hfill I_{01}\equiv {\displaystyle \frac{E_{01}^2}{2}},I_{02}\equiv {\displaystyle \frac{E_{02}^2}{2}}.\end{array}$$

The total intensity detected for an event pair is the sum of the individual impacts:

$$\begin{array}{c}\hfill \begin{array}{|c|}\hline I_{\mathrm{new}}(x,t_1,t_2)=I_1\left(x\right)\left[1+cos(2\omega t_1+2{\varphi }_1\left(x\right))\right]+I_2\left(x\right)\left[1+cos(2\omega t_2+2{\varphi }_2\left(x\right))\right].\\ \hline\end{array}\end{array}$$

The simplified form (modulated product):

$$\begin{array}{c}\hfill \begin{array}{|c|}\hline I_{\mathrm{new}}(x,T_{\mathrm{sum}},\Delta t)\simeq (I_1\left(x\right)+I_2\left(x\right))\left[1+cos(\omega T_{\mathrm{sum}}+{\varphi }_{\mathrm{sum}})cos(\omega \Delta t+\Delta \varphi )\right].\\ \hline\end{array}\end{array}$$

This formulation explicitly separates:

• diffraction (${\mathrm{sinc}}^2$),
• temporal modulation $V\left(T_{\mathrm{sum}}\right)=cos(\omega T_{\mathrm{sum}}+{\varphi }_{\mathrm{sum}})$,
• relative phase including geometric phase $\Delta \varphi \left(x\right)$ and temporal shift $\omega \Delta t$.

3.4. Particular symmetric case $I_{01}=I_{02}=I_0$

$$\begin{array}{c}\hfill I(t_1,t_2)=2I_0+I_0\left[cos(2\omega t_1+2{\varphi }_1)+cos(2\omega t_2+2{\varphi }_2)\right].\end{array}$$

Using the trigonometric identity $cosa+cosb=2cos\left({\displaystyle \frac{a+b}{2}}\right)cos\left({\displaystyle \frac{a-b}{2}}\right)$, and defining the sum and difference of times $T_{\mathrm{sum}}=t_1+t_2$ , $\Delta t=t_1-t_2$ and ${\varphi }_{\mathrm{sum}}={\varphi }_1+{\varphi }_2$ we obtain:

$$\begin{array}{c}\hfill \begin{array}{|c|}\hline I(t_1,t_2)=2I_0+2I_{0}cos(\omega T_{\mathrm{sum}}+{\varphi }_{\mathrm{sum}})cos(\omega \Delta t+\Delta \varphi ).\\ \hline\end{array}\end{array}$$

This result is obtained without interaction between fields and without any $E_1{E}_2$ cross-term.

3.5. Role of time and randomness

In the single-slit case, a single phase term $cos(\omega t+\varphi )$ appears, whose mean $\langle cos\rangle$ can cancel. In the double-slit case, the two conjugate terms $cos(2\omega t_1+2{\varphi }_1)$ and $cos(2\omega t_2-2{\varphi }_2)$ preserve a relative structure protected by the geometric offset.

$$\begin{array}{ccc}\hfill cos(2\omega t_1+2{\varphi }_1)=0& \Rightarrow 2\omega t_1+2{\varphi }_1={\displaystyle \frac{\pi }{2}}+n\pi ,cos(2\omega t_2+2{\varphi }_2)=0\hfill & \hfill \Rightarrow 2\omega t_2+2{\varphi }_2={\displaystyle \frac{\pi }{2}}+m\pi .\end{array}$$

These conditions cannot be satisfied statistically under temporal noise. Thus, randomness cannot suppress the whole modulation.

3.6. Conclusion

In a single slit, the absence of a relative structure makes the modulation fully vulnerable to temporal noise. In the double slit, the temporary term introduces a robust relative structure, allowing fringes to exist, at least transiently, even in the presence of temporal randomness. Simultaneity is not required; it is the relative geometry of the paths that protects the spatial modulation.

4. Model Validation via Statistical Classical Mechanics

4.1. Virial Relation and Statistical Force Balance

The Virial Theorem establishes a general statistical relation between kinetic and potential energies for bound systems. For a central Coulomb potential $V\left(r\right)\propto 1/r$, it reads

$$2\langle T\rangle =-\langle V\rangle ,$$

where brackets denote ensemble or time averages. Using classical expressions, this relation becomes :

$$〈\frac{1}{2}m{v}^2〉=〈\frac{1}{2}{\displaystyle \frac{e^2}{4\pi {ϵ}_{0}r}}〉.$$

Rewriting it in radial form yields the equivalent statistical identity

$$〈{\displaystyle \frac{m{v}^2}{r}}〉=〈{\displaystyle \frac{e^2}{4\pi {ϵ}_0{r}^2}}〉,$$

which formally corresponds to an equality between average centrifugal and electrostatic forces. This relation characterizes a condition of statistical equilibrium and does not rely on the existence of well-defined particle trajectories.

Interpretative Implications.

The emergence of such equilibrium relations from purely statistical considerations highlights a key conceptual point: constraints governing ensemble averages can reproduce structures often interpreted as intrinsically quantum. When these statistical relations are implicitly promoted to ontological statements about individual particles or single events, interpretative paradoxes naturally arise.

Within the present framework, quantum mechanical formalisms are viewed as providing accurate statistical predictions for measurement outcomes, while individual physical events remain local and realist. The appearance of non-locality, superposition, or indeterminism is thus traced back to the attribution of physical reality to quantities that are fundamentally statistical in nature.