This is a serious question that has never been answered before.

Cairo Statistical Theory of Techniques shows a rigorous way to combine classical and quantum physics!

Below we prove beyond doubt how any classical energy density (such as EM energy density, gravitational energy density, etc.) evolves spontaneously and inevitably towards an initial quantum steady state.

By quantum steady state here we mean a stationary solution at ψ^2 (ψ^2=ψ. ψ*) rather than ψ itself and in a secondary step, ψ can be found as the SQRT of the solution.

Note that the quantum vector V1 is the projection of the initial unit space of the energy density situation of classical physics which means,

V1 is a 3D spatial state quantum vector for a classical physical situation of 4D energy density.

V1 is a 1D spatial state quantum vector for a classical 2D initial physical situation, as shown in the presented figures.

Here, the starting conditions of the classical physics situation are:

Infinite free space that has neither BC nor source term.

The classic system is contained within a closed control volume with a classic inward-pointing tangle.

For more clarity, we can compare the formation and explosion of the numerous Big Bangs which are subject to the same physical process.

In the following, we present the correct answer to this question through two illustrative examples A (3D geometric shape) and B (2D geometric shape) [1].

1-Example (A) 3D-Cube discretized into 27 equidistant free nodes as shown in fig.1.

Fig,1-Cube discretized into 27 equidistant free nodes as shown in fig.1.

The statistical theory of Cairo techniques defines the transition matrix B of classical physics (27x27) for the control volume.

B(27x27) for RO=0 (RO=0 means free space) is given by,

.0 1/6   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0

1/6   0 1/6   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0

0 1/6   0   0   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0

1/6   0   0   0 1/6   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0   0   0   0   0   0

0 1/6   0 1/6   0 1/6   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0   0   0   0   0

0   0 1/6   0 1/6   0   0   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0   0   0   0

0   0   0 1/6   0   0   0 1/6   0   0   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0   0   0

0   0   0   0 1/6   0 1/6   0 1/6   0   0   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0   0

0   0   0   0   0 1/6   0 1/6   0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0

1/6   0   0   0   0   0   0   0   0   0 1/6   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0

0 1/6   0   0   0   0   0   0   0 1/6   0 1/6   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0   0   0

0   0 1/6   0   0   0   0   0   0   0 1/6   0   0   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0   0

0   0   0 1/6   0   0   0   0   0 1/6   0   0   0 1/6   0 1/6   0   0   0   0   0 1/6   0   0   0   0   0

0   0   0   0 1/6   0   0   0   0   0 1/6   0 1/6   0 1/6   0 1/6   0   0   0   0   0 1/6   0   0   0   0

0   0   0   0   0 1/6   0   0   0   0   0 1/6   0 1/6   0   0   0 1/6   0   0   0   0   0 1/6   0   0   0

0   0   0   0   0   0 1/6   0   0   0   0   0 1/6   0   0   0 1/6   0   0   0   0   0   0   0 1/6   0   0

0   0   0   0   0   0   0 1/6   0   0   0   0   0 1/6   0 1/6   0 1/6   0   0   0   0   0   0   0 1/6   0

0   0   0   0   0   0   0   0 1/6   0   0   0   0   0 1/6   0 1/6   0   0   0   0   0   0   0   0   0 1/6

0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0   0   0   0   0 1/6   0 1/6   0   0   0   0   0

0   0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0   0   0 1/6   0 1/6   0 1/6   0   0   0   0

0   0   0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0   0   0 1/6   0   0   0 1/6   0   0   0

0   0   0   0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0 1/6   0   0   0 1/6   0 1/6   0   0

0   0   0   0   0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0 1/6   0 1/6   0 1/6   0 1/6   0

0   0   0   0   0   0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0 1/6   0 1/6   0   0   0 1/6

0   0   0   0   0   0   0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0 1/6   0   0   0 1/6   0

0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0 1/6   0 1/6   0 1/6

0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0 1/6   0   0   0   0   0 1/6   0 1/6   0

And the time-independent stationary transfer matrix D(N) for a sufficiently large time N (N is the number of iterations or time jumps dt not to be confused with n the number of free nodes) is given by:

D(N)={1/(I-B) -I}

The numerical values ​​for the inputs of D(N) are,

. . . . . . .  . . . . .Line1- 0.11217139924042430 0.39097856453321000 4.6769216489284893E-2 0.39109945997946949 9.3594337527814389E-2 2.8917725571791415E-2 4.6825121038529496E-2 2.8936273137263536E-2 1.1071538930203321E-2Lin2-  0.39245121669051708 9.3427323090059911E-2 2.8815705697913047E-2 9.4099292136797519E-2 5.7873202288345083E-2 2.2118458029200764E-2 2.9057496590432032E-2 2.2174362578445368E-2 1.0013027641488021E-2

. . .  . . . .

. . . . . . .  .

The matrix D(N) when multiplied by any classical energy density state GT zero will yield the quantum stationary state vector V1 as given below,Line14- 5.9522930308185402E-2 0.11488249791384746 5.7197832143438897E-2 0.11653613666887269 0.45840470771955916 0.11532751395682590 5.8069610117386164E-2 0.11560704495552804 5.7678086265005757E-20.12615384982071848 0.45295668203854378 0.11322885915882226 0.46218312607510498 0.28858852722237699 0.45753292974561188 0.11653613666887269 0.45840470771955916 0.115327513956825900.12138552253472995 7.5702048174876219E-2 5.0296486271624162E-2 0.12615384982071848 0.45295668203854378 0.11322885915882226 5.9522930308185402E-2 0.11488249791384747 5.7197832143438897E-2Etc . . .“To be completed with a double symmetry”

V1=[27/272 37/272 27/272 37/272 25/136 37/272 27/272 37/272 27/272 37/272 25/136 37/272 25/136 67/272 25/136 37/272 25/136 37/272 27/272 37/272 27/272 37/272 25/136 37/272 27/272 37/272 27/272}^T

Note again that:

1- Any classical physical system of energy density contained in free space will evolve towards the quantum state vector provided it is contained in a closed control volume and driven inwards towards the center of mass CM (midpoint MP)

2-From this state it evolves towards another vector depending on the applied external potential V(x,y,z,t) included in the Schrödinger equation (central potential field, finite or infinite potential well, etc.).

3- The quantum vector V1 is the projection of the initial unit space of the energy density situation of classical physics which means,

V1 is a 3D spatial state quantum vector for a classical physical situation of 4D energy density.

V1 is a 1D spatial state quantum vector for a classical 2D initial physical situation, as shown in the presented figures.

1-Useless Math -The Complex Untold Story

• International Journal of Innovative Science and

Research Technology ,

• October 2024
• DOI:
• 10.38124/ijisrt/IJISRT24OCT1091

Dear colleagues,

The idea that quantum transition matrix models could be projections of classical statistical physics models is intriguing and has been explored in several frameworks, particularly in statistical mechanics and renormalization group methods. However, Pure Time Theory (PTT) suggests that the relationship between classical and quantum physics is not merely a geometric projection, but rather a consequence of how time structures the evolution of physical states.

---

1. The Projection Hypothesis: A Geometric or Temporal Effect?

The proposal that 2D quantum mechanics models are projections of 3D classical statistical models is based on the idea that Hilbert space representations in QM could emerge from a higher-dimensional classical framework.

However, this assumes that quantum phenomena are purely geometric projections of classical structures, rather than emerging from a deeper temporal evolution process.

PTT suggests an alternative:

Quantum mechanics does not emerge from classical physics via simple dimensional reduction. Instead, the transition from classical to quantum descriptions occurs due to time-based constraints on information accessibility.

The reason quantum states appear "projected" from a statistical mechanics framework may be because quantum systems are governed by local relaxation times , which dictate how probabilistic distributions evolve.

---

2. PTT’s View: Quantum Transition Matrices as Temporal Structures

Rather than viewing quantum transition matrices as static projections of classical models, PTT suggests they emerge from the time-dependent nature of quantum systems:

🔹 Quantum transitions are structured by the relaxation dynamics of time, not just by statistical projections.

Classical statistical mechanics treats probabilities as evolving deterministically over time.

Quantum mechanics introduces an additional constraint: the evolution of probabilities is tied to local time distortions, not just statistical averaging.

This explains why quantum probabilities do not behave like classical probabilities—they evolve based on a structured, non-uniform time metric rather than a fixed, universal clock.

🔹 Hilbert Space as an Emergent Structure

In classical physics, phase space evolution is smooth and deterministic.

In quantum mechanics, Hilbert space introduces non-commutativity and complex amplitudes.

PTT suggests that this is not because QM is a "projection" of classical physics, but because quantum evolution depends on an underlying structured time fabric.

---

3. Testable Predictions from PTT

If quantum mechanics is not just a projection of classical statistical physics but instead an emergent consequence of time evolution constraints, we should see:

✅ Temporal correlations in quantum transitions that cannot be explained by statistical mechanics alone.

✅ Quantum interference effects that depend on the local relaxation time , rather than just phase-space symmetries.

✅ Deviations from classical statistical expectations in systems with highly controlled time dynamics (e.g., in optical lattices or ultracold atomic gases).

---

4. Conclusion: A Deeper Connection Between Classical and Quantum Physics?

While quantum mechanics and classical statistical physics share deep mathematical similarities, PTT suggests that the true bridge between them is not just projection, but time.

Rather than treating QM as a lower-dimensional projection of classical mechanics, we should view quantum states as dynamically constrained by the evolution of time itself.

This approach preserves the insights from statistical mechanics while offering a new way to understand quantum transitions—one based on time-structured evolution rather than geometric projection alone.

Would love to hear your thoughts on whether temporal constraints could explain the connection between classical and quantum physics better than pure statistical projection models!

Best regards,

Essam Allou