Scale Physics – 2

The Scale Schrödinger Equation and the Scale Uncertainty Principle

Let us now proceed by fully accepting the Scale Physics axioms that I have established.

Our main assumption will be:

𝑥 = 𝑥(𝑡, 𝜆)

and

𝑢 = 𝜕𝑥 / 𝜕𝜆

Furthermore, let us define the scale momentum as:

𝑃_𝜆 = −𝑖Σ 𝜕/𝜕𝜆

Here, Σ is the fundamental constant in scale space, analogous to Planck’s constant.

1. Scale Operators

In quantum mechanics, we have:

𝑝̂ = −𝑖ℏ 𝜕/𝜕𝑥

In scale physics, its analog is:

𝑝̂_𝜆 = −𝑖Σ 𝜕/𝜕𝜆

This operator measures the scale momentum.

2. Scale Kinetic Energy

Normal kinetic energy is:

𝐸_k = 𝑝² / 2𝑚

Its scale analog becomes:

𝐸_k = 𝑃_𝜆² / 2𝑀_𝜆

Here, 𝑀_𝜆 is the scale inertia.

3. Scale Hamiltonian

The total energy of the system becomes:

𝐻 = − (Σ² / 2𝑀_𝜆) 𝜕²/𝜕𝜆² + 𝑈(𝜆)

Here, 𝑈(𝜆) is the scale potential.

4. Scale Schrödinger Equation

The standard Schrödinger equation was:

𝑖ℏ (𝜕𝜓/𝜕𝑡) = 𝐻𝜓

In scale physics, if we use scale evolution instead of time:

𝑖Σ (𝜕Ψ/ 𝜕𝜆) = − (Σ²/2𝑀_𝜆) (𝜕² Ψ/𝜕𝑥²) + 𝑈(𝜆)Ψ

The meaning of this equation:

The wave function evolves along the scale, not in time.

5. Full Scale Schrödinger Equation

If both time and scale are present:

𝑖ℏ (𝜕Ψ/𝜕𝑡) + 𝑖Σ (𝜕Ψ/𝜕𝜆) = − (ℏ²/2𝑚) ∇²Ψ − (Σ²/2𝑀_𝜆)(𝜕² Ψ/𝜕𝜆²) + 𝑉Ψ

I believe this is very important for my theory.

Because here:

The first part is time dynamics,

The second part is scale dynamics.

6. Scale Uncertainty

The canonical commutator is defined as:

[𝜆, 𝑃_𝜆] = 𝑖Σ

When Heisenberg’s logic is applied, we obtain:

Δ𝜆Δ𝑃_𝜆 ≥ Σ/2

This is the Scale Uncertainty Principle.

7. Physical Meaning

This equation states that:

We cannot simultaneously know exactly at what scale a structure is and its tendency to change scale with absolute precision.

For example:

If you determine the exact fractal scale of a system, its scale momentum becomes uncertain.

8. Second Uncertainty

Because 𝑃_𝜆 = 𝑀_𝜆 𝑢, we obtain:

Δ𝜆Δ𝑢 ≥ Σ/2𝑀_𝜆

This is a more physical form.

9. Scale Energy Uncertainty

By the same logic, we can write:

Δ𝐸_𝜆Δ𝜆 ≥ Σ/2

This expression means:

The more precisely the scale is defined, the more uncertain the scale energy becomes.

10. The Fundamental Quantum Equation of Fractal Relativity

The entire structure is unified in a single equation:

𝑖ℏ (𝜕Ψ/𝜕𝑡) + 𝑖Σ (𝜕Ψ/𝜕𝜆) = − (ℏ²/2𝑚)∇² Ψ − (Σ²/2𝑀_𝜆) (𝜕²Ψ/𝜕𝜆²) + 𝑉(𝑥, 𝜆)Ψ

We can read this equation as follows:

The Schrödinger equation governs probabilities in space.

The new term governs transitions between scales.

The wave function propagates not only in position but also in scale space.

At this point, the most important new concepts that emerge are actually the scale momentum 𝑃_𝜆 and the scale constant Σ. If we can clarify their physical meanings, the theory can begin to evolve from a mathematical draft into an original physical theory.

Scale momentum $P_\lambda$ and scale constant $\Sigma$. Let’s clarify their physical meanings

I believe we have reached the most critical point of the theory here. Because a theory gains strength not when it defines a new symbol, but when the measurable physical meaning of that symbol is revealed.

First, let’s recall existing physics:

𝑝 = 𝑚𝑣

What is momentum?

It is the tendency of an object to maintain its state of motion.

Meaning, momentum is actually “inertia of motion in space.”

If we carry the same logic to scale space:

𝑃_𝜆 = 𝑀_𝜆 𝑢

Here, 𝑢 = 𝜕𝑥 / 𝜕𝜆.

What is Scale Momentum?

I define it as follows:

𝑃_𝜆 = A system’s resistance to scale change.

Physical meaning:

An atom cannot easily transform into another scale.

A cell cannot suddenly act like a galaxy.

A galaxy cannot suddenly shrink to the atomic scale.

Every structure tries to preserve its own scale.

This tendency of preservation is represented by:

𝑃_𝜆

Fractal Interpretation

In my approach:

If 𝜆 = ln(𝐿/𝐿₀), then 𝑃_𝜆 is the answer to this question:

How much does the structure change when it scales up or down?

For example:

A sphere: 𝑃_𝜆 ≈ 0 is possible.

Because it looks similar across various scales.

But a fractal structure yields: 𝑃_𝜆 ≠ 0.

Because new details emerge at every new scale.

Scale Inertia

Normal inertia: 𝑚 is the resistance to change in motion.

We can define scale inertia as 𝑀_𝜆.

𝑀_𝜆 = Resistance to scale change.

For example:

A crystal structure: high 𝑀_𝜆.

A cloud: low 𝑀_𝜆 is possible.

Now Let’s Come to the Constant Σ

This is even more important.

In quantum mechanics, what does ℏ represent?

Actually: The smallest scale of action in nature.

This is the source of uncertainties.

What is Σ in Scale Physics?

I propose the following definition:

Σ = The smallest meaningful scale transformation.

Meaning:

A system cannot change scale with infinite precision.

Scale is also quantized.

Scale Quantum

In space: Δ𝑥 can be a minimum.

In energy: Δ𝐸 can be a minimum.

In scale: Let Δ𝜆_min ∼ Σ.

In this case:

Scale is not continuous; it can change in very small packets.

Why Does Uncertainty Arise?

Because we set:

[𝜆, 𝑃_𝜆] = 𝑖Σ

From this, it emerges that:

Δ𝜆Δ𝑃_𝜆 ≥ Σ/2

Meaning:

If you determine the exact scale of a structure, you lose its scale transformation tendency.

Connection with Fractal Dimension

In my work, a more interesting interpretation could be:

Σ = 𝑓(𝐷)

Here, 𝐷 is the fractal dimension.

For instance:

Like Σ = Σ₀ (𝐷 − 1).

Thus:

On a straight line, the scale quantum is small.

In high fractality, the scale quantum is large.

Applying to G Measurements

According to my hypothesis:

Let 𝐺 = 𝐺(𝜆).

Consequently, the definition:

𝑃_𝜆 = 𝜕𝐺 / 𝜕𝜆

can also be made.

This provides a very interesting interpretation:

𝑃_𝜆 = A physical constant’s sensitivity to scale.

If 𝑃_𝜆 = 0, the constant is truly constant.

If 𝑃_𝜆 ≠ 0, there is flow in the scale.

Conclusion

I think the most natural physical interpretation of the theory is this:

𝑃_𝜆 = scale momentum = a structure’s tendency to change scale

and

Σ = scale quantum = the smallest physical scale transformation.

In this case, corresponding to Heisenberg’s [𝑥, 𝑝] = 𝑖ℏ relationship, the fundamental commutator of Scale Physics becomes [𝜆, 𝑃_𝜆] = 𝑖Σ, and this relationship sits at the center of the theory. This can be considered the first real fundamental equation that connects my idea of “derivative = scale down, integral = scale up” to a quantum-like mathematical structure.