How Quantum Traction Theory Derives the Principle of Least Action

https://doi.org/10.5281/zenodo.17594186 (reference)

Why does Nature seem to “prefer” the path of least action? Most textbooks quietly start from this as an axiom:

A physical system evolves along the path that makes the action S stationary (usually “least”).

Quantum Traction Theory (QTT) takes a very different stance: it does not accept the principle of least action as a primitive rule of the universe. Instead, it shows how “least action” emerges from deeper ingredients:

• An internal phase dial attached to each piece of reality
• Additivity of that dial’s rotation
• A finite capacity for how much “quantum stuff” can be represented at once

Out of this, the familiar statement

δS = 0  ⇒  Euler–Lagrange equations

appears not as a decree, but as a consequence.

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1. The hidden dial behind every quantum system

In QTT, every fundamental “world-cell” – the basic locus where reality can live – carries an internal phase dial, a little U(1) clock hand that can rotate. This dial is not a metaphor. It encodes the phase that shows up in quantum amplitudes.

Two simple rules fix how this dial behaves:

1. Additivity: If you follow a history from A to B, then B to C, the total dial rotation from A to C is the sum of the rotations on each segment.
2. Loop consistency: If you go around a closed loop in configuration space and come back to the same physical state, predictions can depend only on the total angle you’ve turned the dial by (e.g. a holonomy), not on how you sliced the loop.

These conditions force the existence of a real-valued functional along a path, call it S_tot, such that the dial angle changes in proportion to it. QTT expresses this as the Action–Phase Law:

dθ = dS_tot/ℏ

Integrating along a history γ from time t₀ to t₁ gives

θ[γ] = S_tot[γ]/ℏ,

S_tot[γ] = ∫_t0^t₁ L_tot(x, \dot x, t)\,dt.

Here L_tot is the total Lagrangian (mechanical, gauge, plus any geometric/Berry pieces). The key point is conceptual:

In QTT, “action” is not guessed; it is whatever quantity must generate the phase of the internal dial.

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2. From dial rotations to path weights

Once the dial’s phase along a path is fixed, the quantum amplitude attached to that path is essentially forced by three requirements:

• Dial rotations compose additively in the exponent
• Amplitudes for successive segments multiply
• Evolution is unitary (phases live on the unit circle)

Put together, that leaves the familiar weight:

&mathcal;A[γ] = e^{iθ[γ]} = \exp\big(i S_tot[γ]/ℏ\big).

QTT calls this the Path–Phase Law:

Each kinematically allowed path contributes an amplitude whose phase is the total action (in units of ℏ) recorded by the internal dial along that path.

Summing over all admissible paths from an initial point (x₀, t₀) to a final point (x₁, t₁) gives:

Ψ(x₁, t₁) ∝ ∑_γ \exp\big(i S_tot[γ]/ℏ\big).

This looks like the usual Feynman path integral, but the logic is reversed: QTT doesn’t assume “sum over histories with e^{iS/ℏ}”; it derives the weight from the dial.

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3. Finite capacity: not all paths are equally real

Standard quantum theory formally speaks of “all paths.” QTT adds a physical constraint that changes the story:

Reality has a finite quantum capacity per world-cell.

That means:

• The ledger of possible histories is discrete and capacity-limited.
• Extremely wild, high-frequency, wildly curving paths are doubly disfavored:
  • They contribute rapidly oscillating phases that tend to cancel in interference.
  • They are inefficient in capacity: encoding them eats up “space” in the ledger that could support smoother, more coherent histories.

The macroscopic histories that actually survive and show up as “classical trajectories” are those that:

1. Don’t self-destruct by destructive interference, and
2. Can be represented efficiently within the finite capacity budget.

In practice, those are the histories that live in narrow tubes where the action changes only very slightly under small deformations of the path.

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4. Stationary action as the classical shadow

Now consider a system where the typical scale of action is huge compared to ℏ:

S_tot » ℏ.

In that regime:

• If we slightly vary a path γ to a nearby path γ + δγ, the action changes by ΔS, and the phase changes by Δθ = ΔS/ℏ.
• Unless ΔS is very small, nearby paths acquire wildly different phases and cancel when you sum over them.

The mathematically precise version of this is a stationary phase argument: for the sum of contributions to add up constructively, the action functional must be stationary under small variations:

δS_tot = 0.

Provided you fix the endpoints in time, this condition leads directly to the Euler–Lagrange equations for L_tot:

\frac{d}{dt}\left(\frac{\partial L_tot}{\partial \dot x}\right) – \frac{\partial L_tot}{\partial x} = 0.

So in QTT, the logical chain is:

1. Dial rules ⇒ Action–Phase Law (dθ = dS/ℏ)
2. Unitarity ⇒ Path–Phase Law (amplitude ∝ e^{iS/ℏ})
3. Finite capacity + interference ⇒ suppress wildly varying paths
4. Surviving macroscopic histories satisfy δS = 0

That last line is the principle of stationary (or “least”) action. But now we see it as a result, not an axiom.

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5. What’s new in the QTT view of Universe??

Formally, the classical equations of motion look the same as in ordinary mechanics:

• The particle still follows a geodesic in the appropriate geometry. – w as Reality Dimension and Dial Center included.
• The field equations are still obtained by varying an action functional.

What QTT adds is a mechanism:

• The action is nothing more or less than the generator of the internal dial’s phase.
• The path weight e^{iS/ℏ} is forced by the composition rules of that dial and unitarity.
• The “preference” for stationary action paths comes from the combination of destructive interference and a hard capacity budget for what histories can coexist.

In other words:

The universe is not “lazy” in a poetic sense; it is phase-coherent under capacity constraints. The principle of least action is the classical shadow of that deeper reality:

Reality Dimension.