Обсуждение участника:Touol/Possible Cancellation of Indeterminism in QM

Mathematical Clarification

Question. What is the mathematical model of the stationary phase principle in the effective particle approach? First, we need to determine whether the stationary phase arises from interference of different paths or from some averaging at the "observer". In the limit of effective particle paths, there are only 2 paths and they do not suppress each other due to multiple paths back and forth with different phases. That is, it makes sense to consider averaging. We cannot average over coordinates, since there are no assumptions about the function of coordinates. It makes sense to check time averaging. The question is what averaging period to take? The periods of the 2 paths are different. We can take the larger period and average over it. We can take infinite integration time. In theory, the result may not depend on the integration time greater than the slow period. This needs to be checked. Additionally, the question arises: what physical characteristics should the observer system have in order to average over time.

Let's check the averaging ⟨P⟩\_T = (1/T) · ∫\[0 to T\] P(t) dt

= |A\_A|² + |A\_B|² + (2·|A\_A|·|A\_B|/T) · ∫\[0 to T\] cos(Δω·t + φ) dt

Cosine integral ∫\[0 to T\] cos(Δω·t + φ) dt = \[sin(Δω·T + φ) - sin(φ)\] / Δω

When T ≫ T\_beat = 2π/Δω ⟨P⟩\_T ≈ |A\_A|² + |A\_B|² Interference is suppressed! When T ≫ T\_beat the result **does not depend** on the exact value of T!

However, surprise, with time averaging we have suppression of interference, but no suppression of one of the paths. This caused some degradation of the idea. Suppression of one of the paths is possible if the "observer" is in resonance with the frequency of the path of one of the detectors. But it is doubtful that an arbitrary "observer" can have resonance with such high frequencies. And it is doubtful that all observers have one specific frequency.

Let's check the interference of 2 bundles of paths from the detectors.

Interference of bundles of classical paths from detectors

- Detector A in a definite state |ψ\_A⟩:**

From detector to observer — multiple classical paths
Path 1: through point r₁, time t₁
Path 2: through point r₂, time t₂
...
Path N: through point rₙ, time tₙ

- Each path has its own action S\_i!**

Feynman path integral for effective particle

⟨|Ψ\_A|²⟩ ∝ 1/ω\_A² ⟨|Ψ\_B|²⟩ ∝ 1/ω\_B²

- If ω\_A < ω\_B (lower frequency):**

⟨|Ψ\_A|²⟩ > ⟨|Ψ\_B|²⟩

- The bundle with lower frequency (lower energy) dominates!**

But here's a surprise: the amplitudes decay proportionally to the square of the frequency. This is strange!!! We do get measurement results after all. ω\_A - ω\_B is large, but ω\_A²/ω\_B² is probably around 1, 1/2, or 1/4. Suppression of one amplitude relative to the other does not occur. Either the idea is wrong, or the conditions of the mathematical model I specified are incorrect. Many believe that QM applies to the situation after measurement. Suppression of amplitudes after measurement is generally strange.

Conclusion. At the moment, the idea has degraded :-). Perhaps in the future I will somehow revive it, but clearly not within 2-3 days. For now, I'll rest. And gather strength for new attempts to understand the measurement problem. Usually my ideas don't reach any mathematical models. Simply because no mathematical model can be associated with them. Having some mathematical model is already progress :-).

Thank you all for your attention. The comments are too long with all the calculations. I had to post only the summary.