[Paper Review] The Systematic Measurement Errors and Uncertainty Relation

T. F. Kamalov|arXiv (Cornell University)|Nov 5, 2006

Scientific Measurement and Uncertainty Evaluation3 citations

TL;DR

This paper proposes that systematic measurement errors in non-inertial reference frames—arising from neglected higher-order time derivatives of position—can account for quantum uncertainty relations, not just quantum indeterminacy. By applying Ostrogradski’s Canonical Formalism to include higher derivatives in the Lagrangian, the model derives a Schrödinger-like equation, suggesting that quantum uncertainty may stem from reference frame imperfections rather than intrinsic quantum axioms.

ABSTRACT

Inertial effects in non-inertial reference frames are compared with quantum properties of tests objects. The real space-time and perfect inertial reference frame can be compared accurate to the uncertainty relation. Complexities if describing micro-object in non-inertial reference-frames are avoidable using Ostrogradski's Canonical Formalism.

Motivation & Objective

• To explain the origin of quantum uncertainty not solely through quantum postulates but through systematic errors in real (non-ideal) reference frames.
• To address the limitations of Newtonian mechanics in describing motion in non-inertial frames due to neglected higher-order time derivatives.
• To reformulate classical dynamics using Ostrogradski’s Canonical Formalism to include higher derivatives and thus model inertial forces more accurately.
• To derive a quantum potential term Q from higher-derivative contributions, linking it to the Schrödinger equation in the first approximation.
• To propose that quantum-like behavior, including uncertainty relations, may emerge from classical dynamics in imperfect reference frames.

Proposed method

• Expands the position function $ r(t) $ as a Taylor series including all higher-order time derivatives beyond acceleration.
• Defines $ q_r $ as the hidden variables representing neglected higher-derivative terms, distinguishing real motion from Newtonian approximation.
• Introduces a difference in action $ S - S_{\text{Newton}} = nh $, where $ h $ is an upper bound of hidden action, linking to uncertainty.
• Applies Ostrogradski’s formalism to Lagrangians depending on higher-order derivatives $ \overset{\cdot(n)}{r} $, leading to generalized Euler-Lagrange equations.
• Derives a modified Jacobi-Hamilton equation with an effective quantum potential $ Q = \sum p^{(n)} \overset{\cdot(n)}{a} $, representing non-inertial effects.
• Shows that in the first approximation, $ Q \approx \frac{i\hbar m}{2} \frac{\nabla^2 S}{m^2} $, leading to the Schrödinger equation for $ \psi = e^{iS/\hbar} $.

Experimental results

Research questions

• RQ1 Can systematic measurement errors in non-inertial reference frames account for the uncertainty relation in quantum mechanics?
• RQ2 How do higher-order time derivatives of position contribute to deviations from Newtonian mechanics in real reference frames?
• RQ3 To what extent can Ostrogradski’s Canonical Formalism describe dynamics in non-inertial frames with inertial forces?
• RQ4 What is the role of the quantum potential $ Q $ in unifying classical non-inertial effects with quantum-like behavior?
• RQ5 Is the uncertainty in momentum and position a consequence of reference frame imperfections rather than intrinsic quantum indeterminacy?

Key findings

• Systematic errors from neglecting higher-order derivatives in non-inertial frames lead to a difference in action $ S - S_{\text{Newton}} = nh $, where $ h $ acts as an upper bound for hidden action.
• The model derives a Schrödinger equation in the first approximation by identifying the quantum potential $ Q \approx \frac{i\hbar m}{2} \frac{\nabla^2 S}{m^2} $ from higher-derivative contributions.
• The effective quantum potential $ Q $ arises from momenta conjugate to higher derivatives, $ p^{(n)} = \partial L / \partial \overset{\cdot(n+1)}{r} $, and contributes to the total energy.
• The equation of motion in non-inertial frames becomes $ \frac{dS}{dt} = \frac{\partial S}{\partial t} + \frac{(\nabla S)^2}{2m} + Q $, with $ Q $ capturing non-inertial effects.
• The uncertainty in coordinate and momentum is interpreted as a consequence of reference frame imperfections, not solely quantum axioms.
• The model suggests that relitc or static gravitational fields may induce reference frame fluctuations, contributing to measurement systematic errors linked to quantum uncertainty.