[Paper Review] Amplification of Nonlocal Effects in Nonlinear Quantum Mechanics by Extreme Localization

George Svetlichny|ArXiv.org|Oct 22, 2004

Quantum Mechanics and Applications2 references3 citations

TL;DR

This paper demonstrates that the Doebner-Goldin nonlinear quantum mechanics—derived from diffeomorphism group representations—exhibits amplified nonlocal signaling effects under extreme spatial localization. At Planck-scale localization, nonlocal effects become comparable in magnitude to linear terms, suggesting they could dominate at high energies and challenge linear quantum gravity models like loop quantum gravity or M-theory.

ABSTRACT

Due to its connection to the diffeomorphism group, nonlinear quantum mechanics may play an important role in quantum geometry. The Doebner-Goldin nonlinearity (arising from representations of the diffeomorphism group) amplifies nonlocal signaling effects under extreme localization, suggesting that even if greatly suppressed at low energies, such effects may be significant at the Planck scale. This offers new perspectives on Planck-scale physics.

Motivation & Objective

• To investigate whether nonlocal effects in nonlinear quantum mechanics become significant at high energies, particularly at the Planck scale.
• To assess the viability of nonlinear quantum mechanics in a relativistic, causal framework by analyzing its behavior under extreme spatial localization.
• To explore the implications of the Doebner-Goldin nonlinearity—motivated by diffeomorphism group representations—for quantum geometry and early-universe cosmology.
• To determine whether nonlocal signaling, typically suppressed at low energies, could become observable at high-energy regimes due to localization-induced amplification.
• To evaluate whether such amplification undermines the foundational assumptions of linear quantum gravity approaches like M-theory and loop quantum gravity.

Proposed method

• Analyzes the Doebner-Goldin (DG) nonlinear Schrödinger equation, which arises from representations of the diffeomorphism group and preserves norm via Im(ψ, Fψ) = 0.
• Uses a Gaussian wave packet approximation with increasing localization (scaling parameter r → ∞) to model extreme spatial confinement.
• Computes the nonlocal signal difference Δ(B,t|A,A') between two measurement outcomes A and A' using the expectation value of observable B after nonlinear evolution.
• Derives the leading-order contribution to nonlocal effects as proportional to r × D_b × ⟨ϕ|B|ϕ⟩, showing linear growth with localization strength r.
• Performs dimensional analysis to compare the nonlinear signal rate to the linear kinetic energy rate, yielding a ratio ∝ νrL², where ν = D_b / (ℏ/2m).
• Estimates the suppression factor ν ≈ 10⁻²⁰ from experimental bounds and evaluates the scale at which nonlocal effects dominate (L ≈ 10⁻²³ cm for ν = 10⁻²⁰).

Experimental results

Research questions

• RQ1 Does extreme spatial localization amplify nonlocal signaling effects in the Doebner-Goldin nonlinear quantum mechanics?
• RQ2 Can nonlocal effects in nonlinear quantum mechanics become significant at the Planck scale despite their suppression at low energies?
• RQ3 How does the amplification of nonlocality in the DG equation compare to other nonlinearities (e.g., Bialynicki-Birula-Myckelski or Kostin)?
• RQ4 What is the effective length scale at which nonlocal effects from Planck-scale processes become dominant?
• RQ5 Does the existence of such nonlocal effects at high energies challenge the assumptions of linear quantum gravity models like M-theory or loop quantum gravity?

Key findings

• Nonlocal signaling effects in the Doebner-Goldin nonlinear quantum mechanics are amplified by a factor of order r (localization strength), making them comparable to linear evolution terms under extreme localization.
• The leading-order nonlocal signal difference scales as Δ₁(B|p,q) ≈ 4rnD_b⟨ϕ|B|ϕ⟩ + O(1), demonstrating that any observable with non-zero expectation in the initial state can detect the effect.
• For localization to the Planck length (r ≈ 1/L_p²), nonlocal effects dominate over linear effects when the de Broglie wavelength L exceeds ~10⁻²³ cm, depending on the suppression factor ν.
• The suppression factor ν ≈ 10⁻²⁰ implies that nonlocal effects become significant for states with wavelengths above ~10⁻²³ cm, suggesting a potential observational window at high-energy scales.
• A new fundamental length scale emerges: L_P / √ν, which could influence dynamics of singularities, dark energy, and inflation if ν is as small as 10⁻¹²⁶.
• The amplification is specific to the diffeomorphism-inspired DG nonlinearity; other nonlinearities (e.g., logarithmic terms) do not exhibit this behavior due to their different asymptotic structure on localized states.