Torsion-Driven Nonlinearity in Spinless Quantum Mechanics
Tomoi Koide, Armin van de Venn
TL;DR
The paper investigates non-relativistic spinless particles in curved spaces with torsion and shows that torsion can induce a logarithmic nonlinearity in the Schrödinger equation. By extending the stochastic variational method to spaces with totally antisymmetric torsion, the authors derive a nonlinear term proportional to s^2 ln|Ψ|^2 with coefficient (ħ^2)/(2m) and map the dynamics to a nonlinear Schrödinger equation iħ ∂t Ψ = [ -ħ^2/(2m) Δ + V + (ħ^2/(2m)) s^2 ln|Ψ|^2 ] Ψ, where s encodes torsion. They also obtain an upper bound on the torsion magnitude from hydrogen-scale considerations and discuss implications for fundamental quantum behavior and cosmological models, suggesting future extensions to relativistic fields. The work links geometric properties of spacetime to quantum fluctuations through a variational framework, offering a mechanism for torsion-driven nonlinear quantum effects beyond spin.
Abstract
We investigate the previously unexplored quantum dynamics of non-relativistic, spinless particles propagating in curved spaces with torsion. Our findings demonstrate that while torsion has been predominantly associated with spin, it can also influence the quantum behavior of spinless particles by inducing a logarithmic nonlinearity in the Schroedinger equation through quantum fluctuations, even in flat space. To facilitate quantization in curved spaces, we introduce a novel stochastic variational method. Unlike canonical quantization, this approach is naturally suited to general coordinate systems, with quantum fluctuations arising from a noise term in the stochastic process that is directly influenced by torsion. By requiring consistency with quantum dynamics, we ultimately derive an upper bound on the magnitude of torsion. Our results reveal a previously unrecognized mechanism by which torsion, as predicted in certain extensions of general relativity, can influence quantum systems, with potential implications for early-universe physics and dark matter or energy models.