Torsion-Induced Quantum Fluctuations in Metric-Affine Gravity using the Stochastic Variational Method

TL;DR

This review explores how torsion in Metric-Affine Gravity influences quantum fluctuations using the Stochastic Variational Method, revealing torsion's impact on spinless quantum degrees of freedom and the non-linearity of quantum mechanics.

Contribution

It integrates MAG and SVM frameworks to analyze torsion's effects on quantum fluctuations, extending previous work and highlighting the role of torsion beyond spin interactions.

Findings

Torsion induces non-linearity in quantum mechanics.

Torsion affects spinless degrees of freedom via quantum fluctuations.

The interplay between curvature and torsion influences the Schrödinger equation.

Abstract

This review paper comprehensively examines the influence of spatial torsion on quantum fluctuations from the perspectives of Metric-Affine Gravity (MAG) and the Stochastic Variational Method (SVM). We first outline the fundamental framework of MAG, a generalized theory that includes both torsion and non-metricity, and discuss the geometrical significance of torsion within this context. Subsequently, we summarize SVM, a powerful technique that facilitates quantization while effectively incorporating geometrical effects. By integrating these frameworks, we evaluate how the geometrical structures originating from torsion affect quantum fluctuations, demonstrating that they induce non-linearity in quantum mechanics. Notably, torsion, traditionally believed to influence only spin degrees of freedom, can also affect spinless degrees of freedom via quantum fluctuations. Furthermore, extending beyond the results of previous work [Koide and van de Venn, Phys. Rev. A112, 052217 (2025)], we investigate the competitive interplay between the Levi-Civita curvature and torsion within the non-linearity of the Schr\"odinger equation. Finally, we discuss the structural parallelism between SVM and information geometry, highlighting that the splitting of time derivatives in stochastic processes corresponds to the dual connections in statistical manifolds. These insights pave the way for future extensions to gravity theories involving non-metricity and are expected to deepen our understanding of unresolved cosmological problems.