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

TL;DR

This work investigates how spatial torsion in Metric-Affine Gravity alters quantum fluctuations when quantization is performed via the Stochastic Variational Method. By formulating SVM on curved spaces with torsion using vielbeins and forward/backward stochastic processes, it derives a non-linear Schrödinger equation whose log-nonlinearity is governed by the competition between Levi-Civita curvature and torsion. The analysis shows that torsion can affect even spinless particles and yields a framework where curvature and torsion inequivalently influence quantum dynamics, challenging the classical geometrical trinity. It also highlights deep connections between SVM and information geometry through dual connections and non-metricity, suggesting avenues for extending these ideas to non-mmetric theories and cosmology.

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 Schrödinger 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.

Figures (5)

• Figure S1: Schematic relationship between gauge symmetry, principal bundles, and associated vector bundles (adapted from Hamilton2017). Lie groups define the gauge symmetry and act via representations on vector spaces, giving rise to associated vector bundles that model matter fields. A principal bundle encodes the gauge symmetry geometrically, while a connection on the principal bundle represents the gauge field. This connection induces covariant derivatives on associated bundles, describing the coupling between gauge fields and matter fields.
• Figure S2: An example of the typical trajectory of Brownian motion. The positions at $t- \mathrm{d} t$, $t$ and $t+ \mathrm{d} t$ are denoted by $r^{i}(t-\mathrm{d} t)$, $r^{i}(t)$ and $r^{i}(t+\mathrm{d} t)$ , respectively.
• Figure S3: The ensembles of trajectories fixing ${\cal P}_t$ and ${\cal F}_t$ shown in the left and right panels, respectively. This figure is taken from Fig. 2 of Ref. U_Goncalves_2020.
• Figure S4: Sample stochastic trajectories generated by the numerical simulation of the double-slit experiment within the framework of SVM. The trajectories represent the zigzag motion of Brownian particles governed by the forward stochastic differential equation (\ref{['eqn:fsde']}). The velocity field for the simulation is derived from the analytical solution to the free Schrödinger equation with an initial wave function given by Eq. (\ref{['eqn:ini_wf']}), which describes two Gaussian wave packets centered at the slits ($y=\pm d$). Five sample trajectories are shown originating from the vicinity of each slit. The simulation uses units where $\hbar = 1$, $\mathrm{M} = 1$, and time is measured with scale $\tau_0 = \mathrm{M} d^2 / \hbar$. This figure is adapted from Fig. 1 of Ref. U_Goncalves_2024.
• Figure S5: Numerical validation of SVM by comparing the particle distribution in a double-slit experiment. Left panel: The exact quantum mechanical probability density $|\Psi(y,t)|^2$ at $T = t/ \tau_0 = 5$ with $\tau_0 = \mathrm{M} d^2 / \hbar$, calculated from the analytical solution of the free Schrödinger equation. The characteristic interference fringes are clearly visible. Right panel: The particle distribution at the same time $T = t/ \tau_0 = 5$ shown as a histogram, generated by numerically simulating 10,000 individual Brownian trajectories with $\mathrm{d} t=0.001$. Each trajectory evolves according to the forward SDE (\ref{['eqn:fsde']}), guided by ${\bf u}_+({\bf x},t)$ derived from the wave function. The agreement between the histogram and the analytical curve demonstrates that the ensemble of stochastic trajectories correctly reproduces the quantum interference pattern, thus validating the SVM approach. These figures are adapted from Fig. 2 of Ref. U_Goncalves_2024.