We investigate the previously unexplored quantum dynamics of nonrelativistic, 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 Schr\"odinger equation through quantum fluctuations, even in flat space. To facilitate quantization in curved spaces, we introduce an extended 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 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.
We investigate the previously unexplored quantum dynamics of nonrelativistic, 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 Schr"odinger equation through quantum fluctuations, even in flat space. To facilitate quantization in curved spaces, we introduce an extended 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 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.
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.
We review the relation between space-time geometries with torsion fields (the so-called Riemann-Cartan-Weyl (RCW) )geometries) and their associated Brownian motions. In this setting, the metric conjugate of the tracetorsion one-form is the drift vectorfield of the Brownian motions. Thus, in the present approach, Brownian motions, in distinction with Nelson�s Stochastic Mechanics, are spacetime structures. We extend this to the state-space of non-relativistic quantum mechanics and discuss the relation between a noncanonical quantum RCW geometry in state-space associated with the gradient of the quantum-mechanical expectation value of a self-adjoint operator given by the generalized laplacian operator defined by a RCW geometry. We discuss the reduction of the wave function in terms of a RCW quantum geometry in state-space. We characterize the Schroedinger equation for both an observed and unobserved quantum systems in terms of the RCW geometries and Brownian motions. Thus, in this work, the Schroedinger field is a torsion generating field, and the U and R processes, in the sense of Penrose, are associated, the former to spacetime geometries and their associated Brownian motions, and the latter to their extension to the state-space of nonrelativistic quantum mechanics given by the projective Hilbert space. In this setting, the Schroedinger equation can be either linear or nonlinear. We discuss the problem of the many times variables and the relation with dissipative processes. We present as an additional example of RCW geometries and their Brownian motions counterpart, the dynamics of viscous fluids obeying the invariant Navier-Stokes equations. We introduce in the present setting an extension of R. Kiehn�s approach to dynamical systems starting from the notion of the topological dimension of one-forms, to apply it to the trace-torsion one-form whose metric conjugate is the Brownian motion�s drift vectorfield and discuss the topological notion of turbulence. We discuss the relation between our setting and the Nottale theory of Scale Relativity, and the work of Castro and Mahecha in this volume in nonlinear quantum mechanics, Weyl geometries and the quantum potential.
A deterministic quantum-gravity framework is proposed that embeds Bohmian mechanics within the spin–torsion geometry of Einstein–Cartan–Kibble–Sciama theory. Starting from the Palatini form of the action for a Dirac field, a Gordon decomposition followed by a Foldy–Wouthuysen expansion shows that the familiar Bohm quantum potential, namely the term proportional to the Laplacian of the wave-function amplitude divided by that amplitude, coincides exactly with the axial-torsion contribution to the Ricci scalar. A covariant, spin-augmented guidance law then follows, and the algebraically determined torsion field, sourced by the global spin density of an entangled state, mediates non-local correlations without superluminal signalling. The same spin–torsion couplings produce an effective dark stress–energy tensor, for which observational tests are outlined across a wide range of scales: neutron-star magnetospheres, black-hole jets, gravitational-wave inspirals and spin-polarized interferometry. This geometric mechanism extends to spinless particles via torsion’s universal back-reaction on the connection, yielding a deterministic reformulation of quantum mechanics without metric quantization. A simple one-dimensional toy model reproduces the Tsirelson bound for Bell-inequality violations, underscoring the empirical and conceptual viability of this geometric route to quantum gravity, which dispenses with the need to quantize the metric.
The coupling between internal degrees of freedom of quantum systems and their overall motion in an external gravitational field plays a central role in multiple extensions of Einstein's equivalence principle to quantum physics. While previous models of such effects were predominantly restricted to linearized gravity and often required quantum particles to follow prescribed world-lines, this letter shows how such phenomena can be understood using generally covariant semi-classical approximations in the framework of quantum field theory in curved space-times. This method provides a unification and generalization of previously established results, but also predicts new effects such as an influence of internal energies on field amplitudes, as well as correction terms to the internal Schrödinger equation that give rise to Berry phases.
Abstract This work presents an alternative approach to obtain the quantum field equations in curved spacetime, considering that sufficiently small particles follow stochastic trajectories around geodesic. Our proposal is based on a stochastic differential equation in which the noise term experienced by the quantum particles is a consequence of the stochastic background in spacetime. This fact allows the particles to describe erratic movements and locally the universe exhibits characteristics akin to a lake with gentle ripples rather than a flat unyielding surface. Building upon this foundational understanding, we investigate the influence of this background on quantum-scale particles without considering the metric to be stochastic, rather we let test particles move randomly around the geodesic of macroscopic particles. Their behavior aligns with solutions to the Klein-Gordon (KG) equation specific to this curved spacetime. As the KG equation, in its non-relativistic limit within a flat spacetime, reduces to the Schrödinger equation, consequently, we propose a compelling connection: the Schrödinger equation may emerge directly from a spacetime lacking local smoothness.
A class of diffusion processes controlled by the geometry of the manifold on which they evolve is considered. The kinetic energy of such diffusions is shown to be a geometric object expressed in terms of the curvature and torsion tensors. This gives rise to an action functional leading to a variational principle, from which a nontrivial critical geometry with nonvanishing torsion emerges. The resulting criticality condition is related to the Schrödinger equation in a manner that reproduces the features of Nelson’s stochastic approach to quantum mechanics.