Abstract A geometric approach to quantum mechanics which is formulated in terms of Finsler geometry is developed. It is shown that quantum mechanics can be formulated in terms of Finsler configuration space trajectories which obey Newton-like evolution but in the presence of an additional kind of potential. This additional quantum potential which was obtained first by Bohm has the consequence of contributing to the forces driving the system. This geometric picture accounts for many aspects of quantum dynamics and leads to a more natural interpretation. It is found for example that dynamics can be accounted for by incorporating quantum effects into the geometry of space-time.
Abstract A geometric approach to quantum mechanics which is formulated in terms of Finsler geometry is developed. It is shown that quantum mechanics can be formulated in terms of Finsler configuration space trajectories which obey Newton-like evolution but in the presence of an additional kind of potential. This additional quantum potential which was obtained first by Bohm has the consequence of contributing to the forces driving the system. This geometric picture accounts for many aspects of quantum dynamics and leads to a more natural interpretation. It is found for example that dynamics can be accounted for by incorporating quantum effects into the geometry of space-time.
Understanding quantum theory in terms of a geometric picture sounds great. There are different approaches to this idea. Here we shall present a geometric picture of quantum theory using the de-Broglie--Bohm causal interpretation of quantum mechanics. We shall show that it is possible to understand the key character of de-Broglie--Bohm theory, the quantum potential, as the conformal degree of freedom of the space--time metric. In this way, gravity should give the causal structure of the space--time, while quantum phenomena determines the scale. Some toy models in terms of tensor and scalar--tensor theories will be presented. Then a few essential physical aspects of the idea including the effect on the black holes, the initial Big--Bang singularity and non locality are investigated. We shall formulate a quantum equivalence principle according to which gravitational effects can be removed by going to a freely falling frame while quantum effects can be eliminated by choosing an appropriate scale. And we shall see that the best framework for both quantum and gravity is Weyl geometry. Then we shall show how one can get the de-Broglie--Bohm quantum theory out of a Weyl covariant theory. Extension to the case of many particle systems and spinning particles is discussed at the end.
The de Broglie–Bohm theory allows us to have got a satisfactory geometrodynamic interpretation of quantum mechanics. The fundamental element, which creates a geometrodynamic picture of the quantum world in the non-relativistic domain, a relativistic curved space-time background, and the quantum gravity domain, is the quantum potential. It is shown that, in the non-relativistic domain, the geometrodynamic nature of the quantum potential followsfrom the fact that it is an information potential containing a space-like active information on the environment; the geometric properties of the space expressed by the quantum potential determine non-local correlations between subatomic particles. Moreover, in the de Broglie–Bohm theory in a curved space-time, it is shown that the quantum, as well as the gravitational, effects of matter have geometric nature and are highly related: the quantum potential can be interpreted as the conformal degree of freedom of the space-time metric, and its presence is equivalent to the curved space-time. It is shown on the basis of some recent research that, in quantum gravity, we have a generalized geometric unification of gravitational and quantum effects of matter; Bohm's interpretation shows that the form of a quantum potential and its relation to the conformal degree of freedom of the space-time metric can be derived from the equations of motion.
This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us, in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by the expectation values of the position. The analysis appears to provide an underlying geometry described by a geodesic equation, with a connection form with a second term which is an essentially quantum effect. These dynamical equations provide a new geometric approach to the quantum evolution where we suggest a definition for “local instability” in the quantum theory.
Background: The unification of General Relativity and Quantum Mechanics represents a fundamental challenge in theoretical physics due to their mutually exclusive conceptual foundations: continuous deterministic geometry versus probabilistic wave functions. While the de Broglie-Bohm pilot-wave theory offers a deterministic, trajectory-based alternative to the standard probabilistic framework, it introduces a non-local, ad-hoc "Quantum Potential" (Q) to account for phenomena such as interference and non-locality. The lack of a geometric origin for Q in standard spacetime remains a significant conceptual barrier to a fully unified theory. Purpose: This paper aims to resolve this interpretational crisis by proposing that the Quantum Potential is not a fundamental or mysterious external force, but rather a derived geometric artifact. We postulate that the physical universe is fundamentally a 4-dimensional Complex Manifold (ℂ4), extending the standard Riemannian spacetime manifold to include imaginary coordinates. Methods: To establish this geometric basis, we treat the quantum wave function not as an abstract probabilistic amplitude, but as a physical, holomorphic map describing a particle's deterministic trajectory through this complex spacetime. Utilizing the Cauchy-Riemann conditions, we demonstrate that the Bohmian amplitude (R) is strictly coupled to the particle's location in the imaginary dimension. We then analyze the dynamics of particles following geodesics within ℂ4. By projecting the complex geodesic equation of motion onto the observable real slice (ℝ4), we separate the real and imaginary components of the complex 4-velocity to observe the energy balance. Conclusions: Our derivation reveals that the Quantum Potential (Q) emerges naturally and is mathematically identical to the kinetic energy component associated with a particle's hidden motion in these imaginary dimensions. This formulation successfully recovers the predictions of the Schrödinger equation while removing the ad-hoc nature of Bohmian mechanics. Furthermore, it interprets quantum non-locality—such as the interference observed in the double-slit experiment—as purely local, deterministic geodesic motion around topological singularities in a curved complex manifold. Ultimately, this framework provides a unified geometric description wherein gravity is the curvature of real coordinates, electromagnetism is the torsion of imaginary coordinates, and quantum mechanics is inertial motion through this complex geometry.
We consider a Finslerian-type geometrization of the nonrelativistic quantum mechanics in its hydrodynamical (Madelung) formulation, by also taking into account the effects of the presence of the electromagnetic fields on the particle motion. In the Madelung representation, the Schr\"odinger equation can be reformulated as the classical continuity and Euler equations of classical fluid mechanics in the presence of a quantum potential, representing the quantum hydrodynamical evolution equations. The equation of particle motion can then be obtained from a Lagrangian similarly to its classical counterpart. After the reparametrization of the Lagrangian, it turns out that the Finsler metric describing the geometric properties of quantum hydrodynamics is a Kropina metric. We present and discuss in detail the metric and the geodesic equations describing the geometric properties of the quantum motion in the presence of electromagnetic fields. As an application of the obtained formalism, we consider the Zermelo navigation problem in a quantum hydrodynamical system, whose solution is given by a Kropina metric. The case of the Finsler geometrization of the quantum hydrodynamical motion of spinless particles in the absence of electromagnetic interactions is also considered in detail, and the Zermelo navigation problem for this case is also discussed.
Abstract The interpretation of the quantum mechanics proposed by de Broglie and Bohm postulates that the time evolution of the position and the momentum of a quantum particle can be described by a trajectory in the phase-space. The evolution equation coincides with the classical one except for the presence of a nonlinear correction to the total energy of the particle denoted by Bohm potential. The particle momentum is associated to the derivative of the phase of the quantum wave function. The phase of a quantum wave function ceases to be globally well defined in the presence of zeros (nodes) and the Bohm potential becomes singular. We develop a geometrical interpretation of the Bohm dynamics based on the Ehresmann theory of the fiber bundles and we express the number of total rotations of the quantum phase around a node in terms of holonomy maps.