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.
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.
The topological solutions of the De Broglie–Bohm quantum mechanics are presented. Starting from the Schrödinger equation for one particle system and ϕ-mapping topological current theory, the trajectory of the particle is derived explicitly, and can be used as the world line of the particle. The world line is just at the zero point of the wave function and it is shown that the vorticity of the world line can be expressed by Hopf index and Brouwer degree. The evolution of the world line at the bifurcation point is given.
A de Broglie-Bohm like model of Klein-Gordon equation, that leads to the correct Schrodinger equation in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm interpretation--that the local momentum of particles is given by the gradient of the phase of the wave function--is not but approximately correct. Also, the number of particles is not locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.
This paper and a following one develop a Hamiltonian formulation of the de Broglie-Bohm (deBB) interpretation of quantum mechanics. Drawing upon a unified Hamiltonian treatment of a classical particle and its associated ensemble, it is shown how the interaction of a particle and a spin-0 quantum wave can be consistently expressed in a canonical framework. A formulation is found in which the particle is acted upon by a force determined by the quantum potential but the particle does not react on the Schrodinger field. This requires the introduction of an additional field which the particle does act upon, and allows the deBB theory to be expressed in Hamiltonian terms. An ensemble theory for the wave-particle system based on Liouville's equation is developed, and it is shown that for a pure quantum state the particle obeys its own Liouville equation. The general particle phase space distribution that is compatible with ‖ψ‖ 2 is derived. A special case of this corresponds to the relation p = ∂S/∂q (and thus the guidance law of the deBB theory) but more general distributions, and hence non-deBB motions, are possible. This relation is interpreted as a constraint on the phase space of the composite system. Using the theory of Hamiltonian constraints it is shown that the first-order guidance law and Hamilton's second-order particle equations are dynamically equivalent.
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 apply De Haro's Geometric View of Theories to one of the simplest quantum systems: a spinless particle on a line and on a circle. The classical phase space M = T*Q is taken as the base of a trivial Hilbert bundle E ~ M x H, and the familiar position and momentum representations are realised as different global trivialisations of this bundle. The Fourier transform appears as a fibrewise unitary transition function, so that the standard position-momentum duality is made precise as a change of coordinates on a single geometric object. For the circle, we also discuss twisted boundary conditions and show how a twist parameter can be incorporated either as a fixed boundary condition or as a base coordinate, in which case it gives rise to a flat U(H)-connection with nontrivial holonomy. These examples provide a concrete illustration of how the Geometric View organises quantum-mechanical representations and dualities in geometric terms.
A de Broglie-Bohm like model of Dirac equation, that leads to the correct Pauli equations for electrons and positrons in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm theory--that the local momentum of particles is given by the gradient of the phase of the wave function--wont be accurate. Also, the number of particles wont be locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.