Quantum Trajectories I: Background and Interpretation

T. F. Kamalov

This model is one of the possible geometrical interpretations of Quantum Mechanics where found to every image Path correspondence the geodesic trajectory of classical test particles in the random geometry of the stochastic fields background. We are finding to the imagined Feynman Path a classical model of test particles as geodesic trajectory in the curved space of Projected Hilbert space on Bloch's sphere.

Yeuncheol Jeong

The ontological interpretation of Bohmian quantum mechanics, through a particle's visual trajectory of motion, reconstitutes realism and determinism (and thus causality) in a quantum system. There also exists a similar but distinctive class of interpretation under Bohmian quantum mechanics, called the hydrodynamical interpretation. Here, the trajectories of the probability fluid flow play a significant computational role. The flow of the probability density visually helps chemists and engineers to solve the corresponding quantum hydrodynamical equations of motion. This particular computational scheme for the numerical solutions is called ‘the quantum trajectory method.' The success of the quantum trajectory method demonstrates the power of visual illustrations and metaphors in representing and analyzing quantum systems.

Alon E. Faraggi, Marco Matone

Adaptation of the Hamilton–Jacobi formalism to quantum mechanics leads to a cocycle condition, which is invariant under D -dimensional Mobius transformations with Euclidean or Minkowski metrics. In this paper we aim to provide a pedagogical presentation of the proof of the Möbius symmetry underlying the cocycle condition. The Möbius symmetry implies energy quantization and undefinability of quantum trajectories, without assigning any prior interpretation to the wave function. As such, the Hamilton–Jacobi formalism, augmented with the global Möbius symmetry, provides an alternative starting point, to the axiomatic probability interpretation of the wave function, for the formulation of quantum mechanics and the quantum spacetime. The Möbius symmetry can only be implemented consistently if spatial space is compact, and correspondingly if there exist a finite ultraviolet length scale. Evidence for nontrivial space topology may exist in the cosmic microwave background radiation.