Geometrical Formulation of Quantum Mechanics

Abstract: Hamilton's action principle is formulated and extended in conformity with the gauge transformations underlying Weyl's geometry. The extended principle characterizes infinitely many equally likely trajectories with a particle traveling along a randomly selected one. Available similar formulations do not conform as directly to the gauge transformations as the present one. Also, they have not paid much attention to the path-independent, assigned gauges. The freedom available in assigning these gauges is exploited here by defining them in terms of the configuration, and interactions of the observing system with the observed one. Impact of the method of observation on its outcome is described in terms of the assigned gauges so defined and illustrated with examples. A wavefunction is defined in a simply connected region essentially as an aggregate of the gauge transformations over all trajectories; equivalently, an aggregate of the Weyl-lengths acquired by a unit vector transported along all trajectories from everywhere. This representation is similar to Feynman's path integral representation differing only in that it incorporates the assigned gauges yielding an adjusted wavefunction that includes the impact of an observing system on the observed one. Probability density is shown to be a uniquely defined gauge invariant quantity but at the expense of the information about the observable effects contained in the gauge factors, assigned and otherwise. The particle trajectories defined here are thus shown to provide additional significant information about a system than provided by the wavefunction together with the probability density. Present description of the impact of method of observation on its outcome is compared with the descriptions according to the existing major representations of quantum mechanics.