Geometrical Formulation of Quantum Mechanics

Jesús Clemente-Gallardo

GIOVANNI GIACHETTA, Luigi Mangiarotti, Gennadi Aleksandrovich Sardanashvily

Hoshang Heydari

Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observables are real-valued functions on the manifold, and the quantum evolution is governed by a symplectic flow that is generated by a Hamiltonian function. In this work we will give a compact introduction to main ideas of geometric formulation of quantum mechanics. We will provide the reader with the details of geometrical structures of both pure and mixed quantum states. We will also discuss and review some important applications of geometric quantum mechanics.

Giuseppe Marmo, G. F. Volkert

In this paper, we review a proposed geometrical formulation of quantum mechanics. We argue that this geometrization makes available mathematical methods from classical mechanics to the quantum framework. We apply this formulation to the study of separability and entanglement for states of composite quantum systems.

Vladimír Majerník

Jordan François, Lucrezia Ravera

Abstract A bundle geometric formulation of non‐relativistic many‐particles Quantum Mechanics is presented. A wave function is seen to be a ‐valued cocyclic tensorial 0‐form on configuration space‐time seen as a principal bundle, while the Schrödinger equation flows from its covariant derivative, with the action functional supplying a (flat) cocyclic connection 1‐form on the configuration bundle. In line with the historical motivations of Dirac and Feynman, ours is thus a Lagrangian geometric formulation of QM, in which the Dirac–Feynman path integral arises in a geometrically natural way. Applying the dressing field method , we obtain a relational reformulation of this geometric non‐relativistic QM: a relational wave function is realised as a basic cocyclic 0‐form on the configuration bundle. In this relational QM, any particle position can be used as a dressing field, i.e., as a “physical reference frame.” The dressing field method naturally accounts for the freedom in choosing the dressing field, which is readily understood as a covariance of the relational formulation under changes of physical reference frame.