Geometrizing Relativistic Quantum Mechanics

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.

José F. Cariñena, Jesús Clemente-Gallardo, Giuseppe Marmo

T. W. B. Kibble

Xue You Lee

Abstract：Geometrical optics has high similarity with classical particle mechanics. After first quantization program classical particle mechanics can obtain non-relativistic quantum mechanics that suitable for micro, but quantum mechanics does not describe the photon that its static mass is zero. So whether there is a kind of "quantization " approach for geometrical optics, by which non-relativistic quantum mechanics that can be used to describe the photon can be obtained. It’s quantum mechanics of photon similar to the Schrödinger wave mechanics under the neglection of the formation and annihilation of photon. In this paper, several methods that can be used to find

Alexander D. Popov

Non-relativistic quantum mechanics was originally formulated to describe particles. Using ideas from the geometric quantization approach, we show how the concept of antiparticles can and should be introduced in the non-relativistic case without appealing to quantum field theory. We discuss this in detail using the example of the one-dimensional harmonic oscillator.