Covariant Formulation of Non-Relativistic Quantum Mechanics

Valter Moretti

Valter Moretti

Jean Michel Sellier

U. Kasper, Eckhard Kreisel, H.‐J. Treder

M. Omote, S. Kamefuchi, Yasushi Takahashi +1 more

Galilei transformations are formulated in a 5-dimensional form and on such basis the non-relativistic quantum mechanics is reconstructed. Our approach has the following advantages over others'. (1) The Schrödinger wavefunctions are taken to be vector (not projective) representations of the Galilei group, so that (2) the problem here is considerably simplified. (3) The formulation proceeds in a way quite parallel to that of relativistic quantum mechanics, and what is most important, (4) Galilei covariance is manifest throughout.

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.