On Hamiltonian formulations of the Schr\"{o}dinger system

TL;DR

This paper compares various Hamiltonian and variational formulations of the Schrödinger system, highlighting their equivalences and advantages, and clarifies the quantization process for constrained systems.

Contribution

It systematically analyzes and compares different Hamiltonian formulations, including Dirac-Bergmann and Faddeev-Jackiw methods, for the Schrödinger system, resolving inconsistencies in previous approaches.

Findings

Different variational formulations lead to the same reduced Hamiltonian.

The Faddeev-Jackiw method is a shortcut of the Dirac approach.

Quantization of constrained systems is clarified and inconsistencies are addressed.

Abstract

We review and compare different variational formulations for the Schr\"{o}dinger field. Some of them rely on the addition of a conveniently chosen total time derivative to the hermitic Lagrangian. Alternatively, the Dirac-Bergmann algorithm yields the Schr\"{o}dinger equation first as a consistency condition in the full phase space, second as canonical equation in the reduced phase space. The two methods lead to the same (reduced) Hamiltonian. As a third possibility, the Faddeev-Jackiw method is shown to be a shortcut of the Dirac method. By implementing the quantization scheme for systems with second class constraints, inconsistencies of previous treatments are eliminated.