Schr\"{o}dinger equations in constrained space with several initial constraints

TL;DR

This paper develops a quantization method for systems with multiple initial constraints, including velocity-dependent ones, using Dirac formalism, and explores how these constraints influence the geometry of the physical world.

Contribution

It introduces a generalized approach to quantize constrained systems with velocity-dependent conditions, extending the Dirac formalism.

Findings

Derived Schrödinger equations for systems with multiple initial constraints

Showed hermiticity constraints influence the geometry of the physical world

Extended the Dirac formalism to include velocity-dependent constraints

Abstract

A general system constrained with {\it several} initial constraint conditions is quantized based on the Dirac formalism and the Schr\"{o}dinger equation for this system is obtained. These constraint conditions are now allowed to depend not only on the coordinates but also on the velocities. It is shown that the hermiticity for the observables of the system restricts the geometrical structure of our world.