Quantum mechanics not on manifold

TL;DR

This paper investigates quantum mechanics on a non-manifold space, specifically a Y-junction, deriving unique transmission rules from conservation laws and exploring alternative Hamiltonian approaches.

Contribution

It demonstrates that transmission rules for a quantum scalar field on a Y-junction are uniquely determined by conservation laws, extending to discrete models and discussing constrained Hamiltonian methods.

Findings

Transmission rules follow from energy and charge conservation.

Results apply to both continuous and discrete models.

Alternative constrained Hamiltonian approaches are discussed.

Abstract

The free scalar field is studied on the Y-junction of three semi infinite axes which is the simplest example of a non-manifold space. It is shown that under an assumption that the junction point can not gain a macroscopic amount of energy and charge the transmission rules for this system uniquely follow from conservation of energy and charge. This result is also obtained in the discrete version of the model. Some alternative approaches to the problem based on quantum mechanics of Hamiltonian systems with constrains are discussed.