The Stueckelberg-Kibble Model as an Example of Quantized Symplectic Reduction

TL;DR

This paper applies a novel quantization method based on Rieffel induction to the Stueckelberg-Kibble model, explicitly constructing the physical state space and illustrating the features of constrained quantization.

Contribution

It demonstrates how Rieffel induction can be used to quantize the Stueckelberg-Kibble model, explicitly constructing the physical state space and analyzing its properties.

Findings

Constructed the physical state space ${\

Carried a massive representation of the Poincaré group.

Revealed features of constrained quantization theory.

Abstract

Recently, it has been observed that a certain class of classical theories with constraints can be quantized by a mathematical procedure known as Rieffel induction. After a short exposition of this idea, we apply the new quantization theory to the Stueckelberg-Kibble model. We explicitly construct the physical state space ${\cal H}_{phys}$, which carries a massive representation of the Poincar\'e group. The longitudinal one-particle component arises from a particular Bogoliubov transformation of the five (unphysical) degrees of freedom one has started with. Our discussion exhibits the particular features of the proposed constrained quantization theory in great clarity.