Restriction on types of coherent states due to gauge symmetry

TL;DR

This paper explores how gauge symmetries in SU(2) coherent states restrict the types of permissible quantum states and fiducial vectors, revealing quantization conditions for monopole strength and extending previous theoretical frameworks.

Contribution

It introduces a general framework linking gauge symmetries in Lagrangians to restrictions on quantum states and fiducial vectors in SU(2) coherent states, highlighting monopole charge quantization.

Findings

Fiducial vectors are restricted to eigenstates of ${ m f f f S}_3$ or their orbits.

The monopole strength is quantized as a multiple of 1/2.

Dirac strings are permitted under certain conditions.

Abstract

From the viewpoint of the SU(2) coherent states (CS) and their path integrals (PI) labeled by a full set of Euler angles $(\phi, \theta, \psi)$ which we developed in the previous paper, we study the relations between gauge symmetries of Lagrangians and allowed quantum states; we investigate permissible types of fiducial vectors (FV) in the full quantum dynamics in terms of SU(2) coherent states for typical Lagrangians. We propose a general framework for a Lagrangian having a certain gauge symmetry with respect to one of the Euler angles $\psi$. We find that for the case fiducial vectors are so restricted that they belong to the eigenstates of ${\hat S}_3$ or to the orbits of them under the action of the SU(2); and the strength of a fictitious monopole, which appears in the Lagrangian, is a multiple of $\frac12$. In this case Dirac strings are permitted. Our formulations and results deepen those of the preceding work by Stone that has piloted us; we illustrate the relation between the two methods. The reasoning here does not work for a Lagrangian without the gauge symmetry. This suggests a new possibility about monopole charge quantization. Besides analogies to field theory and entanglements in quantum information (QI) are briefly mentioned.