$h(1) \oplus su(2)$ vector algebra eigenstates with eigenvalues in the matrix domain

TL;DR

This paper introduces a new class of $h(1) 1 su(2)$ vector algebra eigenstates in the matrix domain, using a combined method to classify and derive these states, including coherent states related to quaternion quantization.

Contribution

It develops a novel framework for vector algebra eigenstates with matrix-valued eigenvalues, expanding the understanding of coherent states and their algebraic structures.

Findings

Derived eigenstates for all generator combinations

Classified states based on generalized commutation relations

Connected states to quaternion coherent state quantization

Abstract

A new set of $ h(1) \oplus su(2)$ vector algebra eigenstates on the matrix domain is obtained by defining them as eigenstates of a generalized annihilation operator formed from a linear combination of the generators of this algebra which eigenvalues are distributed as the elements of a square complex normal matrix. A combined method is used to compute these eigenstates, namely, the method of exponential operators and that of a system of first-order linear differential equations. We compute these states for all possible combination of generators and classify them in different categories according to a generalized commutation relation as well as according to the value of a characteristic parameter related to the $su(2)$ algebra eigenvalues. Proceeding in this way, we found a subset of generalized vector coherent states in the matrix domain which can be easily separated from the general set of Schr\"odinger-Robertson minimum uncertainty intelligent states. In particular, for a special choice of the matrix eigenvalue parameters we found the so-called vector coherent states with matrices associated to the Heisenberg-Weyl group as well as a generalized version of them, and also a direct connection with the coherent state quantization of quaternions.