Vector Coherent States on Clifford algebras

TL;DR

This paper introduces new classes of vector coherent states using Clifford matrices, extending the concept from complex numbers to quaternions and octonions, with potential applications in mathematical physics.

Contribution

It generalizes canonical coherent states by replacing complex variables with Clifford matrices and explores their representations on quaternions and octonions.

Findings

Defined vector coherent states with Clifford matrices

Constructed examples on quaternions and octonions

Extended the framework of coherent states to non-commutative algebras

Abstract

The well-known canonical coherent states are expressed as an infinite series in powers of a complex number $z$ together with a positive sequence of real numbers $\rho(m)=m$. In this article, in analogy with the canonical coherent states, we present a class of vector coherent states by replacing the complex variable $z$ by a real Clifford matrix. We also present another class of vector coherent states by simultaneously replacing $z$ by a real Clifford matrix and $\rho(m)$ by a real matrix. As examples, we present vector coherent states on quaternions and octonions with their real matrix representations.