Kronecker product in terms of Hubbard operators and the Clebsch-Gordan decomposition of SU(2)xSU(2)

TL;DR

This paper explores the use of Hubbard operators to express the Kronecker product and derives the Clebsch-Gordan decomposition for the group SU(2)×SU(2), providing explicit formulas for angular momentum addition.

Contribution

It introduces a Hubbard operator framework to analyze Kronecker products and derives closed-form Clebsch-Gordan coefficients for SU(2)×SU(2).

Findings

Hubbard operators simplify the algebra of multipartite quantum systems.

Explicit Clebsch-Gordan coefficients for SU(2)×SU(2) are obtained.

Framework facilitates analysis of angular momentum addition in quantum systems.

Abstract

We review the properties of the Kronecker (direct, or tensor) product of square matrices $A \otimes B \otimes C \cdots$ in terms of Hubbard operators. In its simplest form, a Hubbard operator $X_n^{i,j}$ can be expressed as the $n$-square matrix which has entry 1 in position $(i,j)$ and zero in all other entries. The algebra and group properties of the observables that define a multipartite quantum system are notably straightforward in such a framework. In particular, we use the Kronecker product in Hubbard notation to get the Clebsch-Gordan decomposition of the product group $SU(2) \times SU(2)$. Finally, the $n$-dimensional irreducible representations so obtained are used to derive closed forms of the Clebsch-Gordan coefficients that rule the addition of angular momenta. Our results can be further developed in many different directions.