Title: The Kronecker product in terms of Hubbard operators and the Clebsch–Gordan decomposition of SU(2)×SU(2)

We review the properties of the Kronecker (direct, or tensor) product of square matrices A⊗B⊗C⋯ in terms of Hubbard operators. In its simplest form, a Hubbard operator X{sub n}{sup 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)×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. -- Highlights: •The Kronecker product is studied in terms of Hubbard operators. •Complicated calculations involving large matrices are reduced to simple relations of subscripts. •The algebraic properties of the quantum observables of multipartite systems are studied. •The Clebsch–Gordan coefficients are given in terms of hypergeometric {sub 3}F{sub 2} functions. •The results can be further developed in many different directions.

Journal Name: Annals of Physics (New York); Journal Volume: 339; Other Information: Copyright (c) 2013 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

Country of Publication:

United States

Language:

English

Subject:

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; ANGULAR MOMENTUM; DECOMPOSITION; FUNCTIONS; HUBBARD MODEL; IRREDUCIBLE REPRESENTATIONS; MATRICES; TENSORS