MATRIX ELEMENTS OF A HYPERBOLIC VECTOR OPERATOR UNDER SO(2,1)

Abstract

We deal here with the use of Wigner–Eckart type arguments to calculate the matrix elements of a hyperbolic vector operator [Formula: see text] by expressing them in terms of reduced matrix elements. In particular, we focus on calculating the matrix elements of this vector operator within the basis of the hyperbolic angular momentum [Formula: see text] whose components [Formula: see text], [Formula: see text], [Formula: see text] satisfy an SO(2,1) Lie algebra. We show that the commutation rules between the components of [Formula: see text] and [Formula: see text] can be inferred from the algebra of ordinary angular momentum. We then show that, by analogy to the Wigner–Eckart theorem, we can calculate the matrix elements of [Formula: see text] within a representation where [Formula: see text] and [Formula: see text] are jointly diagonal.