Representations of Local I‐Spin Charge Densities. I. Spaces Where Each I Occurs Only Once

An approach to the problem of representation of the algebra of currents that puts essential emphasis on the study of infinite‐parameter Lie algebras is proposed. As an example, a class of irreducible Hermitian representations of the commutation relations [Vⁱ(φ₁), V^j(φ₂)] = iε^ijkV^k(φ₁φ₂), where the φ's are elements of a commutative algebra with identity, is derived. The dependence of the representations on the algebra {φ} is completely characterized by two functional equations that are explicitly solved, for {φ} an algebra of polynomials. States of well‐defined momentum and rotational properties are constructed using translational and rotational invariance and forming direct integral spaces. The representations so constructed are seen to belong to two distinct subclasses, distinguished by the vanishing or nonvanishing of a length parameter |η|. The subclass with |η| = 0 is unbounded in isospin and has the trivial momentum‐transfer structure characteristic of field‐theoretical point particles. On the other hand, the spaces characterized by |η| ≠ 0 are bounded in isospin and suited to describe particles with structure. A brief discussion on how to derive invariant form factors from the results here presented is included.