The existence of scalar Lie fields

It is shown that the existence of nontrivial scalar Lie fields (i. e. fields whose commutator is linear in the field itself) is not precluded by algebraic consistency arguments. A partial characterization of the simplest algebraic Lie field structures is given. Several examples are presented, one of which may be represented by Hermitian operators in a Hilbert space having a unitary representation of the Poincaré group.