Local conformal-invariance of the wave equation for finite-component fields. I. The conditions for invariance, and fully-reducible fields

Abstract

The conditions for local conformal-invariance of the wave equation are obtained for finite-component fields, of Types Ia and Ib [in the terminology of Mack and Salam, Ann. Phys. 53, 174 (1969).] These conditions generate a set of locally invariant free massless field equations and restrict the relevant representation of the Lie algebra [(k4⊕d)⊕sl(2,C)] in the index space of the field to belong to a certain class. Those fully-reducible representations which are in this class are described in full. The corresponding Type Ia field equations include only the massless scalar field equation, neutrino equations, Maxwell’s equations, and the Bargmann–Wigner equations for massless fields of arbitrary helicity, and no others. In particular, it is confirmed [Bracken, Lett. Nuovo Cimento 2, 574 (1971)] that not all Poincaré-invariant sets of massless Type Ia field equations are conformal-invariant, contrary to some often-quoted results of McLennan [Nuovo Cimento 3, 1360 (1956)], which are shown to be invalid. It is also shown that in the case of a potential, the wave equation is never conformal-invariant in the strong sense (excluding gauge transformations).