W.I. Fushchych, Scientific Works 2000, Vol. 2, 11–22. On the new invariance algebras of relativistic equations for massless particles W.I. FUSHCHYCH, A.G. NIKITIN We show that the massless Dirac equation and Maxwell equations are invariant under a 23-dimensional Lie algebra, which is isomorphic to the Lie algebra of the group C4 ⊗ U (2) ⊗ U (2). It is also demonstrated that any Poincar´ e-invariant equation for a particle of zero mass and of discrete spin provide a unitary representation of the conformal group and that the conformal group generators may be expressed via the generators of the Poincar´ e group. 1. Introduction Bateman [1] and Cunningham [3] discovered that Maxwell’s equations for a free electromagnetic field were invariant under conformal transformations. Nearly fifty years ago the conformal invariance of an arbitrary relativistic equation, for a massless particle with discrete spin was established by Dirac [4] for a spin- 1 2 particle and by McLennan [20] for a particle of any spin. Until now the question of whether the conformal group is the maximally extensive symmetry group for the equations of motion for massless particles remained unsettled. A positive answer to this question has been obtained only in the frame of the classical Sofus Lie approach (Ovsjannicov [24]), but as has been found recently, Lie methods do not permit the possibility to obtain all possible symmetry groups of differential equations. The restriction of the Lie method is that it applies only to those symmetry groups whose generators belong to the class of differential operators of first order. Using the non-Lie approach, in which the group generators may be differential operators of any order and even integro-differential operators, the new invariance groups of relativistic wave equations have been found (Fushchych [6–9]). It was demonstrated that any Poincar´ e-invariant equation for a free particle of spin s ≥ 12 possessed additional invariance under the group SU (2) ⊗ SU (2) (Fushchych [6, 7]); that the Kemmer– Duffin–Petiau equation was invariant under the group SU (3) ⊗ SU (3), and that the Rarita–Schwinger equation was invariant under the group O(6) ⊗ O(6) was demi- nstrated by Nikitin et al [23] and by Fushchych and Nikitin [10]. The non-Lie approach was also used successfully to obtain the symmetry groups of the Dirac and Kemmer– Duffin–Petiau equations describing the particles in an external electromagnetic field (Fushchych and Nikitin [12]). Other examples of symmetries which cannot be obtai- ned in the classical Lie approach are the symmetry groups of the non-relativistic oscillator (Levi–Leblond [16]) and of the hydrogen atom (Fock [5]). In the present paper, we have found the new symmetry groups of the massless Dirac equation and of Maxwell’s equations using a non-Lie approach. These groups are generated not by the transformations of coordinates, but by the transformations J. Phys. A: Math. Gen, 1979, 12, № 6, P. 747–757.