Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations

A unitary operator on the space of spinors that makes it possible to associate each transformation in this space with a transformation in the space of electromagnetic field strengths is found. A connection is established by means of this operator between representations in the space of spinors and the space of field strengths for the Lorentz, Poincaré, and conformal groups. Unusual symmetries of the Dirac equation are found on this basis. It is noted that the Pauli–Gürsey symmetry operators (without the $\gamma_5$ operator) of the Dirac equation with $m=0$ form the same representation $D(1/2,0)\oplus D(0,1/2)$ of the $O(1,3)$ algebra of the Lorentz group as the spin matrices of the standard spinor representation. It is shown that besides the standard (spinor) representation of the Poincaré group, the massless Dirac equation is invariant with respect to two other representations of this group, namely, the vector and tensor representations specified by the generators of the representations $D(1/2,1/2)$ and $D(1,0)\oplus D(0,0)$ of the Lorentz group, respectively. Unusual families of representations of the conformal algebra associated with these representations of the group $O(1,3)$ are investigated. Analogous $O(1,2)$ and $P(1,2)$ invariance algebras are established for the Dirac equation with $m>0$.