Self-Adjoint Second-Order Equations for Fermions and the Foldy-Wouthuysen Representation

Self-adjoint second-order equations with spinor wave functions in an external electromagnetic field are obtained for fermions. The relationship is found between these equations and the Dirac equations in the Foldy-Wouthuysen representation. Closed chiral symmetric equations are written for left and right spinors by using the Dirac matrices in chiral representation. A closed chiral symmetric Foldy-Wouthuysen equation is obtained for an arbitrary electromagnetic field. There are no summands mixing left and right spinors in these equations irrespective of the presence or absence of fermion mass.