Zitterbewegung in Relativistic Spin-0 and -½ Hamiltonian Theories

Abstract

By working in the Heisenberg picture, one of the present authors has previously given a general treatment of the charge-space Zitterbewegung of the coordinate in theories involving a Hamiltonian that factorizes the Klein-Gordon equation. In the present paper, this treatment is expanded, with particular exphasis being placed on the peculiar doubling of the dimension of the Hamiltonian due to Zitterbewegung. The two-component Weyl theory of the neutrino and the four-component Dirac and two-component Sakata-Taketani theories for massive particles of spin \textonehalf{} and 0, respectively, are discussed from the above viewpoint. The recent two-component theory of Biedenharn, Han, and van Dam (BHV) for massive spin-\textonehalf{} particles is analyzed. From our viewpoint, a Poincar\'e-invariant theory of this type necessarily has four components. We offer a two-component interpretation of the BHV theory at the expense of Poincar\'e invariance. This version cannot be generalized for an arbitrary electromagnetic field, in contrast to the four-component version. We also discuss the relation of the (first order in the time derivative) four-component BHV theory to the (second order) two-component Kramers theory.