Unified Hamiltonian theory of relativistic particle equations

Abstract This paper discusses four related topics of the theory of relativistic free particle equations. 1. I. A unified equation is derived for the Zitterbewegung of the space coordinate. Our derivation is based (besides the quantum, e.g., Heisenberg equation of motion and the commutation relations which are valid in nonrelativistic theory also) on the postulate that for a free particle the square of the Hamiltonian is equal to the sum of the squares of the momentum and of the mass. Our results hold also for massless particles. 2. II. Using also the classical, e.g., Hamilton's canonical equations of motion—in addition to the assumptions named under 1—we derive the explicit structure of the linear (but not necessarily in p!) Hamiltonian for spins 0, 1 2 , and 1. Our derivation makes it particularly clear why, for the cases of spins 0 and 1, the Hamiltonian is not Hermitian, in contrast to the spin 1 2 case. We also see why, for these former cases, the Hamiltonian is of the second order in the momentum, again in contrast to the spin 1 2 case. The “hidden assumption” in Dirac's theory of spin 1 2 particles is formulated explicitly: it is the commutativity between the velocity and the space coordinate operators. 3. III. The spin-dependent aspects of the Zitterbewegung and of the Spinbewegung show up in the time-dependence of the hypercomplex operators represented by matrices. For the spin 0 case, the Zitterbewegung appears in its purest form, uncluttered by the Spinbewegung. The equations of motion express a rotation in charge space. This is perhaps the most physical way for the introduction of the concept of the charge space. For a Dirac particle, the Zitterbewegung is exhibited by the ϱk and the Spinbewegung by the σk operators where, f.i., αk = ϱ1 · σk and β = ϱ3. The equations of motion for the σk express a rotation in ordinary space, which is, in general, slower than the rotation in charge space characterizing the Zitterbewegung. It is pointed out that for a particle in a magnetic field “averaging over the Zitterbewegung” leads to the Pauli equations (in the Heisenberg representation). For the massless particles (Weyl neutrino and photon) the Zitterbewegung and Spinbewegung coalesce and the corresponding rotation takes place in ordinary space. 4. IV. The interdependence of the equations of the motion and of the commutation relations is discussed. The commutation relations are derived for a free particle as compatibility conditions for the coexistence of the classical, e.g., Hamilton and the quantum, e.g., Heisenberg equations of motion. The difference in approach between our considerations and Wigner's well-known demonstration of the nonfollowing of the commutation relations for a harmonic oscillator from the nonrelativistic classical and quantum equations of motion is explained. The Appendix A discusses very briefly other unified forms of relativistic particle equations: A—Wigner-Bargmann charge-conjugation violating equations; B—Foldy “canonical” charge-conjugation preserving equations; C—Sakata-Taketani type quasi-unified Hamiltonian equations; D—Duffin-Kemmer redundant equations for spins 0, 1 2 (!), and 1; E—“Covariant canonical” equations. The Appendix B explains very briefly what is meant by an operator (matrix) a and what is the meaning of the statement that an operator (matrix) a depends on another operator (matrix) b.