Lorentz-covariant two-particles quantum relativistic equation in a Lagrangian approach

Eugene Paul Wigner

Edward H. Kerner, Garold Murdachaew

Expanding on earlier work, it is shown how to rewrite the non-relativistic problem of two particles coupled by a spring as one higher-order one-particle (HOOP) problem, which then can be elevated to be Lorentz covariant. The higher-order Lagrangian is then developed into an Ostrogradsky Hamiltonian format, from which canonical commutation rules are invoked and a relativistic wave equation produced. Within the centre-of-momentum frame of reference, the latter is reduced to a radial Schrödinger equation that is solved numerically, with a display of radial eigenfunctions that are compared with non-relativistic ones. The details of this example illustrate conclusively how the HOOP approach to direct-interaction many-particle relativistic dynamics works.

N. Ionescu-Pallas, I. Gottlieb

Gustavo Rigolin

We show that the Schr\"odinger equation can be derived assuming the Galilean covariance of a generic wave equation and the validity of the de Broglie's wave-particle duality hypothesis. We also obtain from this set of assumptions the transformation law for the wave function under a Galilean boost and prove that complex wave functions are unavoidable for a consistent description of a physical system. The extension to the relativistic domain of the above analysis is also provided. We show that Lorentz covariance and wave-particle duality are consistent with two different transformation laws for the wave function under a Lorentz boost. This leads to two different wave equations, namely, the Klein-Gordon equation and the Lorentz covariant Schr\"odinger equation.

Steven Kenneth Kauffmann

Dirac sought a relativistic quantum free-particle Hamiltonian that imposes space-time symmetry on the Schroedinger equation in configuration representation; he ignored the Lorentz covariance of energy-momentum. Dirac free-particle velocity therefore is momentum-independent, breaching relativity basics. Dirac also made solutions of his equation satisfy the Klein-Gordon equation via requirements imposed on its operators. Dirac particle speed is thereby fixed to the unphysical value of c times the square root of three, and anticommutation requirements prevent four observables, including the components of velocity, from commuting when Planck's constant vanishes, a correspondence-principle breach responsible for Dirac free-particle spontaneous acceleration (zitterbewegung) that diverges in the classical limit. Nonrelativistic Pauli theory contrariwise is physically sensible, and its particle rest-frame action can be extended to become Lorentz invariant. The consequent Lagrangian yields the corresponding closed-form relativistic Hamiltonian when magnetic field is absent, otherwise a successive-approximation regime applies.

A. Stern