Authors: Steven Kenneth Kauffmann
A single-particle Hamiltonian independent of the particle's coordinate ensures the particle conserves momentum, i.e., is free. Lorentz-covariance of that Hamiltonian's energy-momentum specifies it up to the particle's rest energy; the free particle it describes has speed below c and constant velocity parallel to its conserved momentum. Dirac took his free-particle Hamiltonian to have the same squared value as that relativistic one, but unwittingly blocked Lorentz-covariance of his Hamiltonian's energy-momentum by requiring it to be inhomogeneously linear in momentum. The Dirac "free particle" badly flouts relativity and even physical cogency; its velocity direction is extremely nonconstant, while its speed is fixed to c times the square root of three even when it interacts electromagnetically. Both its rest energy and total energy can be negative, and its velocity components and rest energy are artificially correlated by being mutually anticommuting; its alleged "spin" is an artifact of the anticommutation of its velocity components. Unlike the Dirac Hamiltonian, the nonrelativistic Pauli Hamiltonian is apparently physically sensible for particle speed far below c. Its relativistic extension is worked out via Lorentz-invariant upgrade of its associated action functional at zero particle velocity, and is obtained in closed form if there is no applied magnetic field; a successive approximation scheme must otherwise be used.
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