A two-state analysis of the one-dimensional relativistic particle

The quantum description of a one-dimensional relativistic particle can be formulated in terms of a Feynman two-state analysis. The formalism presents the main physical features of the relativistic particle in a concise, simple form. A Hamiltonian is developed in analogy with the ammonia molecule in an electric field. Using this Hamiltonian the conditions under which a particle loses its positive definite energy quality can be determined. Zitterbewegung, the Klein paradox, and the symmetry between particles of negative energy and positive energy anti-particles can be developed as a consequence of this condition. A second order propagation equation for the state vector is formulated which may be interpreted in two ways: (1) the state space is flat and the state vectors satisfy a Feynman Gell-Mann propagation equation; (2) the state vectors satisfy a Klein Gordon equation, but the state space is structured or curved. The structure of the manifold, given by a Weyl geometry, is due to the presence of an electromagnetic field.