Explicit Representation of the Transformation Property of Physical Quantity of Dirac Electron with the Hyperbolic Metric of Velocity

Abstract

This paper aims, in the theory of Dirac electron, at the deliberate use of the hyperbolic metric of velocity, the effectiveness of which Smorodinski1 has recently emphasized in rela­ tivity. The metric of velocity is fully used together w~th a proper spin direction vector, to represent explicitly spinor cf; and vector, tensor, etc., of cf;* (matrix) cf;, of free Dirac electron. The 3-vector polarization operator is confirmed to correspond to the proper spin vector. For the electron in a square well potential, cf; is a product of hyperbolic function of the metric with trigonometric or the Bessel function, etc., and the metric turns out to be imaginary or complex in the case of the bound state or free negative energy state. A classical model of Dirac electron is established concretely by obtaining simple time derivatives of the metric and spin vector, from the covariant equations of a spinning particle under general fields, suggested by Frenkel, Kramers and others. The so-called Thomas factor 1/2 is brought inevitably in the equation of the spin vector itself, regardless of the time average motion. These explicit representations may facilitate the visualization of Dirac electron.