On the Bunge-Kalnay position operator for the Dirac electron

Hisaiti Yamasaki

Progress of Theoretical Physics Vol. 31 No. 2 (1964) pp. 324–325 An Extension of Feynman-Bunge-Corben's Relation regarding the Position-Operator of Dirac Electron to Arbitrary Operators. II Hisaiti Yamasaki Progress of Theoretical Physics Vol. 34 No. 3 (1965) pp. 462–472 Explicit Representation of the Transformation Property of Physical Quantity of Dirac Electron with the Hyperbolic Metric of Velocity Hisaiti Yamasaki Progress of Theoretical Physics Vol. 36 No. 1 (1966) pp. 72–85 A New Derivation of Classical Models of the Spinning Electron from the WKB Solutions to the Pauli and Dirac Equations Hisaiti Yamasaki Progress of Theoretical Physics Vol. 39 No. 2 (1968) pp. 372–393 The Frenkel-Kramers Model and the Zitterbewegung in the Extended Heisenberg Picture of the Dirac Electron Hisaiti Yamasaki

Harald Grosse

善郎 西山

Andrés J. Kálnay, Enrique Mac Cotrina

A satisfactory proper time operator is found for Dirac's quantum relativistic electron. A self-consistent scheme is developed in which this operator and Beck's proper time derivative have the properties they should have. As an application of the proper time formalism as a research tool, properties of three position operators (which are generally considered as covariant) are studied in regard to their behaviour under the inhomogeneous Lorentz group and under ordinary and proper time evolution. They are Pryce's class d, Bacry's and R. J. Finkelstein's operators. It results that all three have some unreasonable properties for Dirac's electron. The reason for the difficulty in the case of Finkelstein's operator can be traced back to an incorrect definition of it. By making some changes a satisfactory position operator is obtained but it turns out to be equivalent to Bunge's one (up to an arbitrary and unimportant constant of motion). The results of this paper reinforce those of previous work which show that (for Dirac's electron) Bunge's position operator seems to be in a privileged place as regards the solution of the localization problem. Incidentaly, the interpretation of some relations previously used as auxiliary formulas for computations, allows us to obtain a manifestly covariant expression for Dirac's Hamiltonian and a second manifestly covariant form of Dirac's equation.

Bernd Thaller, Sigrid Thaller

H. Yamasaki

P. 324, in 9th line from the bottom: Q should read Ǭ.