A Covariant Position Operator for the Relativistic Electron

Sz. Farkas, Z. Kurucz, M. Weiner

E.A. De Kerf, G.G.A. Bäuerle

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.

J. Mourad

A relation expressing the covariant transformation properties of a relativistic position operator is derived. This relation differs from the one existing in the literature expressing manifest covariance by some factor ordering. The relation is derived in order for the localization of a particle to represent a space-time event. It is shown that there exists a conflict between this relation and the hermiticity of a positive energy position operator.

Pankaj Sharan

Gordon N. Fleming

A manifestly covariant formalism is developed suitable for a discussion of position operators for relativistic systems. It is shown that the rest-frame center of mass and the frame-dependent center of mass, which were studied previously by Pryce and M\o{}ller, and the local Newton-Wigner operator all transform like the spatial part of a four-vector when appropriately interpreted. The formalism allows an equivalent treatment of the space and time components of a position four-vector operator, and manifestly covariant canonical commutation rules are obtained for the three operators mentioned above. Finally it is suggested that the Newton-Wigner operator be interpreted as a center of spin.