Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Progress of Theoretical Physics

Remarks on the Transformation Properties of the Dirac Equation

Pong Yul Pac

Published Dec 1, 1959• 6 citations
Publisher

Abstract

The physical properties of dynamical variables in the extreme-relativistic representation will be discussed. Especially, it will be pointed out that in the choice of the proper “mean” position we must be careful, and that our results include those of Bose-Gamba-Sudarshan as special cases.
DOI
10.1143/ptp.22.857

Related papers

Observables in the Extreme Relativistic Representation of the Dirac Equation

P. M. Mathews, A. Sankaranarayanan

A new choice of operators is given to represent dynamical variables of a Dirac particle in the extreme relativistic representation.

Progress of Theoretical Physics•Jul 1, 1961•10 citations

Representations of the Dirac Equation

S. K. Bose, A. Gamba, E. C. G. Sudarshan

A representation of the Dirac equation which displays its extreme relativistic properties is discussed. The "spin" appears naturally related to the "position."Received 20 October 1958DOI:https://doi.org/10.1103/PhysRev.113.1661©1959 American Physical Society

Physical Review•Mar 15, 1959•34 citations

Observables of a Dirac Particle

P. M. Mathews, A. Sankaranarayanan

The most general forms, consistent with reasonable physical requirements, for the position and other observables of a Dirac particles are obtained.

Progress of Theoretical Physics•Oct 1, 1961•15 citations

Foundations for proper-time relativistic quantum theory

Tepper L. Gill, Trey Morris, S. K. Kurtz

This paper is a progress report on the foundations for the canonical proper-time approach to relativistic quantum theory. We first review the the standard square-root equation of relativistic quantum theory, followed by a review of the Dirac equation, providing new insights into the physical properties of both. We then introduce the canonical proper-time theory. For completeness, we give a brief outline of the canonical proper-time approach to electrodynamics and mechanics, and then introduce the canonical proper-time approach to relativistic quantum theory. This theory leads to three new relativistic wave equations. In each case, the canonical generator of proper-time translations is strictly positive definite, so that it represents a particle. We show that the canonical proper-time extension of the Dirac equation for Hydrogen gives results that are consistently closer to the experimental data, when compared to the Dirac equation. However, these results are not sufficient to account for either the Lamb shift or the anomalous magnetic moment.

arXiv:1503.02579•Journal of Physics Conference Series•May 14, 2015•1 citation

Dirac equation in a relativistically invariant form

Bipin R. Desai

Cambridge University Press eBooks•Dec 3, 2009•0 citations

On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit

Leslie L. Foldy, Siegfried A. Wouthuysen

By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and $c$. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.

Physical Review•Apr 1, 1950•2,183 citations
This paper is a progress report on the foundations for the canonical proper-time approach to relativistic quantum theory. We first review the the standard square-root equation of relativistic quantum theory, followed by a review of the Dirac equation, providing new insights into the physical properties of both. We then introduce the canonical proper-time theory. For completeness, we give a brief outline of the canonical proper-time approach to electrodynamics and mechanics, and then introduce the canonical proper-time approach to relativistic quantum theory. This theory leads to three new relativistic wave equations. In each case, the canonical generator of proper-time translations is strictly positive definite, so that it represents a particle. We show that the canonical proper-time extension of the Dirac equation for Hydrogen gives results that are consistently closer to the experimental data, when compared to the Dirac equation. However, these results are not sufficient to account for either the Lamb shift or the anomalous magnetic moment.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.