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Progress of Theoretical Physics

An Extension of Feynman-Bunge-Corben's Relation regarding the Position-Operator of Dirac Electron to Arbitrary Operators. I

Hisaiti Yamasaki

Published Feb 1, 1964• 4 citations
Publisher

Abstract

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

DOI
10.1143/ptp.31.322

Related papers

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

Progress of Theoretical Physics•Feb 1, 1964•4 citations

The Frenkel-Kramers Model and the Zitterbewegung in the Extended Heisenberg Picture of the Dirac Electron

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With respect to the principal operators including the sixteen 4-4 matrices of the Dirac electron, two covariant forms of the extended Heisenberg picture can be presented with the “two covariant Hamiltonians HI and HII” of the first and second order, respectively. One form with HII involving the proper electromagnetic moment corresponds perfectly to the Frenkel-Kramers model which was recently treated by the author; the other form with HI containing the Zitterbewegung corresponds partly to the so-called Weyssenhoff-like model. That an amalgamation of the former form and the Zitterbewegung brings immediately the latter is explicitly shown by virtue of a decomposition of the operator of the Dirac electron into two corresponding parts. This decomposition is derived from the author's extension of the Feynman-Bunge-Corben formula for the position operator of the electron and, in case without external fields, it corresponds to the covariant form of the so-called ‘gerade-ungerade’ decomposition. In the case of the free state mixed with positive and negative energy, the Zitterbewegung of operators derived from HI can properly be realized by the expectation values of the operators. In this process, the author's complex hyperbolic metric of velocity which represents a new negative energy state plays an essential role. These results give an important clue to solve the puzzling problem of the construction of a consistent classical model for the Dirac electron, and also may be useful for investigation of ‘internal structure’ of elementary particles as well as their spin.

Progress of Theoretical Physics•Feb 1, 1968•4 citations

An Extension of Feynman-Bunge-Corben's Relation regarding the Position-Operator of Dirac Electron to Arbitrary Operators. II

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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
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
With respect to the principal operators including the sixteen 4-4 matrices of the Dirac electron, two covariant forms of the extended Heisenberg picture can be presented with the “two covariant Hamiltonians HI and HII” of the first and second order, respectively. One form with HII involving the proper electromagnetic moment corresponds perfectly to the Frenkel-Kramers model which was recently treated by the author; the other form with HI containing the Zitterbewegung corresponds partly to the so-called Weyssenhoff-like model. That an amalgamation of the former form and the Zitterbewegung brings immediately the latter is explicitly shown by virtue of a decomposition of the operator of the Dirac electron into two corresponding parts. This decomposition is derived from the author's extension of the Feynman-Bunge-Corben formula for the position operator of the electron and, in case without external fields, it corresponds to the covariant form of the so-called ‘gerade-ungerade’ decomposition. In the case of the free state mixed with positive and negative energy, the Zitterbewegung of operators derived from HI can properly be realized by the expectation values of the operators. In this process, the author's complex hyperbolic metric of velocity which represents a new negative energy state plays an essential role. These results give an important clue to solve the puzzling problem of the construction of a consistent classical model for the Dirac electron, and also may be useful for investigation of ‘internal structure’ of elementary particles as well as their spin.
The functional form of the wave function with respect to variables appropriate for the WKB approximation is chosen by taking into account the transformation properties of the spinor wave-function.This representation makes it possible, using only the wave equation, to pursue directly the time changes in the variables, i.e. the spin direction, velocity and density, etc., characterizing an electron.These changes correspond straightforwardly to a classical model of the non-relativistic Pauli electron with the proper magnetic moment, and the covariant form of this model, which is just "the Frenkel-Kramers model of the Dirac electron" introduced recently by the author.In this derivation, the existence of freedom of • rotation around the .spinaxis and certain definite angular velocity of this rotation are required.These results and the correspondence obtained may clarify the classical-mechanical significance of the spinor wave-function and also of the spin itself.An explanation is appended of the hyperbolic functions of the velocity metric, which help in the complicated calculations for the Dirac electron. § I. Introduction*) The main part of this paper appeared previously in Rep. Seisanken.No. 22 (1957), 55, in Japanese.