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Progress of Theoretical Physics

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

Hisaiti Yamasaki

Published Feb 1, 1968• 4 citations

Abstract

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.

DOI: 10.1143/ptp.39.372

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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 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

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.

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.

The Hamiltonian form of the Klein–Gordon equation is used to study the Zitterbewegung of spin- zero particles. It is shown that the time dependence of the velocity and position operators in the Heisenberg picture is formally the same as that of a spin- 1/2 particle. It is also shown that the Zitterbewegung is the result of the interference between positive and negative energy states. A representation-free expression is found for the mean position operator. This operator is free from Zitterbewegung. The Klein–Gordon equation is solved with an electrostatic potential originally used by Sauter in his investigation of the Klein paradox for spin- 1/2 particles. It is shown that the transmission to states with negative kinetic energy is very small when the electric field is weak.

The Assessment of the interpretation of Zitterbewegung for free particle solution using various models has been carried out in this work where we firstly considered by using the free particle solution for which Dirac equation and projection operator were carried out.. ln this case, it was invariably revealed that the solution have two doubly degenerate eigenvalues representing positive and negative state that was further support by the analysis which was carried out using Heisenberg’s equation and the representation in relativistic velocity and semi-classical equation of motion for acceleration., but in this second case, the results obtained were observed to have two terms, first terms standing in for rapidly oscillatory motion and the second term representing average motion of the particle in x-direction. The first term in the expression is the one deemed to signify zitterbewegung which is one considered to be sas a result of the fluctuation resulting from the interference between negative and positive energy state while the other term is the normal state terms signifying the average motion of the particle. This therefore confirms the explicit nature of using Dirac equation in handling problems involving the behavior of free particles in field as compared to the use of semiclassical equation of motion.