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Unique Relativistic Extension of the Pauli Hamiltonian

Steven Kenneth Kauffmann

Published Jul 1, 2017• 0 citations
OpenAlex

Abstract

Relativistic extension of the Pauli Hamiltonian is ostensibly achieved by minimal coupling of electromagnetism to the free-particle Dirac Hamiltonian. But the free-particle Pauli Hamiltonian is pathology-free in its nonrelativistic domain, while the free-particle Dirac Hamiltonian yields completely fixed particle speed which is greater than c, spin orbit torque whose ratio to kinetic energy tends to infinity in the zero-momentum limit, and mega-violation of Newton's First Law in that limit. Furthermore, relativistic extension of the Pauli Hamiltonian is unique in principle because inertial frame hopping can keep the particle nonrelativistic. That extension is indeed readily achieved by upgrading the terms of the Pauli Hamiltonian's corresponding action to appropriate Lorentz invariants. The resulting relativistic Lagrangian yields a canonical momentum that can't be analytically inverted in general, but a physically-sensible successive-approximation scheme applies. For hydrogen and simpler systems approximation isn't needed, and the result, which includes spin-orbit coupling, is as transparently physically sensible as the relativistic Lorentz Hamiltonian is, a far cry from the Dirac Hamiltonian pathologies.

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A single-particle Hamiltonian independent of the particle's coordinate ensures the conserves momentum, i.e., is free. This free-particle Hamiltonian is completely determined by Lorentz covariance of its energy-momentum and the particle's rest-energy value; such a has velocity which vanishes when its momentum vanishes. Dirac required his free-particle Hamiltonian to be inhomogeneously linear in momentum, which contrariwise produces velocity that is independent of momentum; he also required his Hamiltonian's square to equal the above relativistic Hamiltonian's square, forcing many observables to anticommute and breach the quantum correspondence principle, as well as forcing the speed of any Dirac free particle to be c times the square root of three, which remains true when the interacts electromagnetically. The quantum correspondence principle breach causes a Dirac free particle to exhibit spontaneous acceleration that becomes unbounded in the classical limit; an artificial spin is also made available. Unlike the Dirac Hamiltonian, the nonrelativistic Pauli Hamiltonian is of unphysical anomalies. Its relativistic extension is worked out via Lorentz-invariant upgrade of its associated action functional at zero velocity, and is obtained in closed form when there is no applied magnetic field; when there is, a successive approximation scheme must be used.

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A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys Newton's second law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac's velocity operators having magnitude c. The local spin motion corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron's point charge. The corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. By defining a spin tensor as the angular momentum of the electron's total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut-Zanghi electron theory. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.

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Equivalence of Maxwell's source-free equations to the time-dependent Schroedinger equation for a solitary particle with two polarizations and Hamiltonian |cp|

Steven Kenneth Kauffmann

It was pointed out in a previous paper that although neither the Klein-Gordon equation nor the Dirac Hamiltonian produces sound solitary free-particle relativistic quantum mechanics, the natural square-root relativistic Hamiltonian for a nonzero-mass free particle does achieve this. Failures of the Klein-Gordon and Dirac theories are reviewed: the solitary Dirac free particle has, inter alia, an invariant speed well in excess of c and staggering spontaneous Compton acceleration, but no pathologies whatsoever arise from the square-root relativistic Hamiltonian. Dirac's key misapprehension of the underlying four-vector character of the time-dependent, configuration-representation Schroedinger equation for a solitary particle is laid bare, as is the invalidity of the standard "proof" that the nonrelativistic limit of the Dirac equation is the Pauli equation. Lorentz boosts from the particle rest frame point uniquely to the square-root Hamiltonian, but these don't exist for a massless particle. Instead, Maxwell's equations are dissected in spatial Fourier transform to separate nondynamical longitudinal from dynamical transverse field degrees of freedom. Upon their decoupling in the absence of sources, the transverse field components are seen to obey two identical time-dependent Schroedinger equations (owing to two linear polarizations), which have the massless free-particle diagonalized square-root Hamiltonian. Those fields are readily modified to conform to the attributes of solitary-photon wave functions. The wave functions' relations to the potentials in radiation gauge are also worked out. The exercise is then repeated without the considerable benefit of the spatial Fourier transform.

arXiv:1004.1820•Apr 11, 2010•2 citations
Relativistic extension of the Pauli Hamiltonian is ostensibly achieved by minimal coupling of electromagnetism to the free-particle Dirac Hamiltonian. But the free-particle Pauli Hamiltonian is pathology-free in its nonrelativistic domain, while the free-particle Dirac Hamiltonian yields completely fixed particle speed which is greater than c, spin orbit torque whose ratio to kinetic energy tends to infinity in the zero-momentum limit, and mega-violation of Newton's First Law in that limit. Furthermore, relativistic extension of the Pauli Hamiltonian is unique in principle because inertial frame hopping can keep the particle nonrelativistic. That extension is indeed readily achieved by upgrading the terms of the Pauli Hamiltonian's corresponding action to appropriate Lorentz invariants. The resulting relativistic Lagrangian yields a canonical momentum that can't be analytically inverted in general, but a physically-sensible successive-approximation scheme applies. For hydrogen and simpler systems approximation isn't needed, and the result, which includes spin-orbit coupling, is as transparently physically sensible as the relativistic Lorentz Hamiltonian is, a far cry from the Dirac Hamiltonian pathologies.
A single-particle Hamiltonian independent of the particle's coordinate ensures the conserves momentum, i.e., is free. Lorentz-covariance of that Hamiltonian's energy-momentum specifies it up to the particle's rest energy; the it describes has speed below c and constant velocity parallel to its conserved momentum. Dirac took his free-particle Hamiltonian to have the same squared value as that relativistic one, but unwittingly blocked Lorentz-covariance of his Hamiltonian's energy-momentum by requiring it to be inhomogeneously linear in momentum. The Dirac free particle badly flouts relativity and even physical cogency; its velocity direction is extremely nonconstant, while its speed is fixed to c times the square root of three even when it interacts electromagnetically. Both its rest energy and total energy can be negative, and its velocity components and rest energy are artificially correlated by being mutually anticommuting; its alleged spin is an artifact of the anticommutation of its velocity components. Unlike the Dirac Hamiltonian, the nonrelativistic Pauli Hamiltonian is apparently physically sensible for speed far below c. Its relativistic extension is worked out via Lorentz-invariant upgrade of its associated action functional at zero velocity, and is obtained in closed form if there is no applied magnetic field; a successive approximation scheme must otherwise be used.
A single-particle Hamiltonian independent of the particle's coordinate ensures the conserves momentum, i.e., is free. This free-particle Hamiltonian is completely determined by Lorentz covariance of its energy-momentum and the particle's rest-energy value; such a has velocity which vanishes when its momentum vanishes. Dirac required his free-particle Hamiltonian to be inhomogeneously linear in momentum, which contrariwise produces velocity that is independent of momentum; he also required his Hamiltonian's square to equal the above relativistic Hamiltonian's square, forcing many observables to anticommute and breach the quantum correspondence principle, as well as forcing the speed of any Dirac free particle to be c times the square root of three, which remains true when the interacts electromagnetically. The quantum correspondence principle breach causes a Dirac free particle to exhibit spontaneous acceleration that becomes unbounded in the classical limit; an artificial spin is also made available. Unlike the Dirac Hamiltonian, the nonrelativistic Pauli Hamiltonian is of unphysical anomalies. Its relativistic extension is worked out via Lorentz-invariant upgrade of its associated action functional at zero velocity, and is obtained in closed form when there is no applied magnetic field; when there is, a successive approximation scheme must be used.
Dirac sought a relativistic quantum free-particle Hamiltonian that imposes space-time symmetry on the Schroedinger equation in configuration representation; he ignored the Lorentz covariance of energy-momentum. Dirac free-particle velocity therefore is momentum-independent, breaching relativity basics. Dirac also made solutions of his equation satisfy the Klein-Gordon equation via requirements imposed on its operators. Dirac particle speed is thereby fixed to the unphysical value of c times the square root of three, and anticommutation requirements prevent four observables, including the components of velocity, from commuting when Planck's constant vanishes, a correspondence-principle breach responsible for Dirac free-particle spontaneous acceleration (zitterbewegung) that diverges in the classical limit. Nonrelativistic Pauli theory contrariwise is physically sensible, and its particle rest-frame action can be extended to become Lorentz invariant. The consequent Lagrangian yields the corresponding closed-form relativistic Hamiltonian when magnetic field is absent, otherwise a successive-approximation regime applies.
A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys Newton's second law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac's velocity operators having magnitude c. The local spin motion corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron's point charge. The corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. By defining a spin tensor as the angular momentum of the electron's total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut-Zanghi electron theory. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.
It is generally acknowledged that neither the Klein-Gordon equation nor the Dirac Hamiltonian can produce sound solitary-particle relativistic quantum mechanics due to the ill effects of their negative-energy solutions; instead their field-quantized wavefunctions are reinterpreted as dealing with particle and antiparticle simultaneously--despite the clear physical distinguishability of antiparticle from particle and the empirically known slight breaking of the underlying CP invariance. The natural square-root Hamiltonian of the free relativistic solitary particle is iterated to obtain the Klein-Gordon equation and linearized to obtain the Dirac Hamiltonian, steps that have calculational but not physical motivation, and which generate the above-mentioned problematic negative-energy solutions as extraneous artifacts. Since the natural square root Hamiltonian for the free relativistic solitary particle contrariwise produces physically unexceptionable quantum mechanics, this article focuses on extending that Hamiltonian to describe a solitary particle (of either spin 0 or spin one-half) in relativistic interaction with an external electromagnetic field. That is achieved by use of Lorentz-covariant solitary-particle four momentum techniques together with the assumption that well-known nonrelativistic dynamics applies in the particle's rest frame. Lorentz-invariant solitary particle actions, whose formal Hamiltonization is an equivalent alternative approach, are as well explicitly displayed. It is proposed that two separate solitary-particle wavefunctions, one for a particle and the other for its antiparticle, be independently quantized in lieu of "reinterpreting" negative energy solutions--which indeed don't even afflict proper solitary particles.
It was pointed out in a previous paper that although neither the Klein-Gordon equation nor the Dirac Hamiltonian produces sound solitary free-particle relativistic quantum mechanics, the natural square-root relativistic Hamiltonian for a nonzero-mass free particle does achieve this. Failures of the Klein-Gordon and Dirac theories are reviewed: the solitary Dirac free particle has, inter alia, an invariant speed well in excess of c and staggering spontaneous Compton acceleration, but no pathologies whatsoever arise from the square-root relativistic Hamiltonian. Dirac's key misapprehension of the underlying four-vector character of the time-dependent, configuration-representation Schroedinger equation for a solitary particle is laid bare, as is the invalidity of the standard "proof" that the nonrelativistic limit of the Dirac equation is the Pauli equation. Lorentz boosts from the particle rest frame point uniquely to the square-root Hamiltonian, but these don't exist for a massless particle. Instead, Maxwell's equations are dissected in spatial Fourier transform to separate nondynamical longitudinal from dynamical transverse field degrees of freedom. Upon their decoupling in the absence of sources, the transverse field components are seen to obey two identical time-dependent Schroedinger equations (owing to two linear polarizations), which have the massless free-particle diagonalized square-root Hamiltonian. Those fields are readily modified to conform to the attributes of solitary-photon wave functions. The wave functions' relations to the potentials in radiation gauge are also worked out. The exercise is then repeated without the considerable benefit of the spatial Fourier transform.