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Hamiltonian form of the Kemmer equation for spinless boson

Ronald R. Bousek

Published Jan 1, 1965• 0 citations
OpenAlex

Abstract

It is shown that for a free spinless particle described by the Kemmer equation, the Hamiltonian which Kemmer originally presented yields a velocity operator whose expectation value does not agree with the correspondence principle. Hamiltoniaos for a free spinless particle and a spinless particle in an electromagnetic field are constructed from the Kemmer equation. These Hairoi I toniaos differ from the ones originally presented by Kemmer. The free particle Hamiltonian yields a velocity operator whose expectation value is consistent with the correspondence principle.

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It is shown that for a free spinless particle described by the Kemmer equation, the Hamiltonian which Kemmer originally presented yields a velocity operator whose expectation value does not agree with the correspondence principle. Hamiltoniaos for a free spinless particle and a spinless particle in an electromagnetic field are constructed from the Kemmer equation. These Hairoi I toniaos differ from the ones originally presented by Kemmer. The free particle Hamiltonian yields a velocity operator whose expectation value is consistent with the correspondence principle.
It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.
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 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.
In this paper it is shown that Maxwell's theory of the electromagnetic field in vacuum can be stated in a form closely parallel to Dirac's theory of the electron. The electromagnetic field is described by a three-by-one column matrix whose elements are linear combinations of the three independent spinor components of the field. The Maxwell equations take a form similar to the Dirac equation for a free electron but involving three-by-three matrices. In terms of a wave function normalized so that the integral-square is the number of photons, the classical expressions for the energy, momentum, and angular momentum in the field are related to expected values of the Hamiltonian, momentum, and angular momentum operators. The angular momentum operator consists of an orbital part plus a three-by-three spin-one matrix part. As in Dirac's theory, the contributions from the states of negative Hamiltonian eigenvalue must be subtracted---these are states of opposite circular polarization.
We construct the analog in the quantum mechanics of a free spinless particle, of Dirac’s formula for the generators of space–time translations in his point form of classical dynamics, where one takes as fundamental variables the generators of homogeneous Lorentz transformations and the coordinate 4-vector of the point where the world line of the particle meets one sheet of a two-sheeted hyperboloid in space–time. A 4-vector coordinate operator is determined for such a particle, with commuting Hermitian components. The corresponding observable is the analog of the coordinate 4-vector of the point on the hyperboloid. This operator bears the same relation to such a surface as the Newton–Wigner operator does to an instant.