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OALib

Remarks on the Order of Quantum Equations of Motion

Eliahu Comay

Published Jan 1, 2020• 1 citation
Publisher

Abstract

The paper examines the Dirac-Pauli differences concerning the order of the primary differential equation of an elementary massive quantum particle.The analysis relies on several self-evident constraints that an acceptable quantum theory must satisfy, like conservation laws, compatibility with Maxwellian electrodynamics and the correspondence principle.The dimension of the quantum function that is used in the Lagrangian density of a given quantum theory together with the corresponding differential equations play an important role in the reasoning procedure.The paper proves that Dirac was right and that second-order quantum theories like the Klein-Gordon equation and the electroweak theory of the W ± bosons do not satisfy fundamental constraints.This outcome is inconsistent with the Standard Model of Particle Physics.

DOI
10.4236/oalib.1106335

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The paper examines the Dirac-Pauli differences concerning the order of the primary differential equation of an elementary massive quantum particle.The analysis relies on several self-evident constraints that an acceptable quantum theory must satisfy, like conservation laws, compatibility with Maxwellian electrodynamics and the correspondence principle.The dimension of the quantum function that is used in the Lagrangian density of a given quantum theory together with the corresponding differential equations play an important role in the reasoning procedure.The paper proves that Dirac was right and that second-order quantum theories like the Klein-Gordon equation and the electroweak theory of the W ± bosons do not satisfy fundamental constraints.This outcome is inconsistent with the Standard Model of Particle Physics.
The foundations of the mathematical structure of quantum theories of a massive particle are the basis of this analysis.It proves the coherence of the particle-wave duality of quantum theories and the principle of complementarity as well.Furthermore, the noncommutativity of Hermitian operators proves that quantum theories are inherently indeterministic.This feature does not deny the fact that the classical limit of quantum theories agrees with classical physics.It is also shown that the foundations of the mathematical structure of quantum theories impose constraints on any specific quantum theory.It is proved that the first-order Dirac theory is consistent with all constraints.In contrast, second-order theories, such as the Klein-Gordon, the electroweak theory of the W ± and the Z particles, and the Higgs boson theory fail to do that.An analogous analysis proves that also the Majorana neutrino theory is inconsistent with fundamental requirements.Similarly, inconsistencies of Proca's idea about a massive photon are shown.
The paper shows that the variational principle serves as an element of the mathematical structure of a quantum theory. The experimentally confirmed properties of the corpuscular-wave duality of a quantum particle are elements of the analysis. A Lagrangian density that yields the equations of motion of a given quantum theory of a massive particle is analyzed. It is proved that if this Lagrangian density is a Lorentz scalar whose dimension is then the associated action consistently defines the required phase of the quantum particle. The dimension of this Lagrangian density proves that also the quantum function has dimension. This result provides new criteria for the acceptability of quantum theories. An examination of the first order Dirac equation demonstrates that it satisfies the new criteria whereas the second order Klein-Gordon equation fails to do that.
In this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that had a positive-definite conserved probability density and was first order both in time and space. One can show that these two conditions imply that a matrix equation is required. The reason why the Klein–Gordon equation did not yield a positive-definite probability density is connected with the second-order time derivative in this equation, which arises because the Klein–Gordon equation is related to the relativistic energy–momentum relation E 2 = m 2 + p 2 via the correspondence principle that includes a term E 2 . Thus, a ‘better’ Lorentz covariant wave equation with a positive-definite probability density should have a first-order time derivative only. However, the equivalence of time and space coordinates in Minkowski space requires that such an equation also have only first-order space derivatives.
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.
The fact that the Schr\"odinger and the Dirac equations for the wave function of an electron are differential equations of the first order with respect to time, while in classical theories differential equations of the second order are common, has necessitated a slightly different setup of the canonical theory in both fields. Under influence of the classical methods, second-order equations are also often used in the quantum theory of particles of integer spin, thus causing a difference between the treatment of Fermi-Dirac and of Einstein-Bose particles.It is shown here that conformity between classical and quantum-mechanical methods can be achieved easily by use of first-order equations throughout, thus avoiding a superfluous distinction between integral and half-odd-integral spin fields. The classical theory of a point charge in an electromagnetic field of force is set up here from this point of view.