The basis of quantum mechanics' compatibility with relativity--whose impairment gives rise to the Klein-Gordon and Dirac equations

TL;DR

This paper demonstrates that the fundamental relativistic compatibility of solitary-particle quantum mechanics is rooted in the Lorentz-covariant formulation of wave equations, criticizing Klein-Gordon and Dirac theories for their inconsistencies and proposing more consistent alternatives.

Contribution

It clarifies the relativistic foundation of quantum mechanics and identifies the impairments caused by Klein-Gordon and Dirac equations, offering symmetry-based alternatives consistent with the strong correspondence principle.

Findings

Klein-Gordon theory introduces negative-energy solutions that undermine probability interpretation.

Dirac theory breaches the strong correspondence principle by imposing unphysical momentum-independent velocities.

Proposes physically sensible alternative equations with external electromagnetic fields and symmetry-based antiparticle descriptions.

Abstract

Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.