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arXiv:hep-ph/9911378

Klein-Fock equation, proper-time formalism and symmetry of Hydrogen atom

V. Yu. Novozhilov, Yuri V. Novozhilov

Published Nov 16, 1999• 0 citations
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Abstract

We present main points of some of Fock papers in Quantum Theory, which were not properly followed when published.
DOI
10.48550/arxiv.hep-ph/9911378

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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.
Fock (1935) explained the degeneracy of the energy levels of the Kepler problem (or hydrogen atom) in terms of the dynamical symmetry group SO(4). He showed that the problem is equivalent to the geodesic flow on the sphere S 3 . The 'hidden' symmetry SO(4) is made manifest. The classical n-dimensional Kepler problem can be better understood by enlarging the phase space of the geodesical motion on S n and including time and energy as canonical variables: a following symplectomorphism transforms the motion on S n in the Kepler problem. The authors prove that the Fock procedure is the implementation at 'quantum' level of the above-mentioned symplectomorphism.