Abstract
We are dealing with the Dirac Hamiltonian H = H0 + V with no magnetic field and radially symmetric electrostatic potential V = V(r), preferably the Coulomb potential. While the observable H is precisely predictable, its components H0 (relativistic mass) and V (potential energy) are not. However they both possess precisely predictable approximations H0 ∼ and V∼ which approximate accurately if the particle is not near its nucleus. On the other hand, near 0, H0 and V are practically unpredictable, perhaps in agreement with the fact, that a neutrino also should be in the game. [We have not yet studied the corresponding observables for the (≥ 12-dimensional) problem of electro-weak interaction.] Mathematically we are focusing on the spectral theory of the unbounded self-adjoint operators H0 ∼ and V ∼. We can prove that V ∼ is unitarily equivalent to V(r) again, by a unitary map given as Wiener-Hopf-type singular integral operator in the standard separation of variables for radially symmetric Dirac Hamiltonians. [This is, as far as the continuous spectrum is concerned.] Very similar unitary equivalence holds for H 0 ∼ and H 0. We are tempted to regard this as a form of “renormalization”.
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Cordes, H.O. The Split of the Dirac Hamiltonian into Precisely Predictable Energy Components. Foundations of Physics 34, 1117–1153 (2004). https://doi.org/10.1023/B:FOOP.0000041287.25713.fe
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DOI: https://doi.org/10.1023/B:FOOP.0000041287.25713.fe