The Split of the Dirac Hamiltonian into Precisely Predictable Energy Components

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 H0∼ and H0. We are tempted to regard this as a form of “renormalization”.