On Realistic Field Theories and the Polarization of the Vacuum

The formal prescription for the regularization of divergent expressions in quantum electrodynamics which has been recently suggested by Pauli implies very strongly that these same divergences may be similarly canceled in a realistic theory wherein one has a mixture of fields in the manner of Pais and Sakata. Indeed, it may be readily seen that the Pauli regulator scheme corresponds field-theoretically to a family of spinor fields interacting with a family of neutral vector meson fields; the formalism is mathematically consistent (at least to the second order in the coupling constants) but physically unsound owing to the appearance of imaginary coupling constants. An attempt to remedy this defect by considering the most general mixture which is possible within the framework of ordinary field theory also leads to failure. Specifically, the current density which is induced in the vacuum by an external electromagnetic field is calculated to order $e^2$ for a variety of situations but it turns out that no combination of charged scalar, spinor and vector fields—with or without anomalous magnetic moments—leads to a full compensation of the divergences although the photon self-energy by itself may be made to vanish. It is concluded that, although the notion of a realistic approach to the theory of elementary particles remains an attractive one, the usual linear field theories do not in themselves seem to be adequate for this purpose; there is also always the possibility that it is the perturbation theory which is at fault.