Nonlinear Quantum Field Theory

Nonlinear Quantum Field Theory


the common name for the theories using nonlinear equations for the operators that describe quantized fields. In terms of physics, this means that the self-action of the field is taken into account. In some theories the self-action is postulated as being primordial (such theories are usually termed nonlinear); in others it is assumed to be “induced” by a certain intermediate interaction. For example, in quantum electrodynamics the nonlinearity “induced” by the interaction of photons through virtual electron-positron pairs should result in effects of light scattering by light and by a field of charged particles that are observable but have not yet been detected because of their smallness.

Two trends may be noted in nonlinear quantum field theory. First, the results produced by consideration of nonlinearity for specific physical fields are being studied. Assumptions are being put forth that consideration of nonlinearity that is “induced,” particularly by gravitation, may eliminate divergence in quantum field theory. Consideration of “induced” nonlinearity would operate in the same way as the nonlinear extension of classical electrodynamics proposed by M. Born and L. Infeld, which solved the problem of “Coulomb divergence” (the energy of the Coulomb field of a point particle in ordinary electrodynamics is found to be infinite).

The second trend, which became well known mainly through the efforts of the groups under W. Heisenberg (Federal Republic of Germany) and D. D. Ivanenko (USSR), is broader. Attempts are being made to find nonlinear equations not for specific fields but rather for matter as a whole (“protomatter”), and specific physical fields are regarded as the various possible states of protomatter resulting from its self-action.

These two trends are as yet only being contemplated. Nonlinear quantum field theory is insufficiently developed, although the importance of taking nonlinearities into account in the physics of elementary particles is becoming increasingly evident.


Nelineinaia kvantovaia teoriia polia. Collection of translated articles, edited by D. D. Ivanenko. Moscow, 1959. (Problemy fiziki.)
Nelokal’nye, nelineinye i nerenormiruemye teorii polia. Preprint 2–5400 of the Joint Institute for Nuclear Research. Dubna, 1970.


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